Let's consider integral

$$
\begin{align*}
U(T, \alpha, \hbar_{\varepsilon}) &= \int dx|x| |\tilde{\theta}_{\varepsilon}^{(1)}(T, x, \alpha, \hbar_{\varepsilon})|^2 = \\
&\quad \frac{1}{\sqrt{2\pi\hbar_{\varepsilon}}} \sqrt{\frac{|\alpha_T|}{2\pi |\tilde{S}_{yy}(x)|}} \times \\
&\quad \int dx|x| \exp\left[\frac{1}{\hbar_{\varepsilon}} \operatorname{Re}[\tilde{S}(y_{cr}(x, T), x)]\right] = \tag{2.1.44} \\
&\quad \frac{1}{\sqrt{2\pi\hbar_{\varepsilon}}} \sqrt{\frac{|\alpha_T|}{2\pi |\tilde{S}_{yy}(x)|}} \int dx|x| \exp\left[\frac{1}{\hbar_{\varepsilon}} [\vartheta'_{T}x^2 + \sigma'_{T}x + \eta'_{T}w'^2_1 + \Theta'_{T}]\right], \\
&\quad \vartheta'_{T} = \operatorname{Re}\vartheta_{T}, \sigma'_{T} = \operatorname{Re}\sigma_{T}, \Theta'_{T} = \operatorname{Re}\Theta^{(2)}_{T}.
\end{align*}
$$

$$
\begin{align}
\frac{\partial \tilde{S}(y_{cr}(x, T), x)}{\partial x} &= 2\vartheta'_{T}x + \sigma'_{T} = 0, \tag{2.1.44} \\
x_{cr} &= -\frac{\sigma'_{T}}{2\vartheta'_{T}}. \nonumber
\end{align}
$$

$$
\begin{align}
\tilde{S}(y_{cr}(x_{cr}, T), x_{cr}) &= \vartheta'_{T} x_{cr}^{2} + \sigma'_{T} x_{cr} + \eta'_{T} w'^{2}_{1} + \Theta'_{T} = \nonumber \\
&\quad \frac{\sigma'^{2}_{T}}{4\vartheta'_{T}} - \frac{\sigma'^{2}_{T}}{2\vartheta'_{T}} + \eta'_{T} w'^{2}_{1} + \Theta'_{T} = -\frac{\sigma'^{2}_{T}}{4\vartheta'_{T}} + \eta'_{T} w'^{2}_{1} + \Theta'_{T}. \tag{2.1.1}
\end{align}
$$
Let us calculate Gaussian integral (2.1.44) using Laplace method.
From Eq.(2.2.1.25) and Eq.(2.2.1.43) we obtain

From Eq.(2.2.1.45) we obtain master inequility for the Feynman-Colombeau path integral.

**Theorem 2.2.1.1. (Master Inequelity for the Feynman-Colombeau Path Integral).**

$$ \int_{\mathbb{R}} |x||\tilde{V}_{1,\varepsilon}(T,x,x_0,\alpha,\hbar)|^2 \le 4U(T,x_0). \qquad (2.1.46) $$

## II.2.2. Master Equation for the Feynman-Colombeau path integral corresponding with a Schrödinger equation with Hermitian Hamiltonian.

Let us consider regularized propagator
Let us calculate Gaussian integral (2.1.44) using Laplace method.
From Eq.(2.2.1.25) and Eq.(2.2.1.43) we obtain

From Eq.(2.2.1.45) we obtain master inequility for the Feynman-Colombeau path integral.

**Theorem 2.2.1.1. (Master Inequelity for the Feynman-Colombeau Path Integral).**

$$ \int_{\mathbb{R}} |x||\tilde{V}_{1,\epsilon}(T,x,x_0,\alpha,\hbar)|^2 \le 4U(T,x_0). \qquad (2.1.46) $$

## II.2.2. Master Equation for the Feynman-Colombeau path integral corresponding with a Schrödinger equation with Hermitian Hamiltonian.

Let us consider regularized propagator
$$ (\mathbf{K}_{\varepsilon, \eta, g}(x, T|x_0, 0))_{\varepsilon} = $$

$$
\begin{gather*}
\left( \overline{\int_{\substack{q(T)=x \\ q(0)=y}} dy D^+[q(t), \varepsilon] \Psi_0(q(0))} \times \exp \left[ \frac{i}{\varepsilon} \left( \int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\varepsilon}(t)}{dt} \right)^2 - V(q_{\varepsilon}(t)) \right] \right) \right. \\
\left. - \frac{1}{\varepsilon} \int_0^T dt \left[ \{q(t)\}_{\eta} + \eta g(t, q(T), q(0), \lambda) - \eta \lambda \right]^2 \right] \right)_{\varepsilon}, \tag{2.2.2.1} \\
\tilde{q}_{\varepsilon}(t) = \frac{q(t)}{1 + \varepsilon \left\| \frac{\dot{q}(t)}{q(t)} \right\|_{2,T}^2 + \varepsilon^2 \left\| \frac{\ddot{q}(t)}{q(t)} \right\|_{2,T}^2}, \\
q_{\varepsilon}(t) = \frac{q(t)}{1 + \varepsilon^k q^m(t)}, \quad k \ge 1, m \ge 2, \\
\Psi_0(x) \approx \delta(x - x_0), \\
\varepsilon(\varepsilon) \ll \eta \ll 1, \eta \ll m, \\
\varepsilon(\varepsilon) \ll \varepsilon.
\end{gather*}
$$

Here: (1) $g(0, q(T), q(0), \lambda) = q(0)$, and (2)

$$
\begin{align}
\{q(t)\}_{\eta} &= \{\delta(t)\}_{\eta} + \eta u(t, T, x, y) \notag \\
\text{iff} \qquad & q(t) = \delta(t) + u(0, T, x, y), \tag{2.2.2.2} \\
\delta(t) &= \sum_{n=1}^{\infty} a_n \sin\left(\frac{n\pi t}{T}\right). \notag
\end{align}
$$
$$ (\mathbf{K}_{\varepsilon, \eta, g}(x, T|x_0, 0))_{\varepsilon} = $$

$$
\begin{gather*}
\left( \overline{\int_{\substack{q(T)=x \\ q(0)=y}} dy D^+[q(t), \varepsilon] \Psi_0(q(0))} \times \exp \left[ \frac{i}{\varepsilon} \left( \int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\varepsilon}(t)}{dt} \right)^2 - V(q_{\varepsilon}(t)) \right] \right) \right. \\
\left. - \frac{1}{\varepsilon} \int_0^T dt \left[ \{q(t)\}_{\eta} + \eta g(t, q(T), q(0), \lambda) - \eta \lambda \right]^2 \right] \right)_{\varepsilon}, \tag{2.2.2.1} \\
\tilde{q}_{\varepsilon}(t) = \frac{q(t)}{1 + \varepsilon \left\| \frac{\dot{q}(t)}{q(t)} \right\|_{2,T}^2 + \varepsilon^2 \left\| \frac{\ddot{q}(t)}{q(t)} \right\|_{2,T}^2}, \\
q_{\varepsilon}(t) = \frac{q(t)}{1 + \varepsilon^k q^m(t)}, \quad k \ge 1, m \ge 2, \\
\Psi_0(x) \approx \delta(x - x_0), \\
\varepsilon(\varepsilon) \ll \eta \ll 1, \eta \ll m, \\
\varepsilon(\varepsilon) \ll \varepsilon.
\end{gather*}
$$

Here: (1) $g(0, q(T), q(0), \lambda) = q(0)$, and (2)

$$
\begin{align}
\{q(t)\}_{\eta} &= \{\delta(t)\}_{\eta} + \eta u(t, T, x, y) \nonumber \\
\text{iff} \nonumber \\
q(t) &= \delta(t) + u(0, T, x, y), \tag{2.2.2.2} \\
\delta(t) &= \sum_{n=1}^{\infty} a_n \sin\left(\frac{n\pi t}{T}\right). \nonumber
\end{align}
$$
Here $u(t,x,y,\lambda)$ is any continuous function such that $u(0,x,y,\lambda) = y, u(T,x,y,\lambda) = x$ and

$$ \{\delta(t)\}_{\eta} = \sum_{n=1}^{N(\epsilon)} \sqrt{\eta} a_n \sin\left(\frac{n\pi t}{T}\right) + \sum_{N(\epsilon)}^{\infty} a_n \sin\left(\frac{n\pi t}{T}\right) \quad (2.2.2.3) $$

Let us now consider quantum average $\langle T, x_0; \epsilon, \eta, \epsilon, g \rangle$:

$$ \langle T, x_0; \epsilon, \eta, \epsilon, g \rangle = \int dx \langle \mathbf{K}_{\epsilon, \eta, g}(x, T|x_0, 0) \mathbf{K}_{\epsilon, \eta, g}^*(x, T|x_0, 0) \rangle = \\ \int dx |\mathbf{K}_{\epsilon, \eta, g}(x, T|x_0, 0)|^2 \quad (2.2.2.4) $$

From Eq. (2.2.2.1) we obtain
Here $u(t,x,y,\lambda)$ is any continuous function such that $u(0,x,y,\lambda) = y, u(T,x,y,\lambda) = x$ and

$$ \{\delta(t)\}_{\eta} = \sum_{n=1}^{N(\epsilon)} \sqrt{\eta} a_n \sin\left(\frac{n\pi t}{T}\right) + \sum_{N(\epsilon)}^{\infty} a_n \sin\left(\frac{n\pi t}{T}\right) \quad (2.2.2.3) $$

Let us now consider quantum average $\langle T, x_0; \epsilon, \eta, \epsilon, g \rangle$:

$$ \langle T, x_0; \epsilon, \eta, \epsilon, g \rangle = \int dx \langle \mathbf{K}_{\epsilon, \eta, g}(x, T|x_0, 0) \mathbf{K}^*_{\epsilon, \eta, g}(x, T|x_0, 0) \rangle = \\ \int dx |\mathbf{K}_{\epsilon, \eta, g}(x, T|x_0, 0)|^2 \quad (2.2.2.4) $$

From Eq. (2.2.2.1) we obtain
$$|\langle T, x_0; \epsilon, \eta, \varepsilon, g \rangle| \le$$

$$\bar{\int} dx|x| \times$$

$$\left| \int_{\substack{q(T)=x \\ q(0)=y}} dy D^+[q(t), \varepsilon] \Psi_0(q(0)) \times \right.$$

$$\exp\left[ \frac{i}{\epsilon} \left( \int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\varepsilon}(t)}{dt} \right)^2 - V(q_{\varepsilon}(t)) \right] \right) - \frac{1}{\epsilon} \int_0^T dt \left[ \{q(t)\}_{\eta} + \eta g(t, q(T), q(0), \lambda) - \eta \lambda \right]^2 \right] =$$

$$\bar{\int} dx|x| \times$$

$$\left\{ \left[ \left[ \int_{\substack{q(T)=x \\ q(0)=y}} dy \tilde{D}_{\eta}^{+}[q(t), \varepsilon] \Psi_{0}(q(0)) \cos \left[ \frac{1}{\epsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\varepsilon}(t)}{dt} \right)^2 - V(q_{\varepsilon}(t)) \right] \right) \right] \right]^2 + \right. \quad (2.2.2.5)$$

$$\left[ \left[ \int_{\substack{q(T)=x \\ q(0)=y}} dy \tilde{D}_{\eta}^{+}[q(t), \varepsilon] \Psi_{0}(q(0)) \sin \left[ \frac{1}{\epsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\varepsilon}(t)}{dt} \right)^2 - V(q_{\varepsilon}(t)) \right] \right) \right] \right]^2 \right\} =$$

$$\bar{\int} dx|x| \left[ \left[ \int_{\substack{q(T)=x \\ q(0)=y}} dy \tilde{D}_{\eta}^{+}[q(t), \varepsilon] \Psi_{0}(q(0)) \cos \left[ \frac{1}{\epsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{dq_{\varepsilon}(t)}{dt} \right)^2 - V_{\varepsilon}(q(t)) \right] \right) \right] \right]^2 +$$
$$|\langle T, x_0; \epsilon, \eta, \varepsilon, g \rangle| \le$$

$$\bar{\int} dx|x| \times$$

$$\left| \int_{\substack{q(T)=x \\ q(0)=y}} dy D^+[q(t), \varepsilon] \Psi_0(q(0)) \times \right.$$

$$\exp\left[ \frac{i}{\epsilon} \left( \int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_\epsilon(t)}{dt} \right)^2 - V(q_\epsilon(t)) \right] \right) - \frac{1}{\epsilon} \int_0^T dt \left[ \{q(t)\}_\eta + \eta g(t, q(T), q(0), \lambda) - \eta \lambda \right]^2 \right] =$$

$$\bar{\int} dx|x| \times$$

$$\left\{ \left[ \left[ \int_{\substack{q(T)=x \\ q(0)=y}} dy \tilde{D}_{\eta}^{+}[q(t), \epsilon] \Psi_{0}(q(0)) \cos \left[ \frac{1}{\epsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon}(t)}{dt} \right)^2 - V(q_{\epsilon}(t)) \right] \right) \right] \right]^2 + \right. \quad (2.2.2.5)$$

$$\left[ \left[ \int_{\substack{q(T)=x \\ q(0)=y}} dy \tilde{D}_{\eta}^{+}[q(t), \epsilon] \Psi_{0}(q(0)) \sin \left[ \frac{1}{\epsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon}(t)}{dt} \right)^2 - V(q_{\epsilon}(t)) \right] \right) \right] \right]^2 \right\} =$$

$$\bar{\int} dx|x| \left[ \left[ \int_{\substack{q(T)=x \\ q(0)=y}} dy \tilde{D}_{\eta}^{+}[q(t), \epsilon] \Psi_{0}(q(0)) \cos \left[ \frac{1}{\epsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{dq_{\epsilon}(t)}{dt} \right)^2 - V_{\epsilon}(q(t)) \right] \right) \right] \right]^2 + \right.$$
Here

$$
\begin{array}{l}
\displaystyle \overline{\int_{q(T)=x}^{q(0)=y} \tilde{D}_{\eta}^{+}[q(t), \varepsilon] \Psi_{0}(q(0))(\cdot)} = \\
\displaystyle \quad D^{+}[q(t), \varepsilon] \Psi_{0}(q(0)) \times
\end{array}
$$

$$
\begin{array}{l}
\displaystyle \overline{\int_{q(T)=x} D^{+}[q(t), \varepsilon] \Psi_0(q(0)) \times} \\
\displaystyle \quad q(0)=y
\end{array}
\qquad (2.2.2.6)
$$

$$
\exp\left\{-\frac{1}{\epsilon} \int_{0}^{T} dt \left[ \{q(t)\}_{\eta} + \eta g(t, q(T), q(0), \lambda) - \eta \lambda \right]^2 \right\} \times (\cdot)
$$

and
Here

$$ \overline{\int_{q(T)=x} \tilde{D}_{\eta}^{+}[q(t), \varepsilon] \Psi_{0}(q(0))(\cdot)} = $$

$$ \overline{\int_{q(T)=x} D^{+}[q(t), \varepsilon] \Psi_{0}(q(0)) \times} \qquad (2.2.2.6) $$

$$ \exp\left\{-\frac{1}{\epsilon} \int_{0}^{T} dt \left[ \{q(t)\}_{\eta} + \eta g(t, q(T), q(0), \lambda) - \eta \lambda \right]^2\right\} \times (\cdot) $$

and
$$
\chi_1(T, \epsilon, \varepsilon, \eta) = \int \bar{d}x |x| \times
$$

$$
\begin{align*}
& \left[ \overline{\int_{q(T)=x} \tilde{D}_{\eta}^{+}[q(t), \varepsilon] \Psi_{0}(q(0)) \times} \right. \\
& \qquad \left. \cos \left[ \frac{1}{\epsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\varepsilon}(t)}{dt} \right)^2 - V(q_{\varepsilon}(t)) \right] \right) \right] \right]^2 \\
& = \overline{\int} dx \left\{ \left[ \overline{\int_{q(T)=x} \tilde{D}_{\eta}^{+}[q(t), \varepsilon] \Psi_{0}(q(0)) \sqrt{|q(T)|} \times} \right. \right. \\
& \qquad \left. \left. \cos \left[ \frac{1}{\epsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\varepsilon}(t)}{dt} \right)^2 - V(q_{\varepsilon}(t)) \right] \right) \right] \right]^2 \right\}, \tag{2.2.2.7}
\end{align*}
$$

$$
\chi_2(T, \epsilon, \epsilon, \eta) = \int \bar{d}x |x| \times
$$

$$
\begin{align*}
& \left[ \overline{\int_{q(T)=x} \tilde{D}_{\eta}^{+}[q(t), \epsilon] \Psi_{0}(q(0)) \times} \right. \\
& \qquad \left. \sin \left[ \frac{1}{\epsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon}(t)}{dt} \right)^2 - V(q_{\epsilon}(t)) \right] \right) \right] \right]^2 \\
& = \overline{\int} dx \left\{ \left[ \overline{\int_{q(T)=x} \tilde{D}_{\eta}^{+}[q(t), \epsilon] \Psi_{0}(q(0)) \sqrt{|q(T)|} \times} \right. \right. \\
& \qquad \left. \left. \sin \left[ \frac{1}{\epsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon}(t)}{dt} \right)^2 - V(q_{\epsilon}(t)) \right] \right) \right] \right]^2 \right\}.
\end{align*}
$$
$$
\chi_1(T, \epsilon, \varepsilon, \eta) = \int \bar{d}x |x| \times
$$

$$
\begin{align*}
& \left[ \overline{\int_{q(T)=x} \tilde{D}_{\eta}^{+}[q(t), \varepsilon] \Psi_{0}(q(0)) \times} \right. \\
& \qquad \left. \cos \left[ \frac{1}{\epsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\varepsilon}(t)}{dt} \right)^2 - V(q_{\varepsilon}(t)) \right] \right) \right] \right]^2 \\
& = \overline{\int} dx \left\{ \left[ \overline{\int_{q(T)=x} \tilde{D}_{\eta}^{+}[q(t), \varepsilon] \Psi_{0}(q(0)) \sqrt{|q(T)|} \times} \right. \right. \\
& \qquad \left. \left. \cos \left[ \frac{1}{\epsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\varepsilon}(t)}{dt} \right)^2 - V(q_{\varepsilon}(t)) \right] \right) \right] \right]^2 \right\}, \tag{2.2.2.7}
\end{align*}
$$

$$
\chi_2(T, \epsilon, \epsilon, \eta) = \int \bar{d}x |x| \times
$$

$$
\begin{align*}
& \left[ \overline{\int_{q(T)=x} \tilde{D}_{\eta}^{+}[q(t), \varepsilon] \Psi_{0}(q(0)) \times} \right. \\
& \qquad \left. \sin \left[ \frac{1}{\epsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon}(t)}{dt} \right)^2 - V(q_{\epsilon}(t)) \right] \right) \right] \right]^2 \\
& = \overline{\int} dx \left\{ \left[ \overline{\int_{q(T)=x} \tilde{D}_{\eta}^{+}[q(t), \epsilon] \Psi_{0}(q(0)) \sqrt{|q(T)|} \times} \right. \right. \\
& \qquad \left. \left. \sin \left[ \frac{1}{\epsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon}(t)}{dt} \right)^2 - V(q_{\epsilon}(t)) \right] \right) \right] \right]^2 \right\}.
\end{align*}
$$
and here

$$
\begin{gather*}
\overline{\int_{q(T)=x} \tilde{D}_{\eta}^{+}[q(t), \varepsilon] \Psi_{0}(q(0))(\cdot)} = \\
\int_{q(T)=x} dy D^{+}[q(t), \varepsilon] \Psi_{0}(q(0)) \times \\
\int_{q(0)=y} dy D^{+}[q(t), \varepsilon] \Psi_{0}(q(0)) \times
\end{gather*}
\qquad (2.2.2.8)
$$

$$
\exp\left\{-\frac{1}{\epsilon} \int_{0}^{T} dt \left[ \{q(t)\}_{\eta} + \eta g(t, q(T), q(0), \lambda) - \eta \lambda \right]^2 \right\} \times (\cdot)
$$

Let us now evaluate a quantity $(\chi_1(T, \epsilon, \varepsilon, \eta))_\varepsilon$:

$$
\begin{align*}
(\chi_1(T, \epsilon, \varepsilon, \eta))_{\varepsilon} &= \\
& \quad \left( \int dx |x| \times \right. \\
& \quad \left. \left[ \overline{\int_{q(T)=x} \tilde{D}_{\eta}^{+}[q(t), \varepsilon] \Psi_{0}(q(0)) \times} \right. \right. \\
& \quad \left. \left. \int_{q(T)=x} \tilde{D}_{\eta}^{+}[q(t), \varepsilon] \Psi_{0}(q(0)) \times \right] \right) \tag{2.2.2.9}
\end{align*}
$$

$$
\cos\left[\frac{1}{\epsilon}\left(\int_{0}^{T} dt\left[\frac{m}{2}\left(\frac{d\tilde{q}_{\epsilon}(t)}{dt}\right)^{2}-V(q_{\epsilon}(t))\right]\right)\right]_{\epsilon}^{2} = \\
\left(\int dx|x|\Pi_{1,\epsilon}^{2}(T,x,x_{0},\epsilon,\eta)\right)_{\epsilon}.
$$

Here
and here

$$
\begin{gather*}
\overline{\int_{q(T)=x} \tilde{D}_{\eta}^{+}[q(t), \varepsilon] \Psi_{0}(q(0))(\cdot)} = \\
\int_{q(T)=x} dy D^{+}[q(t), \varepsilon] \Psi_{0}(q(0)) \times \\
\int_{q(0)=y} dy D^{+}[q(t), \varepsilon] \Psi_{0}(q(0)) \times
\end{gather*}
\qquad (2.2.2.8)
$$

$$
\exp\left\{-\frac{1}{\epsilon} \int_{0}^{T} dt \left[ \{q(t)\}_{\eta} + \eta g(t, q(T), q(0), \lambda) - \eta \lambda \right]^2 \right\} \times (\cdot)
$$

Let us now evaluate a quantity $(\chi_1(T, \epsilon, \varepsilon, \eta))_\varepsilon$:

$$
\begin{align*}
(\chi_1(T, \epsilon, \varepsilon, \eta))_{\varepsilon} &= \\
&\quad \left( \int dx |x| \times \right. \\
&\quad \left. \left[ \overline{\int_{q(T)=x} \tilde{D}_{\eta}^{+}[q(t), \varepsilon] \Psi_{0}(q(0)) \times} \right. \right. \\
&\quad \left. \left. \int_{q(T)=x} \tilde{D}_{\eta}^{+}[q(t), \varepsilon] \Psi_{0}(q(0)) \times \right] \right) \tag{2.2.2.9}
\end{align*}
$$

$$
\cos \left[ \frac{1}{\epsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon}(t)}{dt} \right)^2 - V(q_{\epsilon}(t)) \right] \right) \right]^2_{\epsilon} = \\
\left( \int dx |x| \Pi_{1,\epsilon}^2(T,x,x_0,\epsilon,\eta) \right)_{\epsilon}.
$$

Here
$$ (\Pi_{1,\varepsilon}(T,x,x_0,\varepsilon,\eta))_{\varepsilon} = \\ \left( \overline{\int_{q(T)=x} \tilde{D}_{\eta}[q(t),\varepsilon] \Psi_0(q(0)) \cos \left[ \frac{1}{\varepsilon} \left( \int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\varepsilon}(t)}{dt} \right)^2 - V(q_{\varepsilon}(t)) \right) \right] \right)}_{\varepsilon} \quad (2.2.2.10) $$

We assume now that: $(V(q_{\varepsilon}(t),t))_{\varepsilon} = (V_0(q_{\varepsilon}(t),t))_{\varepsilon} + (V_1(q_{\varepsilon}(t),t))_{\varepsilon}$. Here

$$ \begin{gathered} (V_0(q_{\varepsilon}(t),t))_{\varepsilon} = (a_{\varepsilon}(t))_{\varepsilon} q_{\varepsilon}^2(t) + (b_{\varepsilon}(t))_{\varepsilon} (q_{\varepsilon}(t))_{\varepsilon}, \\ V_{1,\varepsilon}(q_{\varepsilon}(t),t) = (a_{3,\varepsilon}(t))_{\varepsilon} (q_{\varepsilon}^3(t))_{\varepsilon} + \dots + (a_{k,\varepsilon}(t))_{\varepsilon} (q_{\varepsilon}^k(t))_{\varepsilon}, \\ (a_{\varepsilon})_{\varepsilon} > 0, \tag{2.2.2.11} \end{gathered} $$

$$ (q_{\varepsilon}(t))_{\varepsilon} = \left( \frac{q(t)}{1 + \varepsilon^k q^m(t)} \right)_{\varepsilon}, \\ k \ge 1, m \ge 2 $$

Substitution Eq.(2.2.2.11) into Eq.(2.2.2.9) gives
$$ (\Pi_{1,\epsilon}(T, x, x_0, \epsilon, \eta))_{\epsilon} = $$

$$ \left( \overline{\int_{q(T)=x} \tilde{D}_{\eta}[q(t), \epsilon] \Psi_0(q(0)) \cos \left[ \frac{1}{\epsilon} \left( \int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon}(t)}{dt} \right)^2 - V(q_{\epsilon}(t)) \right] \right) \right]}_{\epsilon} \right) \quad (2.2.2.10) $$

We assume now that: $(V(q_{\epsilon}(t), t))_{\epsilon} = (V_0(q_{\epsilon}(t), t))_{\epsilon} + (V_1(q_{\epsilon}(t), t))_{\epsilon}$. Here

$$ (V_0(q_{\epsilon}(t), t))_{\epsilon} = (a_{\epsilon}(t))_{\epsilon} q_{\epsilon}^2(t) + (b_{\epsilon}(t))_{\epsilon} (q_{\epsilon}(t))_{\epsilon}, $$

$$ V_{1,\epsilon}(q_{\epsilon}(t),t) = (a_{3,\epsilon}(t))_{\epsilon} (q_{\epsilon}^3(t))_{\epsilon} + \dots + (a_{k,\epsilon}(t))_{\epsilon} (q_{\epsilon}^k(t))_{\epsilon}, $$

$$ (a_{\epsilon})_{\epsilon} > 0, \qquad (2.2.2.11) $$

$$ (q_{\epsilon}(t))_{\epsilon} = \left( \frac{q(t)}{1 + \epsilon^k q^m(t)} \right)_{\epsilon}, $$

$$ k \ge 1, m \ge 2 $$

Substitution Eq.(2.2.2.11) into Eq.(2.2.2.9) gives
$$ (\Pi_{1,\varepsilon}(T,x,x_0,\varepsilon,\eta))_{\varepsilon} = $$

$$ \overline{\int_{q(T)=x} \tilde{D}_{\eta}^{+}[q(t), \varepsilon] \Psi_{0}(q(0)) \times} $$

$$ \cos\left[\frac{1}{\epsilon}\left(\int_{0}^{T} dt\left[\frac{m}{2}\left(\frac{d\tilde{q}_{\varepsilon}(t)}{dt}\right)^{2}-V(q_{\varepsilon}(t),t)\right]\right)\right] = $$

$$ \overline{\int_{q(T)=x} \tilde{D}_{\eta}^{+}[q(t), \varepsilon] \Psi_{0}(q(0)) \times} $$

$$ \cos\left[\frac{1}{\epsilon}\left(\int_{0}^{T} dt\left[\frac{m}{2}\left(\frac{d\tilde{q}_{\varepsilon}(t)}{dt}\right)^{2}-V_{0}(q_{\varepsilon}(t),t)-V_{1}(q_{\varepsilon}(t),t)\right]\right)\right] = $$

$$ \overline{\int_{q(T)=x} \tilde{D}_{\eta}^{+}[q(t), \varepsilon] \Psi_{0}(q(0)) \rho_{q,\varepsilon}(\cdot) \times} $$

$$ \cos\left[\frac{1}{\epsilon}\left(\int_{0}^{T} dt\left[\frac{m}{2}\left(\frac{d\tilde{q}_{\varepsilon}(t)}{dt}\right)^{2}-V_{0}(q_{\varepsilon}(t),t)\right]\right)\right] \times \qquad (2.2.2.12) $$

$$ \cos\left[\frac{1}{\epsilon}\left(\int_{0}^{T} V_{1}(q_{\varepsilon}(t),t) dt\right)\right] + $$

$$ \overline{\int_{q(T)=x} \tilde{D}_{\eta}^{+}[q(t), \varepsilon] \Psi_{0}(q(0)) \times} $$

$$ \sin\left[\frac{1}{\epsilon}\left(\int_{0}^{T} dt\left[\frac{m}{2}\left(\frac{d\tilde{q}_{\varepsilon}(t)}{dt}\right)^{2}-V_{0}(q_{\varepsilon}(t),t)\right]\right)\right] \times $$

$$ \sin\left[-\frac{1}{\epsilon}\left(\int_{0}^{T} V_{1}(q_{\varepsilon}(t),t) dt\right)\right] = $$
$$ (\Pi_{1,\varepsilon}(T,x,x_0,\varepsilon,\eta))_{\varepsilon} = $$

$$ \overline{\int_{q(T)=x} \tilde{D}_{\eta}^{+}[q(t), \varepsilon] \Psi_{0}(q(0)) \times} $$

$$ \cos\left[\frac{1}{\epsilon}\left(\int_{0}^{T} dt\left[\frac{m}{2}\left(\frac{d\tilde{q}_{\varepsilon}(t)}{dt}\right)^{2}-V(q_{\varepsilon}(t),t)\right]\right)\right] = $$

$$ \overline{\int_{q(T)=x} \tilde{D}_{\eta}^{+}[q(t), \varepsilon] \Psi_{0}(q(0)) \times} $$

$$ \cos\left[\frac{1}{\epsilon}\left(\int_{0}^{T} dt\left[\frac{m}{2}\left(\frac{d\tilde{q}_{\varepsilon}(t)}{dt}\right)^{2}-V_{0}(q_{\varepsilon}(t),t)-V_{1}(q_{\varepsilon}(t),t)\right]\right)\right] = $$

$$ \overline{\int_{q(T)=x} \tilde{D}_{\eta}^{+}[q(t), \varepsilon] \Psi_{0}(q(0)) \rho_{q,\varepsilon}(\cdot) \times} $$

$$ \cos\left[\frac{1}{\epsilon}\left(\int_{0}^{T} dt\left[\frac{m}{2}\left(\frac{d\tilde{q}_{\varepsilon}(t)}{dt}\right)^{2}-V_{0}(q_{\varepsilon}(t),t)\right]\right)\right] \times \qquad (2.2.2.12) $$

$$ \cos\left[\frac{1}{\epsilon}\left(\int_{0}^{T} V_{1}(q_{\varepsilon}(t),t) dt\right)\right] + $$

$$ \overline{\int_{q(T)=x} \tilde{D}_{\eta}^{+}[q(t), \varepsilon] \Psi_{0}(q(0)) \times} $$

$$ \sin\left[\frac{1}{\epsilon}\left(\int_{0}^{T} dt\left[\frac{m}{2}\left(\frac{d\tilde{q}_{\varepsilon}(t)}{dt}\right)^{2}-V_{0}(q_{\varepsilon}(t),t)\right]\right)\right] \times $$

$$ \sin\left[-\frac{1}{\epsilon}\left(\int_{0}^{T} V_{1}(q_{\varepsilon}(t),t) dt\right)\right] = $$
$$v_{1,\epsilon}(T, x, x_0, \epsilon, \eta) =$$

$$\overline{\int_{q(T)=x} \tilde{D}_{\eta}^{+}[q(t), \epsilon] \Psi_{0}(q(0)) \times}$$

$$\cos \left[ \frac{1}{\epsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon}(t)}{dt} \right)^{2} - V_{0}(q_{\epsilon}(t), t) \right] \right) \right] \times$$

$$\cos \left[ \frac{1}{\epsilon} \left( \int_{0}^{T} V_{1}(q_{\epsilon}(t), t) dt \right) \right],$$

(2.2.2.13)

$$v_{2,\epsilon}(T, x, x_0, \epsilon, \eta) =$$

$$\overline{\int_{q(T)=x} \tilde{D}_{\eta}^{+}[q(t), \epsilon] \Psi_{0}(q(0)) \times}$$

$$\sin \left[ \frac{1}{\epsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon}(t)}{dt} \right)^{2} - V_{0}(q_{\epsilon}(t), t) \right] \right) \right] \times$$

$$\sin \left[ -\frac{1}{\epsilon} \left( \int_{0}^{T} V_{1}(q_{\epsilon}(t), t) dt \right) \right].$$

Let us now evaluate a quantity $(\chi_1(T, \epsilon, \epsilon, \eta))_\epsilon$ (see Eq.(2.2.9)). By using replacement $q(t) \rightarrow \sqrt{\epsilon} q(t)$, from Eq.(2.2.9) we obtain
$$v_{1,\epsilon}(T, x, x_0, \epsilon, \eta) =$$

$$\overline{\int_{q(T)=x} \tilde{D}_{\eta}^{+}[q(t), \epsilon] \Psi_{0}(q(0)) \times}$$

$$\cos \left[ \frac{1}{\epsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon}(t)}{dt} \right)^{2} - V_{0}(q_{\epsilon}(t), t) \right] \right) \right] \times$$

$$\cos \left[ \frac{1}{\epsilon} \left( \int_{0}^{T} V_{1}(q_{\epsilon}(t), t) dt \right) \right],$$

(2.2.2.13)

$$v_{2,\epsilon}(T, x, x_0, \epsilon, \eta) =$$

$$\overline{\int_{q(T)=x} \tilde{D}_{\eta}^{+}[q(t), \epsilon] \Psi_{0}(q(0)) \times}$$

$$\sin \left[ \frac{1}{\epsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon}(t)}{dt} \right)^{2} - V_{0}(q_{\epsilon}(t), t) \right] \right) \right] \times$$

$$\sin \left[ -\frac{1}{\epsilon} \left( \int_{0}^{T} V_{1}(q_{\epsilon}(t), t) dt \right) \right].$$

Let us now evaluate a quantity $(\chi_1(T, \epsilon, \epsilon, \eta))_\epsilon$ (see Eq.(2.2.9)). By using replacement $q(t) \rightarrow \sqrt{\epsilon} q(t)$, from Eq.(2.2.9) we obtain
$$ (\chi_1(T, \epsilon, \varepsilon, \eta))_{\varepsilon} = $$

$$ \left( \int dx |x| \times \right. $$

$$ \left[ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}[q(t), \epsilon \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times} \right. $$

$$ \cos \left[ \frac{1}{\epsilon} \left( \int_0^T dt \left[ \frac{m}{2} \left( \frac{\sqrt{\epsilon} d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - V(\sqrt{\epsilon} q_{v(\epsilon,\epsilon)}(t)) \right] \right) \right]_{\epsilon}^2 = $$

$$ \left( \int dx |x| \Pi_{1,\epsilon}^2 \left( T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta \right) \right)_{\epsilon}. $$

By using replacement $q(t) \to \sqrt{\epsilon} q(t)$, from Eq.(2.2.2.13) we obtain
$$ (\chi_1(T, \epsilon, \varepsilon, \eta))_{\varepsilon} = $$

$$ \left( \int dx |x| \times \right. $$

$$ \left[ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}[q(t), \epsilon \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times} \right. $$

$$ \cos \left[ \frac{1}{\epsilon} \left( \int_0^T dt \left[ \frac{m}{2} \left( \frac{\sqrt{\epsilon} d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - V(\sqrt{\epsilon} q_{v(\epsilon,\epsilon)}(t)) \right] \right) \right]_{\epsilon}^2 = $$

$$ \left( \int dx |x| \Pi_{1,\epsilon}^2 \left( T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta \right) \right)_{\epsilon}. $$

By using replacement $q(t) \to \sqrt{\epsilon} q(t)$, from Eq.(2.2.2.13) we obtain
$$v_{1,\epsilon}\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) =$$

$$\overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[q(t), \epsilon, \epsilon] \Psi_{0}(\sqrt{\epsilon} q(0)) \times}$$

$$\cos\left[\frac{1}{\epsilon}\left(\int_{0}^{T} dt\left[\frac{m}{2}\left(\frac{\sqrt{\epsilon} d\tilde{q}_{\epsilon\epsilon}(t)}{dt}\right)^{2}-V_{0}(\sqrt{\epsilon} q_{v(\epsilon,\epsilon)}(t),t)\right]\right)\right] \times$$

$$\cos\left[\frac{1}{\epsilon}\left(\int_{0}^{T} V_{1}(\sqrt{\epsilon} q_{v(\epsilon, \epsilon)}(t), t) dt\right)\right] = \qquad (2.2.2.15.a)$$

$$\overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[q(t), \epsilon, \epsilon] \Psi_{0}(\sqrt{\epsilon} q(0)) \times}$$

$$\cos\left(\int_{0}^{T} dt\left[\frac{m}{2}\left(\frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt}\right)^{2}-a_{\epsilon}(t)q_{v(\epsilon,\epsilon)}^{2}(t)+b_{\epsilon}(t)\frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}}\right]\right) \times$$

$$\cos\left(\int_{0}^{T} V_{1}(\sqrt{\epsilon} q_{v(\epsilon, \epsilon)}(t)) dt\right)$$

and
$$v_{1,\epsilon}\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) =$$

$$\overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[q(t), \epsilon, \epsilon] \Psi_{0}(\sqrt{\epsilon} q(0)) \times}$$

$$\cos\left[\frac{1}{\epsilon}\left(\int_{0}^{T} dt\left[\frac{m}{2}\left(\frac{\sqrt{\epsilon} d\tilde{q}_{\epsilon\epsilon}(t)}{dt}\right)^{2}-V_{0}(\sqrt{\epsilon} q_{v(\epsilon,\epsilon)}(t),t)\right]\right)\right] \times$$

$$\cos\left[\frac{1}{\epsilon}\left(\int_{0}^{T} V_{1}(\sqrt{\epsilon} q_{v(\epsilon, \epsilon)}(t), t) dt\right)\right] = \qquad (2.2.2.15.a)$$

$$\overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[q(t), \epsilon, \epsilon] \Psi_{0}(\sqrt{\epsilon} q(0)) \times}$$

$$\cos\left(\int_{0}^{T} dt\left[\frac{m}{2}\left(\frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt}\right)^{2}-a_{\epsilon}(t)q_{v(\epsilon,\epsilon)}^{2}(t)+b_{\epsilon}(t)\frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}}\right]\right) \times$$

$$\cos\left(\int_{0}^{T} V_{1}(\sqrt{\epsilon} q_{v(\epsilon, \epsilon)}(t)) dt\right)$$

and
$$
\begin{gather}
v_{2,\epsilon}\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) = \nonumber \\
\overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times} \tag{2.2.2.15.b} \\
\sin\left[ \frac{1}{\epsilon} \left( \int_0^T dt \left[ \frac{m}{2} \left( \frac{\sqrt{\epsilon} d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - V_0(\sqrt{\epsilon} q_{v(\epsilon,\epsilon)}(t), t) \right] \right) \right] \times \nonumber \\
\sin\left[ -\frac{1}{\epsilon} \left( \int_0^T V_1(\sqrt{\epsilon} q_{v(\epsilon,\epsilon)}(t), t) dt \right) \right]. \nonumber
\end{gather}
$$

Here

$$
\begin{align}
v(\epsilon, \epsilon) &= \epsilon^k \epsilon^{\frac{m}{2}}, \nonumber \\
\frac{1}{\epsilon} V_1(\sqrt{\epsilon} q_{v(\epsilon, \epsilon)}(t)) &= \frac{1}{\epsilon} V_1(\sqrt{\epsilon} q_{v(\epsilon, \epsilon)}(t)) = a_3(t) \epsilon^{1/2} q_{v(\epsilon, \epsilon)}^3(t) + \dots \nonumber \\
&\quad + a_k(t) \epsilon^{(k/2)-1} q_{v(\epsilon, \epsilon)}^k(t) = \hat{V}_1(\sqrt{\epsilon} q_{v(\epsilon, \epsilon)}(t)). \tag{2.2.2.16} \\
\hat{V}_1(\sqrt{\epsilon} q_{v(\epsilon, \epsilon)}(t)) &= \sqrt{\epsilon} \tilde{V}_1(q_{v(\epsilon, \epsilon)}(t)), \nonumber \\
\tilde{V}_1(q_{v(\epsilon, \epsilon)}(t)) &= a_3(t) q_{v(\epsilon, \epsilon)}^3(t) + \dots + a_k(t) \epsilon^{(k/2)-3/2} q_{v(\epsilon, \epsilon)}^k(t). \nonumber
\end{align}
$$

Let us rewrite Eqs.(2.2.15.a,b) in the following equivalent form
$$
\begin{align}
v_{2,\epsilon}\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) &= \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times} \tag{2.2.2.15.b} \\
&\quad \sin\left[\frac{1}{\epsilon}\left(\int_0^T dt \left[\frac{m}{2}\left(\frac{\sqrt{\epsilon} d\tilde{q}_{\epsilon\epsilon}(t)}{dt}\right)^2 - V_0(\sqrt{\epsilon} q_{v(\epsilon,\epsilon)}(t),t)\right]\right)\right] \times \nonumber \\
&\quad \sin\left[-\frac{1}{\epsilon}\left(\int_0^T V_1(\sqrt{\epsilon} q_{v(\epsilon,\epsilon)}(t),t)dt\right)\right]. \nonumber
\end{align}
$$

Here

$$
\begin{align*}
v(\epsilon, \epsilon) &= \epsilon^k \epsilon^{\frac{m}{2}}, \\
\frac{1}{\epsilon} V_1(\sqrt{\epsilon} q_{v(\epsilon, \epsilon)}(t)) &= \frac{1}{\epsilon} V_1(\sqrt{\epsilon} q_{v(\epsilon, \epsilon)}(t)) = a_3(t) \epsilon^{1/2} q_{v(\epsilon, \epsilon)}^3(t) + \dots \\
&\quad + a_k(t) \epsilon^{(k/2)-1} q_{v(\epsilon, \epsilon)}^k(t) = \hat{V}_1(\sqrt{\epsilon} q_{v(\epsilon, \epsilon)}(t)). \tag{2.2.2.16} \\
\hat{V}_1(\sqrt{\epsilon} q_{v(\epsilon, \epsilon)}(t)) &= \sqrt{\epsilon} \tilde{V}_1(q_{v(\epsilon, \epsilon)}(t)), \\
\tilde{V}_1(q_{v(\epsilon, \epsilon)}(t)) &= a_3(t) q_{v(\epsilon, \epsilon)}^3(t) + \dots + a_k(t) \epsilon^{(k/2)-3/2} q_{v(\epsilon, \epsilon)}^k(t).
\end{align*}
$$

Let us rewrite Eqs.(2.2.15.a,b) in the following equivalent form
$$
\begin{align}
& v_{1,\epsilon}\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) = \nonumber \\
& \quad \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\text{sm}}^{(1)}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon}q(0)) \times} \nonumber \\
& \quad \left| \cos\left(\int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t)q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right| \times \nonumber \\
& \quad \cos\left(\int_0^T \hat{V}_1(q_{v(\epsilon,\epsilon)}(t))dt\right), \tag{2.2.2.17}
\end{align}
$$

$$
\begin{align}
v_{2,\epsilon}\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) = & \nonumber \\
& \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\text{sm}}^{(2)}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon}q(0)) \times} \nonumber \\
& \left| \sin\left(\int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t)q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right| \times \nonumber \\
& \sin\left(-\int_0^T \hat{V}_1(q_{v(\epsilon,\epsilon)}(t))dt\right). \tag{2.2.2.17}
\end{align}
$$

Here $\tilde{D}_{\text{sm}}^{(1)}[q(t), \epsilon, \epsilon]$ and $\tilde{D}_{\text{sm}}^{(2)}[q(t), \epsilon, \epsilon]$ is the corresponding signed Feynman's "measures", i.e.
$$
\begin{align}
& v_{1,\epsilon}\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) = \nonumber \\
& \quad \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\text{sm}}^{(1)}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon}q(0)) \times} \nonumber \\
& \quad \left| \cos\left(\int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t)q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right| \times \nonumber \\
& \quad \cos\left(\int_0^T \hat{V}_1(q_{v(\epsilon,\epsilon)}(t))dt\right), \tag{2.2.2.17}
\end{align}
$$

$$
\begin{align}
v_{2,\epsilon}\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) = & \nonumber \\
& \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\text{sm}}^{(2)}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon}q(0)) \times} \nonumber \\
& \left| \sin\left(\int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t)q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right| \times \nonumber \\
& \sin\left(-\int_0^T \hat{V}_1(q_{v(\epsilon,\epsilon)}(t))dt\right). \tag{2.2.2.17}
\end{align}
$$

Here $\tilde{D}_{\text{sm}}^{(1)}[q(t), \epsilon, \epsilon]$ and $\tilde{D}_{\text{sm}}^{(2)}[q(t), \epsilon, \epsilon]$ is the corresponding signed Feynman's "measures", i.e.
$$ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\text{sm}}^{(1)}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times (\cdot)} = $$

$$ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[q(t), \epsilon, \epsilon] \Psi_{0}(\sqrt{\epsilon} q(0)) \times} $$

$$ [\cos^{+}(S_{\epsilon}(T, x, q(t), \epsilon)) + \cos^{-}(S_{\epsilon}(T, x, q(t), \epsilon))] \times (\cdot) = $$

$$ \overline{\int_{\substack{q(T)=\frac{x}{\sqrt{\epsilon}} \\ q(0)=\frac{y}{\sqrt{\epsilon}}}}} dy D^{+}[q(t), \epsilon] \Psi_{0}(\sqrt{\epsilon} q(0)) \times $$

$$ [\cos^{+}(S_{\epsilon}(T, x, q(t), \epsilon)) + \cos^{-}(S_{\epsilon}(T, x, q(t), \epsilon))] \times $$

(2.2.2.18)

$$ \exp\left\{-\frac{1}{\epsilon} \int_{0}^{T} dt \left[ \left\{ \sqrt{\epsilon} q(t) \right\}_{\eta} + \eta g(t, \sqrt{\epsilon} q(T), \sqrt{\epsilon} q(0), \lambda) - \eta \lambda \right]^{2} \right\} \times (\cdot) = $$

$$ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}}} D^{+}[q(t), \epsilon] \Psi_{0}(q(0)) \times $$

$$ [\cos^{+}(S_{\epsilon}(T, x, q(t), \epsilon)) + \cos^{-}(S_{1,\epsilon}(T, x, q(t), \epsilon))] \times $$

$$ \exp\left\{-\frac{1}{\epsilon} \int_{0}^{T} dt \left[ \left\{ \sqrt{\epsilon} q(t) \right\}_{\eta} - \eta g(t, \sqrt{\epsilon} q(T), \sqrt{\epsilon} q(0)) - \eta \lambda \right]^{2} \right\} \times (\cdot), $$

$$ S_{\epsilon}(T, x, q(t), \epsilon) = \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^{2} - a_{\epsilon}(t) q_{\epsilon}^{2}(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon, \epsilon)}(t)}{\sqrt{\epsilon}} \right] $$

and
$$ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\text{sm}}^{(1)}[q(t), \epsilon, \epsilon]} \Psi_0(\sqrt{\epsilon} q(0)) \times (\cdot) = $$

$$ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[q(t), \epsilon, \epsilon]} \Psi_0(\sqrt{\epsilon} q(0)) \times $$

$$ [\cos^+(S_\epsilon(T, x, q(t), \epsilon)) + \cos^-(S_\epsilon(T, x, q(t), \epsilon))] \times (\cdot) = $$

$$ \overline{\int_{\substack{q(T)=\frac{x}{\sqrt{\epsilon}} \\ q(0)=\frac{y}{\sqrt{\epsilon}}}}} dy D^+[q(t), \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times $$

$$ [\cos^+(S_\epsilon(T, x, q(t), \epsilon)) + \cos^-(S_\epsilon(T, x, q(t), \epsilon))] \times $$

(2.2.2.18)

$$ \exp\left\{-\frac{1}{\epsilon} \int_0^T dt \left[ \left\{ \sqrt{\epsilon} q(t) \right\}_\eta + \eta g(t, \sqrt{\epsilon} q(T), \sqrt{\epsilon} q(0), \lambda) - \eta \lambda \right]^2 \right\} \times (\cdot) = $$

$$ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} D^+[q(t), \epsilon]} \Psi_0(q(0)) \times $$

$$ [\cos^+(S_\epsilon(T, x, q(t), \epsilon)) + \cos^-(S_{1,\epsilon}(T, x, q(t), \epsilon))] \times $$

$$ \exp\left\{-\frac{1}{\epsilon} \int_0^T dt \left[ \left\{ \sqrt{\epsilon} q(t) \right\}_\eta - \eta g(t, \sqrt{\epsilon} q(T), \sqrt{\epsilon} q(0)) - \eta \lambda \right]^2 \right\} \times (\cdot), $$

$$ S_{\epsilon}(T, x, q(t), \epsilon) = \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{\epsilon}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon, \epsilon)}(t)}{\sqrt{\epsilon}} \right] $$

and
$$ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\text{sm}}^{(2)}[q(t), \epsilon, \epsilon]} \Psi_0(\sqrt{\epsilon} q(0)) \times (\cdot) = $$

$$ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[q(t), \epsilon]} \Psi_0(q(0)) \times $$

$$ [\sin^+(S_\epsilon(T, x, q(t), \epsilon)) + \sin^-(S_\epsilon(T, x, q(t), \epsilon))] \times (\cdot) = $$

$$ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} dy D^+[q(t), \epsilon]} \Psi_0(\sqrt{\epsilon} q(0)) \times $$

$$ [\sin^+(S_\epsilon(T, x, q(t), \epsilon)) + \sin^-(S_\epsilon(T, x, q(t), \epsilon))] \times \exp\left\{-\frac{1}{\epsilon} \int_0^T dt \left[ \{q(t)\}_\eta - \eta g(t, \sqrt{\epsilon}q(T), \sqrt{\epsilon}q(0)) - \eta \lambda \right]^2 \right\} \times (\cdot) = \tag{2.2.2.19} $$

$$ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} D^+[q(t), \epsilon]} \Psi_0(\sqrt{\epsilon} q(0)) \times $$

$$ [\sin^+(S_{1,\epsilon}(T, x, q(t), \epsilon)) + \sin^-(S_\epsilon(T, x, q(t), \epsilon))] \times $$

$$ \exp\left\{-\frac{1}{\epsilon} \int_0^T dt \left[ \{\sqrt{\epsilon} q(t)\}_\eta + \eta g(t, \sqrt{\epsilon} q(T), \sqrt{\epsilon} q(0), \lambda) - \eta \lambda \right]^2 \right\} \times (\cdot), $$

$$ S_{\epsilon}(T, x, q(t), \epsilon) = \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{\epsilon}^2(t) + b_{\epsilon}(t) \frac{q_{V(\epsilon, \epsilon)}(t)}{\sqrt{\epsilon}} \right]. $$
$$ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\text{sm}}^{(2)}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times (\cdot)} = $$

$$ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[q(t), \epsilon] \Psi_{0}(q(0)) \times} $$

$$ [\sin^+(S_\epsilon(T, x, q(t), \epsilon)) + \sin^-(S_\epsilon(T, x, q(t), \epsilon))] \times (\cdot) = $$

$$ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} dy D^+[q(t), \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times} $$

$$ [\sin^+(S_\epsilon(T, x, q(t), \epsilon)) + \sin^-(S_\epsilon(T, x, q(t), \epsilon))] \times \quad (2.2.2.19) $$

$$ \exp\left\{-\frac{1}{\epsilon} \int_0^T dt \left[ \{q(t)\}_\eta - \eta g(t, \sqrt{\epsilon} q(T), \sqrt{\epsilon} q(0)) - \eta \lambda \right]^2 \right\} \times (\cdot) = $$

$$ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} D^+[q(t), \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times} $$

$$ [\sin^+(S_{1,\epsilon}(T, x, q(t), \epsilon)) + \sin^-(S_\epsilon(T, x, q(t), \epsilon))] \times $$

$$ \exp\left\{-\frac{1}{\epsilon} \int_0^T dt \left[ \{\sqrt{\epsilon} q(t)\}_\eta + \eta g(t, \sqrt{\epsilon} q(T), \sqrt{\epsilon} q(0), \lambda) - \eta \lambda \right]^2 \right\} \times (\cdot), $$

$$ S_{\epsilon}(T, x, q(t), \epsilon) = \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon}(t)}{dt} \right)^{2} - a_{\epsilon}(t) q_{\epsilon}^{2}(t) + b_{\epsilon}(t) \frac{q_{V(\epsilon, \epsilon)}(t)}{\sqrt{\epsilon}} \right]. $$
Let us rewrite Eq.(2.2.18)-Eq.(2.2.19) in the following equivalent form

$$ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\text{sm}}^{(1)}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times (\cdot) =} $$

$$ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[q(t), \epsilon, \epsilon] \Psi_{0}(\sqrt{\epsilon} q(0)) \times} $$

$$ [\cos^{+}(S_{\epsilon}(T, x, q(t), \epsilon)) + \cos^{-}(S_{\epsilon}(T, x, q(t), \epsilon)) + 1 - 1] \times (\cdot) = $$

$$ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\text{m}}^{(1)}[q(t), \epsilon] \Psi_{0}(\sqrt{\epsilon} q(0)) \times (\cdot) - \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[q(t), \epsilon] \Psi_{0}(\sqrt{\epsilon} q(0)) \times (\cdot) =} \quad (2.2.20) $$

$$ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\text{m}}^{(1)}[q(t), \epsilon, \epsilon] \Psi_{0}(q(0)) \times (\cdot) - \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} D^{+}[q(t), \epsilon] \Psi_{0}(q(0)) \times} $$

$$ \exp\left\{-\int_{0}^{T} dt \left[ \{\sqrt{\epsilon} q(t)\}_{\eta} + \eta g(t, q(T), q(0), \lambda) - \eta \lambda \right]^2\right\} \times (\cdot) $$

and
Let us rewrite Eq.(2.2.18)-Eq.(2.2.19) in the following equivalent form

$$ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\text{sm}}^{(1)}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times (\cdot) =} $$

$$ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[q(t), \epsilon, \epsilon] \Psi_{0}(\sqrt{\epsilon} q(0)) \times} $$

$$ [\cos^{+}(S_{\epsilon}(T, x, q(t), \epsilon)) + \cos^{-}(S_{\epsilon}(T, x, q(t), \epsilon)) + 1 - 1] \times (\cdot) = $$

$$ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\text{m}}^{(1)}[q(t), \epsilon] \Psi_{0}(\sqrt{\epsilon} q(0)) \times (\cdot) - \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[q(t), \epsilon] \Psi_{0}(\sqrt{\epsilon} q(0)) \times (\cdot) =} \quad (2.2.20) $$

$$ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\text{m}}^{(1)}[q(t), \epsilon, \epsilon] \Psi_{0}(q(0)) \times (\cdot) - \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} D^{+}[q(t), \epsilon] \Psi_{0}(q(0)) \times} $$

$$ \exp\left\{-\int_{0}^{T} dt \left[\{\sqrt{\epsilon} q(t)\}_{\eta} + \eta g(t, q(T), q(0), \lambda) - \eta \lambda\right]^{2}\right\} \times (\cdot) $$

and
$$ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\text{sm}}^{(2)}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times (\cdot)} = $$

$$ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[q(t), \epsilon, \epsilon] \Psi_{0}(\sqrt{\epsilon} q(0)) \times} $$

$$ [\sin^+(S_\epsilon(T, x, q(t), \epsilon)) + \sin^-(S_\epsilon(T, x, q(t), \epsilon)) + 1 - 1] \times (\cdot) = $$

$$ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\mathbf{m}}^{(2)}[q(t), \epsilon, \epsilon] \Psi_{0}(\sqrt{\epsilon} q(0)) \times (\cdot)} \quad (2.2.21) $$

$$ - \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[q(t), \epsilon, \epsilon] \Psi_{0}(\sqrt{\epsilon} q(0)) \times (\cdot)} = $$

$$ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\mathbf{m}}^{(2)}[q(t), \epsilon] \Psi_{0}(\sqrt{\epsilon} q(0)) \times (\cdot) - \int_{q(T)=\frac{x}{\sqrt{\epsilon}}} D^{+}[q(t), \epsilon] \Psi_{0}(\sqrt{\epsilon} q(0)) \times} $$

$$ \exp\left\{-\frac{1}{\epsilon} \int_{0}^{T} dt \left[ \{\sqrt{\epsilon} q(t)\}_{\eta} + \eta g(t, \sqrt{\epsilon} q(T), \sqrt{\epsilon} q(0), \lambda) - \eta \lambda \right]^2 \right\} \times (\cdot). $$

Here $\tilde{D}_{\mathbf{m}}^{(1)}$ and $\tilde{D}_{\mathbf{m}}^{(2)}$ is the corresponding positive Feynman submeasures, i.e.
$$ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\text{sm}}^{(2)}[q(t), \epsilon, \epsilon]} \Psi_0(\sqrt{\epsilon} q(0)) \times (\cdot) = $$

$$ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[q(t), \epsilon, \epsilon]} \Psi_0(\sqrt{\epsilon} q(0)) \times (\cdot) $$

$$ [\sin^+(S_\epsilon(T, x, q(t), \epsilon)) + \sin^-(S_\epsilon(T, x, q(t), \epsilon)) + 1 - 1] \times (\cdot) = $$

$$ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\mathbf{m}}^{(2)}[q(t), \epsilon, \epsilon]} \Psi_0(\sqrt{\epsilon} q(0)) \times (\cdot) \quad (2.2.21) $$

$$ - \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[q(t), \epsilon, \epsilon]} \Psi_0(\sqrt{\epsilon} q(0)) \times (\cdot) = $$

$$ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\mathbf{m}}^{(2)}[q(t), \epsilon]} \Psi_0(\sqrt{\epsilon} q(0)) \times (\cdot) - \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} D^{+}[q(t), \epsilon]} \Psi_0(\sqrt{\epsilon} q(0)) \times (\cdot) $$

$$ \exp\left\{-\frac{1}{\epsilon} \int_{0}^{T} dt \left[ \{\sqrt{\epsilon} q(t)\}_{\eta} + \eta g(t, \sqrt{\epsilon} q(T), \sqrt{\epsilon} q(0), \lambda) - \eta \lambda \right]^2 \right\} \times (\cdot) $$

Here $\tilde{D}_{\mathbf{m}}^{(1)}$ and $\tilde{D}_{\mathbf{m}}^{(2)}$ is the corresponding positive Feynman submeasures, i.e.
$$ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\text{m}}^{(1)}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times (\cdot) =} \\ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} D^{+}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times} \qquad (2.2.22) $$

$$ [\cos^+(S_\epsilon(T,x,q(t),\epsilon)) + \cos^-(S_\epsilon(T,x,q(t),\epsilon)) + 1] \times \\ \exp\left\{-\frac{1}{\epsilon} \int_0^T dt \left[ \{\sqrt{\epsilon} q(t)\}_\eta + \eta g(t, \sqrt{\epsilon} q(T), \sqrt{\epsilon} q(0), \lambda) - \eta \lambda \right]^2 \right\} \times (\cdot). $$

and

$$ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\text{m}}^{(2)}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times (\cdot) =} \\ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} D^{+}[q(t), \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times} \qquad (2.2.23) $$

$$ [\sin^+(S_\epsilon(T,x,q(t),\epsilon)) + \sin^-(S_{1,\epsilon}(T,x,q(t),\epsilon)) + 1] \times \\ \exp\left\{-\frac{1}{\epsilon} \int_0^T dt \left[ \{\sqrt{\epsilon} q(t)\}_\eta - \eta g(t, \sqrt{\epsilon} q(T), \sqrt{\epsilon} q(0), \lambda) - \eta \lambda \right]^2 \right\} \times (\cdot). $$

From Eq.(2.2.2.18)-Eq.(2.2.2.23) we obtain
$$
\begin{gather*}
\overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\mathbf{m}}^{(1)}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times (\cdot)} = \\
\overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} D^+[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times} \tag{2.2.22}
\end{gather*}
$$

$$
[\cos^+(S_\epsilon(T,x,q(t),\epsilon)) + \cos^-(S_\epsilon(T,x,q(t),\epsilon)) + 1] \times \\
\exp\left\{-\frac{1}{\epsilon} \int_0^T dt \left[\{\sqrt{\epsilon}q(t)\}_\eta + \eta g(t, \sqrt{\epsilon}q(T), \sqrt{\epsilon}q(0), \lambda) - \eta\lambda\right]^2\right\} \times (\cdot).
$$

and

$$
\begin{gather*}
\overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\mathbf{m}}^{(2)}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times (\cdot)} = \\
\overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} D^+[q(t), \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times} \tag{2.2.23}
\end{gather*}
$$

$$
[\sin^+(S_\epsilon(T,x,q(t),\epsilon)) + \sin^-(S_{1,\epsilon}(T,x,q(t),\epsilon)) + 1] \times \\
\exp\left\{-\frac{1}{\epsilon} \int_0^T dt \left[\{\sqrt{\epsilon}q(t)\}_\eta - \eta g(t, \sqrt{\epsilon}q(T), \sqrt{\epsilon}q(0), \lambda) - \eta\lambda\right]^2\right\} \times (\cdot).
$$

From Eq.(2.2.2.18)-Eq.(2.2.2.23) we obtain
$$
\begin{align*}
& v_{1,\epsilon}\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) = \\
& \quad \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\text{sm}}^{(1)}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times} \\
& \quad \left| \cos\left(\int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right| \times \\
& \quad \cos\left(\int_0^T \hat{V}_1(\sqrt{\epsilon} q_{v(\epsilon,\epsilon)}(t)) dt\right) = \\
& \quad \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\text{m}}^{(1)}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times} \\
& \quad \left| \cos\left(\int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right| \times \tag{2.2.2.24} \\
& \quad \cos\left(\int_0^T \hat{V}_1(\sqrt{\epsilon} q_{v(\epsilon,\epsilon)}(t)) dt\right) - \\
& \quad \quad \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[q(t), \epsilon\epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times} \\
& \quad \left| \cos\left(\int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right| \times \\
& \quad \cos\left(\int_0^T \hat{V}_1(\sqrt{\epsilon} q_{v(\epsilon,\epsilon)}(t)) dt\right)
\end{align*}
$$
$$
\begin{align*}
& v_{1,\epsilon}\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) = \\
& \qquad \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\text{sm}}^{(1)}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon}q(0)) \times} \\
& \qquad \left| \cos\left(\int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t)q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right| \times \\
& \qquad \cos\left(\int_0^T \hat{V}_1(\sqrt{\epsilon}q_{v(\epsilon,\epsilon)}(t))dt\right) = \\
& \qquad \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\text{m}}^{(1)}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon}q(0)) \times} \\
& \qquad \left| \cos\left(\int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t)q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right| \times \tag{2.2.2.24} \\
& \qquad \cos\left(\int_0^T \hat{V}_1(\sqrt{\epsilon}q_{v(\epsilon,\epsilon)}(t))dt\right) - \\
& \qquad \qquad - \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[q(t), \epsilon\epsilon] \Psi_0(\sqrt{\epsilon}q(0)) \times} \\
& \qquad \left| \cos\left(\int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t)q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right| \times \\
& \qquad \cos\left(\int_0^T \hat{V}_1(\sqrt{\epsilon}q_{v(\epsilon,\epsilon)}(t))dt\right)
\end{align*}
$$
and
and
$$v_{2,\epsilon}\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) =$$

$$\overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\text{sm}}^{(2)}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times}$$

$$\left| \sin\left(\int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{V(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{V(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right| \times$$

$$\cos\left(\int_{0}^{T} \hat{V}_{1}(\sqrt{\epsilon} q_{V(\epsilon, \epsilon)}(t)) dt\right) =$$

$$\overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\mathbf{m}}^{(2)}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times}$$

$$\left| \sin\left(\int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{V(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{V(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right| \times \quad (2.2.2.25)$$

$$\sin\left(-\int_{0}^{T} \hat{V}_{1}(\sqrt{\epsilon} q_{V(\epsilon, \epsilon)}(t)) dt\right) -$$

$$- \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[q(t), \epsilon\epsilon] \Psi_{0}(\sqrt{\epsilon} q(0)) \times}$$

$$\left| \sin\left(\int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{V(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{V(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right| \times$$

$$\sin\left(\int_{0}^{T} \hat{V}_{1}(\sqrt{\epsilon} q_{V(\epsilon, \epsilon)}(t)) dt\right).$$
$$
\begin{align*}
& v_{2,\epsilon}\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) = \\
& \qquad \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\text{sm}}^{(2)}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times} \\
& \qquad \left| \sin\left(\int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right| \times \\
& \qquad \cos\left(\int_0^T \hat{V}_1(\sqrt{\epsilon} q_{v(\epsilon,\epsilon)}(t)) dt\right) = \\
& \qquad \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\mathbf{m}}^{(2)}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times} \\
& \qquad \left| \sin\left(\int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right| \times \tag{2.2.2.25} \\
& \qquad \sin\left(-\int_0^T \hat{V}_1(\sqrt{\epsilon} q_{v(\epsilon,\epsilon)}(t)) dt\right) - \\
& \qquad \qquad - \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[q(t), \epsilon\epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times} \\
& \qquad \left| \sin\left(\int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right| \times \\
& \qquad \sin\left(\int_0^T \hat{V}_1(\sqrt{\epsilon} q_{v(\epsilon,\epsilon)}(t)) dt\right).
\end{align*}
$$
From Eq.(2.2.2.25) and Eq.(2.2.2.26) we obtain

$$
\begin{align}
v_{1,\epsilon}\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) &= \tilde{v}_{1,\epsilon}\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) - v_{1,\epsilon}^*\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right), \nonumber \\
\tilde{v}_{1,\epsilon}\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) &= \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\mathbf{m}}^{(1)}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times} \nonumber \\
&\quad \left| \cos\left(\int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right| \times \nonumber \\
&\quad \cos\left(\int_0^T \hat{V}_1(\sqrt{\epsilon} q_{v(\epsilon,\epsilon)}(t)) dt\right), \tag{2.2.2.26}
\end{align}
$$

$$
v_{1,\epsilon}^*(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta) = \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times}
$$

$$
\left| \cos \left( \int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right| \times \\
\cos \left( \int_0^T \hat{V}_1(\sqrt{\epsilon} q_{v(\epsilon,\epsilon)}(t)) dt \right)
$$

and
From Eq.(2.2.2.25) and Eq.(2.2.2.26) we obtain

$$v_{1,\epsilon}\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) = \tilde{v}_{1,\epsilon}\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) - v_{1,\epsilon}^*\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right),$$

$$\tilde{v}_{1,\epsilon}\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) = \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\mathbf{m}}^{(1)}[q(t), \epsilon, \epsilon]} \Psi_0(\sqrt{\epsilon} q(0)) \times$$

$$\left| \cos\left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right| \times \\ \cos\left( \int_{0}^{T} \hat{V}_1(\sqrt{\epsilon} q_{v(\epsilon,\epsilon)}(t)) dt \right), \qquad (2.2.2.26)$$

$$v_{1,\epsilon}^*(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta) = \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^+[q(t), \epsilon, \epsilon]} \Psi_0(\sqrt{\epsilon} q(0)) \times$$

$$\left| \cos\left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right| \times \\ \cos\left( \int_{0}^{T} \hat{V}_1(\sqrt{\epsilon} q_{v(\epsilon,\epsilon)}(t)) dt \right)$$

and
$$v_{2,\epsilon}\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) = \tilde{v}_{2,\epsilon}\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) - v_{2,\epsilon}^*\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right),$$

$$\tilde{\tilde{v}}_{2,\epsilon}\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) = \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\mathbf{m}}^{(2)}[q(t), \epsilon\epsilon]\Psi_0(\sqrt{\epsilon}q(0)) \times}$$

$$\left| \sin\left(\int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right| \times \sin\left(-\int_0^T V_{1,\epsilon}(\sqrt{\epsilon}q_{v(\epsilon,\epsilon)}(t))dt\right), \qquad (2.2.2.27)$$

$$v_{2,\epsilon}^*(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta) = \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^+[q(t), \epsilon, \epsilon]\Psi_0(\sqrt{\epsilon}q(0)) \times}$$

$$\left| \sin\left(\int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right| \times \sin\left(-\int_0^T \hat{V}_1(\sqrt{\epsilon}q_{v(\epsilon,\epsilon)}(t))dt\right).$$

Let us evaluate now path integral $v_{1,\epsilon}^*(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta)$ for the case

$$g(t, q(T), q(0), \lambda) \equiv 0, \quad \lambda = 0 :$$
$$v_{2,\epsilon}\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) = \tilde{v}_{2,\epsilon}\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) - v_{2,\epsilon}^*\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right),$$

$$\tilde{\tilde{v}}_{2,\epsilon}\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) = \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\mathbf{m}}^{(2)}[q(t), \epsilon\epsilon]\Psi_0(\sqrt{\epsilon}q(0)) \times}$$

$$\left| \sin\left(\int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right| \times \sin\left(-\int_{0}^{T} V_{1,\epsilon}(\sqrt{\epsilon}q_{v(\epsilon,\epsilon)}(t))dt\right), \qquad (2.2.2.27)$$

$$v_{2,\epsilon}^*(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta) = \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^+[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon}q(0)) \times}$$

$$\left| \sin\left(\int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right| \times \sin\left(-\int_{0}^{T} \hat{V}_1(\sqrt{\epsilon}q_{v(\epsilon,\epsilon)}(t))dt\right).$$

Let us evaluate now path integral $v_{1,\epsilon}^*(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta)$ for the case

$$g(t, q(T), q(0), \lambda) \equiv 0, \quad \lambda = 0 :$$
$$
\begin{gather*}
V_{1,\epsilon}^*(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta) = \overline{\int_{\substack{q(T)=\frac{x}{\sqrt{\epsilon}} \\ q(0)=\frac{y}{\sqrt{\epsilon}}}} dy D^+[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \Xi_1(q(t), \epsilon, \epsilon) \times \\
\cos\left(\int_0^T \hat{V}_1(\sqrt{\epsilon} q_{v(\epsilon,\epsilon)}(t)) dt\right) \exp\left\{-\frac{1}{\epsilon} \int_0^T dt \left[\{\sqrt{\epsilon} q(t)\}_\eta\right]^2\right\} \quad (2.2.2.28) \\
\Xi_1(q(t), \epsilon, \epsilon) =
\end{gather*}
$$

$$
\left| \cos \left( \int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right|.
$$

From Eqs.(2.2.2.28) we obtain
$$
\begin{gather*}
V_{1,\epsilon}^*(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta) = \overline{\int_{\substack{q(T)=\frac{x}{\sqrt{\epsilon}} \\ q(0)=\frac{y}{\sqrt{\epsilon}}}} dy D^+[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \Xi_1(q(t), \epsilon, \epsilon) \times \\
\cos\left(\int_0^T \hat{V}_1(\sqrt{\epsilon} q_{v(\epsilon,\epsilon)}(t)) dt\right) \exp\left\{-\frac{1}{\epsilon} \int_0^T dt \left[\{\sqrt{\epsilon} q(t)\}_\eta\right]^2\right\} \quad (2.2.2.28) \\
\Xi_1(q(t), \epsilon, \epsilon) =
\end{gather*}
$$

$$
\left| \cos \left( \int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right|.
$$

From Eqs.(2.2.2.28) we obtain
$$v_{1,\epsilon}^*(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta) = \overline{\int_{\substack{q(T)=\frac{x}{\sqrt{\epsilon}} \\ q(0)=\frac{y}{\sqrt{\epsilon}}}} dy D^+[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \Xi_1(q(t), \epsilon, \epsilon) \times$$

$$\cos\left(\int_{0}^{T} \sqrt{\epsilon} \tilde{V}_{1}(q_{V(\epsilon, \epsilon)}(t)) dt\right) \exp\left\{-\int_{0}^{T} dt\left[\{q(t)\}_{\eta}\right]^{2}\right\} = \\ \overline{\int_{\substack{q(T)=\frac{x}{\sqrt{\epsilon}} \\ q(0)=\frac{y}{\sqrt{\epsilon}}}} dy D^{+}[q(t), \epsilon, \epsilon] \Psi_{0}(\sqrt{\epsilon} q(0)) \Xi_{1}(q(t), \epsilon, \epsilon) \times$$

$$\exp\left\{-\int_{0}^{T} dt \left[\{q(t)\}_{\eta}\right]^{2}\right\} \left[\sum_{n=0}^{M} \frac{(-1)^{n} \epsilon^{n}}{(2n)!} \left(\int_{0}^{T} \tilde{V}_{1}(q_{V(\epsilon, \epsilon)}(t)) dt\right)^{2n} + \epsilon^{n+1} O\left(\left(\int_{0}^{T} \tilde{V}_{1}(q_{V(\epsilon, \epsilon)}(t)) dt\right)^{2n+1}\right)\right] = \qquad (2.2.2.29)$$

$$\overline{\int_{\substack{q(T)=\frac{x}{\sqrt{\epsilon}} \\ q(0)=\frac{y}{\sqrt{\epsilon}}}} dy D^+[q(t), \epsilon, \epsilon] \Psi_0(y) \Xi_1(q(t), \epsilon, \epsilon) \exp\left\{-\int_{0}^{T} dt \left[\{q(t)\}_{\eta}\right]^2\right\} +$$

$$\sum_{n=1}^{M} \frac{(-1)^n \epsilon^n}{(2n)!} \left[ \overline{\int_{\substack{q(T)=\frac{x}{\sqrt{\epsilon}} \\ q(0)=\frac{y}{\sqrt{\epsilon}}}} dy D^+[q(t), \epsilon, \epsilon] \Psi_0(y) \Xi_1(q(t), \epsilon, \epsilon) \times \right. \\ \left. \exp\left\{-\int_0^T dt \left[\{q(t)\}_\eta\right]^2\right\} \left(\int_0^T \tilde{V}_1(q_{V(\epsilon, \epsilon)}(t)) dt\right)^{2n} \right] \right]$$
$$v_{1,\epsilon}^*(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta) = \overline{\int_{\substack{q(T)=\frac{x}{\sqrt{\epsilon}} \\ q(0)=\frac{y}{\sqrt{\epsilon}}}} dy D^+[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \Xi_1(q(t), \epsilon, \epsilon) \times$$

$$\cos\left(\int_{0}^{T} \sqrt{\epsilon} \tilde{V}_{1}(q_{V(\epsilon, \epsilon)}(t)) dt\right) \exp\left\{-\int_{0}^{T} dt\left[\{q(t)\}_{\eta}\right]^{2}\right\} = \\ \overline{\int_{\substack{q(T)=\frac{x}{\sqrt{\epsilon}} \\ q(0)=\frac{y}{\sqrt{\epsilon}}}} dy D^{+}[q(t), \epsilon, \epsilon] \Psi_{0}(\sqrt{\epsilon} q(0)) \Xi_{1}(q(t), \epsilon, \epsilon) \times$$

$$\exp\left\{-\int_{0}^{T} dt \left[\{q(t)\}_{\eta}\right]^{2}\right\} \left[\sum_{n=0}^{M} \frac{(-1)^{n} \epsilon^{n}}{(2n)!} \left(\int_{0}^{T} \tilde{V}_{1}(q_{V(\epsilon, \epsilon)}(t)) dt\right)^{2n} + \epsilon^{n+1} O\left(\left(\int_{0}^{T} \tilde{V}_{1}(q_{V(\epsilon, \epsilon)}(t)) dt\right)^{2n+1}\right)\right] = \tag{2.2.2.29}$$

$$\overline{\int_{\substack{q(T)=\frac{x}{\sqrt{\epsilon}} \\ q(0)=\frac{y}{\sqrt{\epsilon}}}} dy D^+[q(t), \epsilon, \epsilon] \Psi_0(y) \Xi_1(q(t), \epsilon, \epsilon) \exp\left\{-\int_{0}^{T} dt \left[\{q(t)\}_{\eta}\right]^2\right\} +$$

$$\sum_{n=1}^{M} \frac{(-1)^n \epsilon^n}{(2n)!} \left[ \overline{\int_{\substack{q(T)=\frac{x}{\sqrt{\epsilon}} \\ q(0)=\frac{y}{\sqrt{\epsilon}}}} dy D^+[q(t), \epsilon, \epsilon] \Psi_0(y) \Xi_1(q(t), \epsilon, \epsilon) \times \right. \\ \left. \exp\left\{-\int_{0}^{T} dt \left[\{q(t)\}_{\eta}\right]^2\right\} \left(\int_{0}^{T} \tilde{V}_1(q_{V(\epsilon, \epsilon)}(t)) dt\right)^{2n} \right] \right]$$
Let us evaluate now path integral $v_{2,\epsilon}^*(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta)$ for the case:

$g(t, q(T), q(0), \lambda) \equiv 0, \lambda = 0:$

$$
\begin{align}
\tilde{v}_{2,\epsilon}^*(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta) &= \overline{\int_{q(0)=\frac{y}{\sqrt{\epsilon}}}^{q(T)=\frac{x}{\sqrt{\epsilon}}} dy D^+[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \Xi_2(q(t), \epsilon, \epsilon) \times} \nonumber \\
&\quad \sin\left(-\int_0^T \hat{V}_1(\sqrt{\epsilon} q_{v(\epsilon,\epsilon)}(t)) dt\right) \exp\left\{-\int_0^T dt \left[\{q(t)\}_\eta\right]^2\right\} \tag{2.2.2.30}
\end{align}
$$

$$
\Xi_2(q(t), \epsilon, \epsilon) =
$$

$$
\left| \sin\left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right| \times
$$

From Eq.(2.2.2.30) we obtain
Let us evaluate now path integral $v_{2,\epsilon}^*(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta)$ for the case:

$g(t, q(T), q(0), \lambda) \equiv 0, \lambda = 0:$

$$ \tilde{v}_{2,\epsilon}^*(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta) = \overline{\int_{q(0)=\frac{y}{\sqrt{\epsilon}}}^{q(T)=\frac{x}{\sqrt{\epsilon}}} dy D^+[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \Xi_2(q(t), \epsilon, \epsilon) \times} \\ \sin\left(-\int_0^T \hat{V}_1(\sqrt{\epsilon} q_{V(\epsilon,\epsilon)}(t)) dt\right) \exp\left\{-\int_0^T dt \left[\{q(t)\}_\eta\right]^2\right\} \quad (2.2.2.30) $$

$$ \Xi_2(q(t), \epsilon, \epsilon) = $$

$$ \left| \sin\left(\int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{V(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{V(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right| \times $$

From Eq.(2.2.2.30) we obtain
$$v_{2,\epsilon}^*(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta) = \overline{\int_{\substack{q(T)=\frac{x}{\sqrt{\epsilon}} \\ q(0)=\frac{y}{\sqrt{\epsilon}}}} dy D^+[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \Xi_2(q(t), \epsilon, \epsilon) \times$$

$$\sin\left(-\int_{0}^{T} \sqrt{\epsilon} \tilde{V}_{1}(q_{v(\epsilon, \epsilon)}(t)) dt\right) \exp\left\{-\int_{0}^{T} dt\left[\{q(t)\}_{\eta}\right]^{2}\right\} = \\ \overline{\int_{\substack{q(T)=\frac{x}{\sqrt{\epsilon}} \\ q(0)=\frac{y}{\sqrt{\epsilon}}}} dy D^{+}[q(t), \epsilon, \epsilon] \Psi_{0}(\sqrt{\epsilon} q(0)) \Xi_{2}(q(t), \epsilon, \epsilon) \times$$

$$\exp\left\{-\int_{0}^{T} dt \left[\{q(t)\}_{\eta}\right]^{2}\right\} \left[\sum_{n=0}^{M} \frac{(-1)^{n} \epsilon^{n+1/2}}{(2n+1)!} \left(\int_{0}^{T} \tilde{V}_{1}(q_{v(\epsilon, \epsilon)}(t)) dt\right)^{2n+1} + \epsilon^{n+1} O\left(\left(\int_{0}^{T} \tilde{V}_{1}(q_{v(\epsilon, \epsilon)}(t)) dt\right)^{2n+2}\right)\right] = \tag{2.2.2.31}$$

$$\overline{\int_{\substack{q(T)=\frac{x}{\sqrt{\epsilon}} \\ q(0)=\frac{y}{\sqrt{\epsilon}}}} dy D^+[q(t), \epsilon, \epsilon] \Psi_0(y) \Xi_2(q(t), \epsilon, \epsilon) \exp\left\{-\int_{0}^{T} dt \left[\{q(t)\}_{\eta}\right]^2\right\} +$$

$$\sum_{n=1}^{M} \frac{(-1)^n \epsilon^n}{(2n)!} \left[ \overline{\int_{\substack{q(T)=\frac{x}{\sqrt{\epsilon}} \\ q(0)=\frac{y}{\sqrt{\epsilon}}}} dy D^+[q(t), \epsilon, \epsilon] \Psi_0(y) \Xi_2(q(t), \epsilon, \epsilon) \times \right. \\ \left. \exp\left\{-\int_{0}^{T} dt \left[\{q(t)\}_{\eta}\right]^2\right\} \left(\int_{0}^{T} \tilde{V}_1(q_{v(\epsilon, \epsilon)}(t)) dt\right)^{2n} \right].$$
$$v_{2,\epsilon}^*(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta) = \overline{\int_{\substack{q(T)=\frac{x}{\sqrt{\epsilon}} \\ q(0)=\frac{y}{\sqrt{\epsilon}}}} dy D^+[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \Xi_2(q(t), \epsilon, \epsilon) \times$$

$$\sin\left(-\int_{0}^{T} \sqrt{\epsilon} \tilde{V}_{1}(q_{v(\epsilon, \epsilon)}(t)) dt\right) \exp\left\{-\int_{0}^{T} dt\left[\{q(t)\}_{\eta}\right]^{2}\right\} = \\ \overline{\int_{\substack{q(T)=\frac{x}{\sqrt{\epsilon}} \\ q(0)=\frac{y}{\sqrt{\epsilon}}}} dy D^{+}[q(t), \epsilon, \epsilon] \Psi_{0}(\sqrt{\epsilon} q(0)) \Xi_{2}(q(t), \epsilon, \epsilon) \times$$

$$\exp\left\{-\int_{0}^{T} dt \left[\{q(t)\}_{\eta}\right]^{2}\right\} \left[ \sum_{n=0}^{M} \frac{(-1)^{n} \epsilon^{n+1/2}}{(2n+1)!} \left(\int_{0}^{T} \tilde{V}_{1}(q_{v(\epsilon, \epsilon)}(t)) dt\right)^{2n+1} + \epsilon^{n+1} O\left(\left(\int_{0}^{T} \tilde{V}_{1}(q_{v(\epsilon, \epsilon)}(t)) dt\right)^{2n+2}\right) \right] = \tag{2.2.2.31}$$

$$\overline{\int_{\substack{q(T)=\frac{x}{\sqrt{\epsilon}} \\ q(0)=\frac{y}{\sqrt{\epsilon}}}} dy D^+[q(t), \epsilon, \epsilon] \Psi_0(y) \Xi_2(q(t), \epsilon, \epsilon) \exp\left\{-\int_{0}^{T} dt \left[\{q(t)\}_{\eta}\right]^2\right\} +$$

$$\sum_{n=1}^{M} \frac{(-1)^n \epsilon^n}{(2n)!} \left[ \overline{\int_{\substack{q(T)=\frac{x}{\sqrt{\epsilon}} \\ q(0)=\frac{y}{\sqrt{\epsilon}}}} dy D^+[q(t), \epsilon, \epsilon] \Psi_0(y) \Xi_2(q(t), \epsilon, \epsilon) \times \exp\left\{-\int_{0}^{T} dt \left[\{q(t)\}_{\eta}\right]^2\right\} \left(\int_{0}^{T} \tilde{V}_1(q_{v(\epsilon, \epsilon)}(t)) dt\right)^{2n} \right].$$
**Lemma.2.2.2.1. (Generalized Hölder's inequality for the Feynman-Colombeau path integral)** Let $\frac{1}{p_1} + \frac{1}{p_2} = 1$ with $p_1, p_2 > 1$. Assume that

$$ \left( F_{3,\varepsilon} \left[ \frac{d\tilde{q}_{\varepsilon}(t)}{dt}, q_{\varepsilon}(t), t_i, t_f \right] \right)_{\varepsilon} > 0, $$ 
$$ \left( \left. \int_{\substack{q(t_f)=x \\ q(t_i)=y}} F_{1,\varepsilon} \left[ \frac{d\tilde{q}_{\varepsilon}(t)}{dt}, q_{\varepsilon}(t), t_i, t_f \right] \right|^{p_1} \times \right. $$ 
$$ \left. \left\{ F_{3,\varepsilon} \left[ \frac{d\tilde{q}_{\varepsilon}(t)}{dt}, q_{\varepsilon}(t), t_i, t_f \right] \right\} dy D^+[q(t)] \right)_{\varepsilon \in (0,1]} < \infty, \quad (2.2.2.32) $$

$$ \left( \left. \int_{\substack{q(t_f)=x \\ q(t_i)=y}} F_{2,\varepsilon} \left[ \frac{d\tilde{q}_{\varepsilon}(t)}{dt}, q_{\varepsilon}(t), t_i, t_f \right] \right|^{p_2} \times \right. $$ 
$$ \left. \left\{ F_{3,\varepsilon} \left[ \frac{d\tilde{q}_{\varepsilon}(t)}{dt}, q_{\varepsilon}(t), t_i, t_f \right] \right\} dy D^+[q(t)] \right)_{\varepsilon \in (0,1]} < \infty. $$

Then Hölder's inequality for Feinman-Colombeau path integral states that:
**Lemma.2.2.2.1. (Generalized Hölder's inequality for the Feynman-Colombeau path integral)** Let $\frac{1}{p_1} + \frac{1}{p_2} = 1$ with $p_1, p_2 > 1$. Assume that

$$ \left( F_{3,\varepsilon} \left[ \frac{d\tilde{q}_{\varepsilon}(t)}{dt}, q_{\varepsilon}(t), t_i, t_f \right] \right)_{\varepsilon} > 0, $$ 
$$ \left( \left. \int_{\substack{q(t_f)=x \\ q(t_i)=y}} F_{1,\varepsilon} \left[ \frac{d\tilde{q}_{\varepsilon}(t)}{dt}, q_{\varepsilon}(t), t_i, t_f \right] \right|^{p_1} \times \right. $$ 
$$ \left. \left\{ F_{3,\varepsilon} \left[ \frac{d\tilde{q}_{\varepsilon}(t)}{dt}, q_{\varepsilon}(t), t_i, t_f \right] \right\} dy D^+[q(t)] \right)_{\varepsilon \in (0,1]} < \infty, \quad (2.2.2.32) $$

$$ \left( \left. \int_{\substack{q(t_f)=x \\ q(t_i)=y}} F_{2,\varepsilon} \left[ \frac{d\tilde{q}_{\varepsilon}(t)}{dt}, q_{\varepsilon}(t), t_i, t_f \right] \right|^{p_2} \times \right. $$ 
$$ \left. \left\{ F_{3,\varepsilon} \left[ \frac{d\tilde{q}_{\varepsilon}(t)}{dt}, q_{\varepsilon}(t), t_i, t_f \right] \right\} dy D^+[q(t)] \right)_{\varepsilon \in (0,1]} < \infty. $$

Then Hölder's inequality for Feinman-Colombeau path integral states that:
$$
\left( \begin{array}{l} \overline{\displaystyle\int_{\substack{q(t_f)=x \\ q(t_i)=y}} \left| \left( F_{1,\epsilon} \left[ \frac{d\tilde{q}_\epsilon(t)}{dt}, q_\epsilon(t), t_i, t_f \right] F_{2,\epsilon} \left[ \frac{d\tilde{q}_\epsilon(t)}{dt}, q_\epsilon(t), t_i, t_f \right] \right) \right| \times} \\ \\ \left\{ \left( F_{3,\epsilon} \left[ \frac{d\tilde{q}_\epsilon(t)}{dt}, q_\epsilon(t), t_i, t_f \right] \right) \right\} dyD^+[q(t)] \right)_{\epsilon \in (0,1]} = \\ \\ \overline{\displaystyle\int_{\substack{q(t_f)=x \\ q(t_i)=y}} \left| \left( F_{1,\epsilon} \left[ \frac{d\tilde{q}_\epsilon(t)}{dt}, q_\epsilon(t), t_i, t_f \right] F_{2,\epsilon} \left[ \frac{d\tilde{q}_\epsilon(t)}{dt}, q_\epsilon(t), t_i, t_f \right] \right) \right|_{\epsilon \in (0,1]} \times} \\ \\ \left\{ \left( F_{3,\epsilon} \left[ \frac{d\tilde{q}_\epsilon(t)}{dt}, q_\epsilon(t), t_i, t_f \right] \right)_{\epsilon \in (0,1]} \right\} dyD^+[q(t)] \le \\ \\ \left( \left| \left( \overline{\displaystyle\int_{\substack{q(t_f)=x \\ q(t_i)=y}} F_{1,\epsilon} \left[ \frac{d\tilde{q}_\epsilon(t)}{dt}, q_\epsilon(t) \right] \right)^{p_1} \times \right. \right. \\ \\ \left. \left. \left\{ F_{3,\epsilon} \left[ \frac{d\tilde{q}_\epsilon(t)}{dt}, q_\epsilon(t), t_i, t_f \right] \right\} dyD^+[q(t)] \right)^{\frac{1}{p_1}} \right)_{\epsilon \in (0,1]} \times \\ \\ \left( \left| \left( \overline{\displaystyle\int_{\substack{q(t_f)=x \\ q(t_i)=y}} F_{2,\epsilon} \left[ \frac{d\tilde{q}_\epsilon(t)}{dt}, q_\epsilon(t), t_i, t_f \right] \right)^{p_2} \times \right. \right. \\ \\ \left. \left. \left\{ F_{3,\epsilon} \left[ \frac{d\tilde{q}_\epsilon(t)}{dt}, q_\epsilon(t), t_i, t_f \right] \right\} dyD^+[q(t)] \right)^{\frac{1}{p_2}} \right)_{\epsilon \in (0,1]} ,
\end{array} \right.
\quad (2.2.2.33)
$$
$$
\left( \begin{array}{l} \displaystyle \overline{\displaystyle\int_{\substack{q(t_f)=x \\ q(t_i)=y}} \left| \left( F_{1,\epsilon} \left[ \frac{d\tilde{q}_\epsilon(t)}{dt}, q_\epsilon(t), t_i, t_f \right] F_{2,\epsilon} \left[ \frac{d\tilde{q}_\epsilon(t)}{dt}, q_\epsilon(t), t_i, t_f \right] \right) \right|} \times \\[1.5ex] \displaystyle \left\{ \left( F_{3,\epsilon} \left[ \frac{d\tilde{q}_\epsilon(t)}{dt}, q_\epsilon(t), t_i, t_f \right] \right) \right\} dyD^+[q(t)] \bigg|_{\epsilon \in (0,1]} = \\[1.5ex] \displaystyle \overline{\displaystyle\int_{\substack{q(t_f)=x \\ q(t_i)=y}} \left| \left( F_{1,\epsilon} \left[ \frac{d\tilde{q}_\epsilon(t)}{dt}, q_\epsilon(t), t_i, t_f \right] F_{2,\epsilon} \left[ \frac{d\tilde{q}_\epsilon(t)}{dt}, q_\epsilon(t), t_i, t_f \right] \right) \right|}_{\epsilon \in (0,1]} \times \\[1.5ex] \displaystyle \left\{ \left( F_{3,\epsilon} \left[ \frac{d\tilde{q}_\epsilon(t)}{dt}, q_\epsilon(t), t_i, t_f \right] \right) \right\} dyD^+[q(t)] \le \\[1.5ex] \displaystyle \left( \left( \overline{\displaystyle\int_{\substack{q(t_f)=x \\ q(t_i)=y}} |F_{1,\epsilon}\left[\frac{d\tilde{q}_\epsilon(t)}{dt}, q_\epsilon(t)\right]|^{p_1} \right) \times \right. \\[1.5ex] \displaystyle \left. \left[ \left\{ F_{3,\epsilon} \left[ \frac{d\tilde{q}_\epsilon(t)}{dt}, q_\epsilon(t), t_i, t_f \right] \right\} dyD^+[q(t)] \right]^{\frac{1}{p_1}} \right) \times \\[1.5ex] \displaystyle \left( \left( \overline{\displaystyle\int_{\substack{q(t_f)=x \\ q(t_i)=y}} |F_{2,\epsilon}\left[\frac{d\tilde{q}_\epsilon(t)}{dt}, q_\epsilon(t), t_i, t_f\right]|^{p_2} \right) \times \right. \\[1.5ex] \displaystyle \left. \left\{ F_{3,\epsilon} \left[ \frac{d\tilde{q}_\epsilon(t)}{dt}, q_\epsilon(t), t_i, t_f \right] \right\} dyD^+[q(t)] \right]^{\frac{1}{p_2}} \bigg|_{\epsilon \in (0,1]} ,
\end{array} \right.
\quad (2.2.2.33)
$$
Let us evaluate path integral $\tilde{v}_{1,\epsilon}\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right)$ and $\tilde{v}_{2,\epsilon}\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right)$.

Assume now that $p_2 \gg \epsilon^{-1}$, i.e. $p_1 \approx 1$. Then from Eqs.(2.2.2.26)-(2.2.2.27) and Hölder's inequality (2.2.2.33) we obtain:
Let us evaluate path integral $\tilde{v}_{1,\epsilon}\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right)$ and $\tilde{v}_{2,\epsilon}\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right)$.

Assume now that $p_2 \gg \epsilon^{-1}$, i.e. $p_1 \approx 1$. Then from Eqs.(2.2.2.26)-(2.2.2.27) and Hölder's inequality (2.2.2.33) we obtain:
$$|\tilde{V}_{1,\epsilon}(T, x, x_0, \epsilon, \eta)| \le$$

$$\overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\mathbf{m}}^{(1)}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times}$$

$$\left| \cos\left( \int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right| \times$$

$$\left| \cos\left( \int_0^T \hat{V}_1(\sqrt{\epsilon} q_{v(\epsilon,\epsilon)}(t)) dt \right) \right| \le$$

$$\left[ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\mathbf{m}}^{(1)}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times} \right. \quad (2.2.2.34)$$

$$\left| \cos\left( \int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right|^{1/p_1} \times$$

$$\left[ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\mathbf{m}}^{(1)}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times} \right.$$

$$\left| \cos\left( \int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right| \times$$

$$\left| \cos\left( \int_0^T \hat{V}_1(\sqrt{\epsilon} q_{v(\epsilon,\epsilon)}(t)) dt \right) \right|^{p_2} \left[ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\mathbf{m}}^{(1)}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times} \right]^{1/p_2}.$$

and
$$|\tilde{V}_{1,\epsilon}(T, x, x_0, \epsilon, \eta)| \le$$

$$\overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\mathbf{m}}^{(1)}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times}$$

$$\left| \cos\left( \int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right| \times$$

$$\left| \cos\left( \int_0^T \hat{V}_1(\sqrt{\epsilon} q_{v(\epsilon,\epsilon)}(t)) dt \right) \right| \le$$

$$\left[ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\mathbf{m}}^{(1)}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times} \right. \quad (2.2.2.34)$$

$$\left| \cos\left( \int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right|^{1/p_1} \times$$

$$\left[ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\mathbf{m}}^{(1)}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times} \right.$$

$$\left| \cos\left( \int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right| \times$$

$$\left| \cos\left( \int_0^T \hat{V}_1(\sqrt{\epsilon} q_{v(\epsilon,\epsilon)}(t)) dt \right) \right|^{p_2} \quad \left[ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\mathbf{m}}^{(1)}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times} \right]^{1/p_2}.$$

and
$$|\tilde{V}_{2,\epsilon}(T, x, x_0, \epsilon, \eta)| \le$$

$$\overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\mathbf{m}}^{(2)}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times}$$

$$\left| \sin\left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right| \times$$

$$\left| \sin\left( -\int_{0}^{T} \hat{V}_{1}(\sqrt{\epsilon} q_{v(\epsilon,\epsilon)}(t)) dt \right) \right| \le$$

$$\left[ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\mathbf{m}}^{(2)}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times} \right. \quad (2.2.2.35)$$

$$\left| \sin\left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right|^{1/p_1} \times$$

$$\left[ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\mathbf{m}}^{(2)}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times} \right.$$

$$\left| \sin\left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right| \times$$

$$\left| \sin\left( -\int_{0}^{T} \hat{V}_{1}(\sqrt{\epsilon} q_{v(\epsilon,\epsilon)}(t)) dt \right) \right|^{p_2} \quad \right]^{1/p_2}.$$

Therefore we obtain
$$|\tilde{V}_{2,\epsilon}(T, x, x_0, \epsilon, \eta)| \le$$

$$\overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\mathbf{m}}^{(2)}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times}$$

$$\left| \sin\left(\int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right| \times$$

$$\left| \sin\left(-\int_{0}^{T} \hat{V}_{1}(\sqrt{\epsilon} q_{v(\epsilon,\epsilon)}(t)) dt\right) \right| \le$$

$$\left[ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\mathbf{m}}^{(2)}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times} \right. \quad (2.2.2.35)$$

$$\left| \sin\left(\int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right|^{1/p_1} \times$$

$$\left[ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\mathbf{m}}^{(2)}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times} \right.$$

$$\left| \sin\left(\int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right| \times$$

$$\left| \sin\left(-\int_{0}^{T} \hat{V}_{1}(\sqrt{\epsilon} q_{v(\epsilon,\epsilon)}(t)) dt\right) \right|^{p_2} \quad \right]^{1/p_2}.$$

Therefore we obtain
$$|\tilde{V}_{1,\epsilon}(T, x, x_0, \epsilon, \eta)| \le$$

$$ \left[ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\mathbf{m}}^{(1)}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times} \right. \\ \left. \left| \cos\left(\int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right| \right] \times \quad (2.2.2.36.a) $$

$$ \left[ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\mathbf{m}}^{(1)}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times} \right. \\ \left. \left| \cos\left(\int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right| \right]^{1/p_2} . $$

and
$$|\tilde{V}_{1,\epsilon}(T, x, x_0, \epsilon, \eta)| \le$$

$$ \left[ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\mathbf{m}}^{(1)}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times} \right. \\ \left. \left| \cos\left(\int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right| \right] \times \quad (2.2.2.36.a) $$

$$ \left[ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\mathbf{m}}^{(1)}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times} \right. \\ \left. \left| \cos\left(\int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right| \right]^{1/p_2} . $$

and
$$|\tilde{\nabla}_{2,\epsilon}(T, x, x_0, \epsilon, \eta)| \le$$

$$ \left| \sin \left( \int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{\nu(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{\nu(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right| \times \quad (2.2.2.36.b) $$

$$ \left| \sin \left( \int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{\nu(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{\nu(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right|^{1/p_2} . $$

From Eqs.(2.2.2.36.a,b) we obtain

$$ \left| \tilde{\nabla}_{1,\epsilon} \left( T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta \right) \right| \le \mathbf{J}_{1,\epsilon}(T, x, x_0, \epsilon, \eta) \times \Gamma_{1,\epsilon}(T, x, x_0, \epsilon, \eta, p_2), \quad (2.2.2.37) $$

$$ \left| \tilde{\nabla}_{2,\epsilon} \left( T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta \right) \right| \le \mathbf{J}_{2,\epsilon}(T, x, x_0, \epsilon, \eta) \times \Gamma_{2,\epsilon}(T, x, x_0, \epsilon, \eta, p_2). $$

Here
$$|\tilde{\nabla}_{2,\epsilon}(T, x, x_0, \epsilon, \eta)| \le$$

$$ \left| \sin \left( \int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{\nu(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{\nu(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right| \times \quad (2.2.2.36.b) $$

$$ \left| \sin \left( \int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{\nu(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{\nu(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right|^{1/p_2} . $$

From Eqs.(2.2.2.36.a,b) we obtain

$$ \left| \tilde{\nabla}_{1,\epsilon} \left( T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta \right) \right| \le \mathbf{J}_{1,\epsilon}(T, x, x_0, \epsilon, \eta) \times \Gamma_{1,\epsilon}(T, x, x_0, \epsilon, \eta, p_2), \quad (2.2.2.37) $$

$$ \left| \tilde{\nabla}_{2,\epsilon} \left( T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta \right) \right| \le \mathbf{J}_{2,\epsilon}(T, x, x_0, \epsilon, \eta) \times \Gamma_{2,\epsilon}(T, x, x_0, \epsilon, \eta, p_2). $$

Here
$$ \mathbf{J}_{1,\epsilon}(T, x, x_0, \epsilon, \eta) = $$

$$ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\mathbf{m}}^{(1)}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times} $$

$$ \left| \cos\left(\int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right|, \qquad (2.2.2.38.a) $$

$$ \Gamma_{1,\epsilon}(T, x, x_0, \epsilon, \eta, p_2) = $$

$$ \left[ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\mathbf{m}}^{(1)}[q(t), \epsilon\epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times} \right. $$

$$ \left| \cos\left(\int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right|^{1/p_2} . $$

and
$$ \mathbf{J}_{1,\epsilon}(T, x, x_0, \epsilon, \eta) = $$

$$ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\mathbf{m}}^{(1)}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times} $$

$$ \left| \cos\left(\int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right|, \qquad (2.2.2.38.a) $$

$$ \Gamma_{1,\epsilon}(T, x, x_0, \epsilon, \eta, p_2) = $$

$$ \left[ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\mathbf{m}}^{(1)}[q(t), \epsilon\epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times} \right. $$

$$ \left| \cos\left(\int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right|^{1/p_2} . $$

and
$$ \mathbf{J}_{2,\varepsilon}(T, x, x_0, \varepsilon, \eta) = $$

$$ \overline{\int_{q(T)=\frac{x}{\sqrt{\varepsilon}}} \tilde{D}_{\mathbf{m}}^{(2)}[q(t), \varepsilon\varepsilon]\Psi_0(\sqrt{\varepsilon}q(0)) \times} $$

$$ \left| \sin\left(\int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\varepsilon\varepsilon}(t)}{dt} \right)^2 - a_{\varepsilon}(t) q_{v(\varepsilon,\varepsilon)}^2(t) + b_{\varepsilon}(t) \frac{q_{v(\varepsilon,\varepsilon)}(t)}{\sqrt{\varepsilon}} \right] \right) \right|, \qquad (2.2.2.38.b) $$

$$ \Gamma_{2,\varepsilon}(T, x, x_0, \varepsilon, \eta, p_2) = $$

$$ \left[ \overline{\int_{q(T)=\frac{x}{\sqrt{\varepsilon}}} \tilde{D}_{\mathbf{m}}^{(2)}[q(t), \varepsilon, \varepsilon]\Psi_0(\sqrt{\varepsilon}q(0)) \times} \right. $$

$$ \left| \sin\left(\int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\varepsilon\varepsilon}(t)}{dt} \right)^2 - a_{\varepsilon}(t) q_{v(\varepsilon,\varepsilon)}^2(t) + b_{\varepsilon}(t) \frac{q_{v(\varepsilon,\varepsilon)}(t)}{\sqrt{\varepsilon}} \right] \right) \right|^{1/p_2} . $$

From Eq.(2.2.2.38.a,b) by using definition (see Eq.(2.2.2.23)) of the submeasures $\tilde{D}_{\mathbf{m}}^{(1)}[q(t), \varepsilon, \varepsilon]$, $\tilde{D}_{\mathbf{m}}^{(2)}[q(t), \varepsilon, \varepsilon]$ we obtain
$$ \mathbf{J}_{2,\varepsilon}(T, x, x_0, \varepsilon, \eta) = $$

$$ \overline{\int_{q(T)=\frac{x}{\sqrt{\varepsilon}}} \tilde{D}_{\mathbf{m}}^{(2)}[q(t), \varepsilon\varepsilon]\Psi_0(\sqrt{\varepsilon}q(0)) \times} $$

$$ \left| \sin\left(\int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\varepsilon\varepsilon}(t)}{dt} \right)^2 - a_{\varepsilon}(t) q_{v(\varepsilon,\varepsilon)}^2(t) + b_{\varepsilon}(t) \frac{q_{v(\varepsilon,\varepsilon)}(t)}{\sqrt{\varepsilon}} \right] \right) \right|, \qquad (2.2.2.38.b) $$

$$ \Gamma_{2,\varepsilon}(T, x, x_0, \varepsilon, \eta, p_2) = $$

$$ \left[ \overline{\int_{q(T)=\frac{x}{\sqrt{\varepsilon}}} \tilde{D}_{\mathbf{m}}^{(2)}[q(t), \varepsilon, \varepsilon]\Psi_0(\sqrt{\varepsilon}q(0)) \times} \right. $$

$$ \left| \sin\left(\int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\varepsilon\varepsilon}(t)}{dt} \right)^2 - a_{\varepsilon}(t) q_{v(\varepsilon,\varepsilon)}^2(t) + b_{\varepsilon}(t) \frac{q_{v(\varepsilon,\varepsilon)}(t)}{\sqrt{\varepsilon}} \right] \right) \right|^{1/p_2} . $$

From Eq.(2.2.2.38.a,b) by using definition (see Eq.(2.2.2.23)) of the submeasures $\tilde{D}_{\mathbf{m}}^{(1)}[q(t), \varepsilon, \varepsilon]$, $\tilde{D}_{\mathbf{m}}^{(2)}[q(t), \varepsilon, \varepsilon]$ we obtain
$$ \mathbf{J}_{1,\epsilon}(T, x, x_0, \epsilon, \eta) \le $$

$$ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\text{sm}}^{(1)}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times} $$

$$ \left| \cos\left( \int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{V(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{V(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right| + \qquad (2.2.2.39.a) $$

$$ + \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times} $$

$$ \left| \cos\left( \int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{V(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{V(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right| = $$

$$ \Theta_{1,\epsilon}(T, x, x_0, \epsilon, \eta) + \Lambda_{1,\epsilon}(T, x, x_0, \epsilon, \eta) $$

and
$$ \mathbf{J}_{1,\varepsilon}(T, x, x_0, \varepsilon, \eta) \le $$

$$ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\text{sm}}^{(1)}[q(t), \varepsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times} $$

$$ \left| \cos\left(\int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\varepsilon\varepsilon}(t)}{dt} \right)^2 - a_{\varepsilon}(t) q_{v(\varepsilon,\varepsilon)}^2(t) + b_{\varepsilon}(t) \frac{q_{v(\varepsilon,\varepsilon)}(t)}{\sqrt{\varepsilon}} \right] \right) + \quad (2.2.2.39.a) $$

$$ + \int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[q(t), \varepsilon, \varepsilon] \Psi_{0}(\sqrt{\varepsilon} q(0)) \times $$

$$ \left| \cos\left(\int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\varepsilon\varepsilon}(t)}{dt} \right)^2 - a_{\varepsilon}(t) q_{v(\varepsilon,\varepsilon)}^2(t) + b_{\varepsilon}(t) \frac{q_{v(\varepsilon,\varepsilon)}(t)}{\sqrt{\varepsilon}} \right] \right) = $$

$$ \Theta_{1,\varepsilon}(T, x, x_0, \varepsilon, \eta) + \Lambda_{1,\varepsilon}(T, x, x_0, \varepsilon, \eta) $$

and
$$ \mathbf{J}_{2,\varepsilon}(T, x, x_0, \varepsilon, \eta) \le $$

$$ \overline{\int_{q(T)=\frac{x}{\sqrt{\varepsilon}}} \tilde{D}_{\text{sm}}^{(2)}[q(t), \varepsilon, \varepsilon] \Psi_0(\sqrt{\varepsilon} q(0)) \times} $$

$$ \left| \sin\left(\int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\varepsilon\varepsilon}(t)}{dt} \right)^2 - a_{\varepsilon}(t) q_{v(\varepsilon,\varepsilon)}^2(t) + b_{\varepsilon}(t) \frac{q_{v(\varepsilon,\varepsilon)}(t)}{\sqrt{\varepsilon}} \right] \right) \right| + \quad (2.2.2.39.b) $$

$$ + \overline{\int_{q(T)=\frac{x}{\sqrt{\varepsilon}}} \tilde{D}_{\eta}^{+}[q(t), \varepsilon, \varepsilon] \Psi_{0}(\sqrt{\varepsilon} q(0)) \times} $$

$$ \left| \sin\left(\int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\varepsilon\varepsilon}(t)}{dt} \right)^2 - a_{\varepsilon}(t) q_{v(\varepsilon,\varepsilon)}^2(t) + b_{\varepsilon}(t) \frac{q_{v(\varepsilon,\varepsilon)}(t)}{\sqrt{\varepsilon}} \right] \right) \right| \times $$

$$ \Theta_{2,\varepsilon}(T, x, x_0, \varepsilon, \eta) + \Lambda_{2,\varepsilon}(T, x, x_0, \varepsilon, \eta). $$

Here
$$ \mathbf{J}_{2,\varepsilon}(T, x, x_0, \varepsilon, \eta) \le $$

$$ \overline{\int_{q(T)=\frac{x}{\sqrt{\varepsilon}}} \tilde{D}_{\text{sm}}^{(2)}[q(t), \varepsilon, \varepsilon] \Psi_0(\sqrt{\varepsilon} q(0)) \times} $$

$$ \left| \sin\left(\int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\varepsilon\varepsilon}(t)}{dt} \right)^2 - a_{\varepsilon}(t) q_{V(\varepsilon,\varepsilon)}^2(t) + b_{\varepsilon}(t) \frac{q_{V(\varepsilon,\varepsilon)}(t)}{\sqrt{\varepsilon}} \right] \right) + \quad (2.2.2.39.b) $$

$$ + \overline{\int_{q(T)=\frac{x}{\sqrt{\varepsilon}}} \tilde{D}_{\eta}^{+}[q(t), \varepsilon, \varepsilon] \Psi_{0}(\sqrt{\varepsilon} q(0)) \times} $$

$$ \left| \sin\left(\int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\varepsilon\varepsilon}(t)}{dt} \right)^2 - a_{\varepsilon}(t) q_{V(\varepsilon,\varepsilon)}^2(t) + b_{\varepsilon}(t) \frac{q_{V(\varepsilon,\varepsilon)}(t)}{\sqrt{\varepsilon}} \right] \right) \times $$

$$ \Theta_{2,\varepsilon}(T, x, x_0, \varepsilon, \eta) + \Lambda_{2,\varepsilon}(T, x, x_0, \varepsilon, \eta). $$

Here
$$
\Theta_{1,\epsilon}(T, x, x_0, \epsilon, \eta) = \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\text{sm}}^{(1)}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times}
$$

$$
\left| \cos\left( \int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{V(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{V(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right| = \tag{2.2.2.40.a}
$$

$$
\overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[q(t), \epsilon, \epsilon] \Psi_{0}(\sqrt{\epsilon} q(0)) \times}
$$

$$
\cos\left(\int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{V(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{V(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right),
$$

$$
\Theta_{2,\epsilon}(T, x, x_0, \epsilon, \eta) = \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\text{sm}}^{(2)}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times}
$$

$$
\left| \sin\left( \int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt}\right)^2 - a_\epsilon(t) q_{V(\epsilon,\epsilon)}^2(t) + b_\epsilon(t) \frac{q_{V(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right| = \tag{2.2.2.40.b}
$$

$$
\overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[q(t), \epsilon, \epsilon] \Psi_{0}(\sqrt{\epsilon} q(0)) \times}
$$

$$
\sin\left(\int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{V(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{V(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right),
$$

and
$$
\Theta_{1,\epsilon}(T, x, x_0, \epsilon, \eta) = \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\text{sm}}^{(1)}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times}
$$

$$
\left| \cos\left( \int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{V(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{V(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right| = \tag{2.2.2.40.a}
$$

$$
\overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[q(t), \epsilon, \epsilon] \Psi_{0}(\sqrt{\epsilon} q(0)) \times}
$$

$$
\cos\left(\int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{V(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{V(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right),
$$

$$
\Theta_{2,\epsilon}(T, x, x_0, \epsilon, \eta) = \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\text{sm}}^{(2)}[q(t), \epsilon, \epsilon] \Psi_0(\sqrt{\epsilon} q(0)) \times}
$$

$$
\left| \sin\left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{V(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{V(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right| = \tag{2.2.2.40.b}
$$

$$
\overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[q(t), \epsilon, \epsilon] \Psi_{0}(\sqrt{\epsilon} q(0)) \times}
$$

$$
\sin\left(\int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{V(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{V(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right),
$$

and
$$
\Lambda_{1,\epsilon}(T,x,x_0,\epsilon,\eta) = \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[q(t),\epsilon\epsilon]\Psi_0(\sqrt{\epsilon}q(0)) \times} \tag{2.2.2.41.a} \\
\left| \cos\left(\int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t)q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right|,
$$

$$
\Lambda_{2,\epsilon}(T,x,x_0,\epsilon,\eta) = \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[q(t),\epsilon,\epsilon]\Psi_0(\sqrt{\epsilon}q(0)) \times} \tag{2.2.2.41.b} \\
\left| \sin\left(\int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t)q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right|.
$$

From (2.2.2.37.a,b) and (2.2.2.39.a,b) we obtain the inequalities:

$$
\begin{align}
\left| \tilde{v}_{1,\epsilon} \left( T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta \right) \right| &\le \mathbf{J}_{1,\epsilon} (T, x, x_0, \epsilon, \eta) \le \nonumber \\
&\qquad \Theta_{1,\epsilon} (T, x, x_0, \epsilon, \eta) + \Lambda_{1,\epsilon} (T, x, x_0, \epsilon, \eta), \tag{2.2.2.42} \\
\left| \tilde{v}_{2,\epsilon} \left( T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta \right) \right| &\le \mathbf{J}_{2,\epsilon} (T, x, x_0, \epsilon, \eta) \le \nonumber \\
&\qquad \Theta_{2,\epsilon} (T, x, x_0, \epsilon, \eta) + \Lambda_{2,\epsilon} (T, x, x_0, \epsilon, \eta). \nonumber
\end{align}
$$

Hence
$$
\Lambda_{1,\epsilon}(T,x,x_0,\epsilon,\eta) = \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[q(t),\epsilon\epsilon]\Psi_0(\sqrt{\epsilon}q(0)) \times} \tag{2.2.2.41.a} \\
\left| \cos\left(\int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t)q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right|,
$$

$$
\Lambda_{2,\epsilon}(T,x,x_0,\epsilon,\eta) = \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[q(t),\epsilon,\epsilon]\Psi_0(\sqrt{\epsilon}q(0)) \times} \tag{2.2.2.41.b} \\
\left| \sin\left(\int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t)q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \right|.
$$

From (2.2.2.37.a,b) and (2.2.2.39.a,b) we obtain the inequalities:

$$
\begin{align}
\left| \tilde{v}_{1,\epsilon} \left( T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta \right) \right| &\le \mathbf{J}_{1,\epsilon} (T, x, x_0, \epsilon, \eta) \le \nonumber \\
&\qquad \Theta_{1,\epsilon} (T, x, x_0, \epsilon, \eta) + \Lambda_{1,\epsilon} (T, x, x_0, \epsilon, \eta), \tag{2.2.2.42} \\
\left| \tilde{v}_{2,\epsilon} \left( T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta \right) \right| &\le \mathbf{J}_{2,\epsilon} (T, x, x_0, \epsilon, \eta) \le \nonumber \\
&\qquad \Theta_{2,\epsilon} (T, x, x_0, \epsilon, \eta) + \Lambda_{2,\epsilon} (T, x, x_0, \epsilon, \eta). \nonumber
\end{align}
$$

Hence
$$
\begin{gather*}
\nu_{1,\epsilon}\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) = \tilde{\nu}_{1,\epsilon}\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) - \nu_{1,\epsilon}^*\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) \le \\
\mathbf{J}_{1,\epsilon}(T, x, x_0, \epsilon, \eta) - \nu_{1,\epsilon}^*\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) \le \tag{2.2.2.43.a}
\end{gather*}
$$

$$
\Theta_{1,\epsilon}(T,x,x_0,\epsilon,\eta) + \Lambda_{1,\epsilon}(T,x,x_0,\epsilon,\eta) - \nu_{1,\epsilon}^*(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta)
$$

and

$$
\begin{gather*}
\nu_{2,\epsilon}\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) = \tilde{\nu}_{2,\epsilon}\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) - \nu_{2,\epsilon}^*\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) \le \\
\mathbf{J}_{2,\epsilon}(T, x, x_0, \epsilon, \eta) - \nu_{2,\epsilon}^*\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) \le \tag{2.2.2.43.b}
\end{gather*}
$$

$$
\Theta_{2,\epsilon}(T,x,x_0,\epsilon,\eta) + \Lambda_{2,\epsilon}(T,x,x_0,\epsilon,\eta) - \nu_{2,\epsilon}^*(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta).
$$

Substitution Eqs.(2.2.2.30,31) and Eqs.(2.2.2.41.a,b) into Eqs.(2.2.2.43.a,b) gives
$$v_{1,\epsilon}\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) = \tilde{v}_{1,\epsilon}\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) - v_{1,\epsilon}^*\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) \le \\ \mathbf{J}_{1,\epsilon}(T, x, x_0, \epsilon, \eta) - v_{1,\epsilon}^*\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) \le \qquad (2.2.2.43.a)$$

$$\Theta_{1,\epsilon}(T, x, x_0, \epsilon, \eta) + \Lambda_{1,\epsilon}(T, x, x_0, \epsilon, \eta) - v_{1,\epsilon}^*\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right)$$

and

$$v_{2,\epsilon}\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) = \tilde{v}_{2,\epsilon}\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) - v_{2,\epsilon}^*\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) \le \\ \mathbf{J}_{2,\epsilon}(T, x, x_0, \epsilon, \eta) - v_{2,\epsilon}^*\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) \le \qquad (2.2.2.43.b)$$

$$\Theta_{2,\epsilon}(T, x, x_0, \epsilon, \eta) + \Lambda_{2,\epsilon}(T, x, x_0, \epsilon, \eta) - v_{2,\epsilon}^*\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right).$$

Substitution Eqs.(2.2.2.30,31) and Eqs.(2.2.2.41.a,b) into Eqs.(2.2.2.43.a,b) gives
$$v_{1,\epsilon}\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) \le$$

$$\Theta_{1,\epsilon}(T,x,x_0,\epsilon,\eta) + \Lambda_{1,\epsilon}(T,x,x_0,\epsilon,\eta) -$$

$$\left\{ \begin{array}{l} \displaystyle \int_{\substack{q(T)=\frac{x}{\sqrt{\epsilon}} \\ q(0)=\frac{y}{\sqrt{\epsilon}}}} dy D^+[q(t), \epsilon, \epsilon] \Psi_0(y) \Xi_1(q(t), \epsilon, \epsilon) \exp \left\{ -\int_0^T dt \left[ \{q(t)\}_\eta \right]^2 \right\} + \\ \\ \displaystyle \sum_{n=1}^M \frac{(-1)^n \epsilon^n}{(2n)!} \left[ \begin{array}{l} \displaystyle \int_{\substack{q(T)=\frac{x}{\sqrt{\epsilon}} \\ q(0)=\frac{y}{\sqrt{\epsilon}}}} dy D^+[q(t), \epsilon, \epsilon] \Psi_0(y) \Xi_1(q(t), \epsilon, \epsilon) \times \\ \\ \exp \left\{ -\int_0^T dt \left[ \{q(t)\}_\eta \right]^2 \right\} \left( \int_0^T \tilde{V}_1(q_{v(\epsilon,\epsilon)}(t)) dt \right)^{2n} \end{array} \right] = \end{array} \right.$$

(2.2.2.44.a)

$$\Theta_{1,\epsilon}(T,x,x_0,\epsilon,\eta) + \sum_{n=1}^{M} \frac{(-1)^n \epsilon^n}{(2n)!} \left[ \begin{array}{l} \displaystyle \int_{\substack{q(T)=\frac{x}{\sqrt{\epsilon}} \\ q(0)=\frac{y}{\sqrt{\epsilon}}}} dy D^+[q(t),\epsilon,\epsilon] \Psi_0(y) \Xi_1(q(t),\epsilon,\epsilon) \times \\ \\ \exp \left\{ -\int_0^T dt \left[ \{q(t)\}_\eta \right]^2 \right\} \left( \int_0^T \tilde{V}_1(q_{v(\epsilon,\epsilon)}(t)) dt \right)^{2n} \end{array} \right]$$

and
$$v_{1,\epsilon}\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) \le$$

$$\Theta_{1,\epsilon}(T,x,x_0,\epsilon,\eta) + \Lambda_{1,\epsilon}(T,x,x_0,\epsilon,\eta) -$$

$$\left\{ \begin{array}{l} \displaystyle \int_{\substack{q(T)=\frac{x}{\sqrt{\epsilon}} \\ q(0)=\frac{y}{\sqrt{\epsilon}}}} dy D^+[q(t), \epsilon, \epsilon] \Psi_0(y) \Xi_1(q(t), \epsilon, \epsilon) \exp \left\{ -\int_0^T dt \left[ \{q(t)\}_\eta \right]^2 \right\} + \\ \\ \displaystyle \sum_{n=1}^M \frac{(-1)^n \epsilon^n}{(2n)!} \left[ \int_{\substack{q(T)=\frac{x}{\sqrt{\epsilon}} \\ q(0)=\frac{y}{\sqrt{\epsilon}}}} dy D^+[q(t), \epsilon, \epsilon] \Psi_0(y) \Xi_1(q(t), \epsilon, \epsilon) \times \right. \\ \\ \left. \exp \left\{ -\int_0^T dt \left[ \{q(t)\}_\eta \right]^2 \right\} \left( \int_0^T \tilde{V}_1(q_{v(\epsilon,\epsilon)}(t)) dt \right)^{2n} \right] \right\} = \end{array} \right.$$

(2.2.2.44.a)

$$\Theta_{1,\epsilon}(T,x,x_0,\epsilon,\eta) + \sum_{n=1}^{M} \frac{(-1)^n \epsilon^n}{(2n)!} \left[ \int_{\substack{q(T)=\frac{x}{\sqrt{\epsilon}} \\ q(0)=\frac{y}{\sqrt{\epsilon}}}} dy D^+[q(t),\epsilon,\epsilon] \Psi_0(y) \Xi_1(q(t),\epsilon,\epsilon) \times \right. \\ \left. \exp\left\{ -\int_0^T dt \left[ \{q(t)\}_\eta \right]^2 \right\} \left( \int_0^T \tilde{V}_1(q_{v(\epsilon,\epsilon)}(t)) dt \right)^{2n} \right] \right]$$

and
$$v_{2,\epsilon}\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) \le$$

$$\Theta_{2,\epsilon}(T, x, x_0, \epsilon, \eta) + \Lambda_{2,\epsilon}(T, x, x_0, \epsilon, \eta) -$$

$$\left\{ \begin{array}{l} \displaystyle \int_{\substack{q(T)=\frac{x}{\sqrt{\epsilon}} \\ q(0)=\frac{y}{\sqrt{\epsilon}}}} dy D^+[q(t), \epsilon, \epsilon] \Psi_0(y) \Xi_2(q(t), \epsilon, \epsilon) \exp \left\{ -\int_0^T dt \left[ \{q(t)\}_\eta \right]^2 \right\} + \\ \\ \displaystyle \sum_{n=1}^M \frac{(-1)^n \epsilon^{n+1/2}}{(2n)!} \left[ \begin{array}{l} \displaystyle \int_{\substack{q(T)=\frac{x}{\sqrt{\epsilon}} \\ q(0)=\frac{y}{\sqrt{\epsilon}}}} dy D^+[q(t), \epsilon, \epsilon] \Psi_0(y) \Xi_2(q(t), \epsilon, \epsilon) \times \\ \\ \displaystyle \exp \left\{ -\int_0^T dt \left[ \{q(t)\}_\eta \right]^2 \right\} \left( \int_0^T \tilde{V}_1(q_{v(\epsilon,\epsilon)}(t)) dt \right)^{2n} \end{array} \right\} = \end{array} \right.$$

(2.2.2.44.b)

$$\Theta_{2,\epsilon}(T, x, x_0, \epsilon, \eta) + \sum_{n=1}^{M} \frac{(-1)^n \epsilon^n}{(2n+1)!} \left[ \begin{array}{l} \displaystyle \int_{\substack{q(T)=\frac{x}{\sqrt{\epsilon}} \\ q(0)=\frac{y}{\sqrt{\epsilon}}}} dy D^+[q(t), \epsilon, \epsilon] \Psi_0(y) \Xi_2(q(t), \epsilon, \epsilon) \times \\ \\ \displaystyle \exp \left\{ -\int_0^T dt \left[ \{q(t)\}_\eta \right]^2 \right\} \left( -\int_0^T \tilde{V}_1(q_{v(\epsilon,\epsilon)}(t)) dt \right)^{2n+1} \end{array} \right].$$

Therefore
$$v_{2,\epsilon}\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) \le$$

$$\Theta_{2,\epsilon}(T, x, x_0, \epsilon, \eta) + \Lambda_{2,\epsilon}(T, x, x_0, \epsilon, \eta) -$$

$$\left\{ \begin{array}{l} \displaystyle \int_{\substack{q(T)=\frac{x}{\sqrt{\epsilon}} \\ q(0)=\frac{y}{\sqrt{\epsilon}}}} dy D^+[q(t), \epsilon, \epsilon] \Psi_0(y) \Xi_2(q(t), \epsilon, \epsilon) \exp \left\{ -\int_0^T dt \left[ \{q(t)\}_\eta \right]^2 \right\} + \\ \\ \displaystyle \sum_{n=1}^M \frac{(-1)^n \epsilon^{n+1/2}}{(2n)!} \left[ \begin{array}{l} \displaystyle \int_{\substack{q(T)=\frac{x}{\sqrt{\epsilon}} \\ q(0)=\frac{y}{\sqrt{\epsilon}}}} dy D^+[q(t), \epsilon, \epsilon] \Psi_0(y) \Xi_2(q(t), \epsilon, \epsilon) \times \\ \\ \displaystyle \exp \left\{ -\int_0^T dt \left[ \{q(t)\}_\eta \right]^2 \right\} \left( \int_0^T \tilde{V}_1(q_{v(\epsilon,\epsilon)}(t)) dt \right)^{2n} \end{array} \right\} = \end{array} \right.$$

(2.2.2.44.b)

$$\Theta_{2,\epsilon}(T, x, x_0, \epsilon, \eta) + \sum_{n=1}^{M} \frac{(-1)^n \epsilon^n}{(2n+1)!} \left[ \begin{array}{l} \displaystyle \int_{\substack{q(T)=\frac{x}{\sqrt{\epsilon}} \\ q(0)=\frac{y}{\sqrt{\epsilon}}}} dy D^+[q(t), \epsilon, \epsilon] \Psi_0(y) \Xi_2(q(t), \epsilon, \epsilon) \times \\ \\ \displaystyle \exp \left\{ -\int_0^T dt \left[ \{q(t)\}_\eta \right]^2 \right\} \left( -\int_0^T \tilde{V}_1(q_{v(\epsilon,\epsilon)}(t)) dt \right)^{2n+1} \end{array} \right].$$

Therefore
$$v_{1,\epsilon}\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) \le \Theta_{1,\epsilon}(T, x, x_0, \epsilon, \eta) +$$

$$\sum_{n=1}^{M} \frac{(-1)^n \epsilon^n}{(2n)!} \left[ \begin{array}{l} \displaystyle \int_{q(T)=\frac{x}{\sqrt{\epsilon}}} dy D^+[q(t), \epsilon, \epsilon] \Psi_0(y) \Xi_1(q(t), \epsilon, \epsilon) \times \\ \displaystyle \int_{q(0)=\frac{y}{\sqrt{\epsilon}}} dy D^+[q(t), \epsilon, \epsilon] \Psi_0(y) \Xi_2(q(t), \epsilon, \epsilon) \times \end{array} \right. \quad (2.2.2.45.a)$$

$$\exp\left\{-\int_{0}^{T} dt \left[\{q(t)\}_{\eta}\right]^{2}\right\} \left(\int_{0}^{T} \tilde{V}_{1}(q_{v(\epsilon, \epsilon)}(t)) dt\right)^{2n}].$$

and

$$v_{2,\epsilon}\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) \le \Theta_{2,\epsilon}(T, x, x_0, \epsilon, \eta) +$$

$$\sum_{n=1}^{M} \frac{(-1)^n \epsilon^{n+1/2}}{(2n+1)!} \left[ \begin{array}{l} \displaystyle \int_{q(T)=\frac{x}{\sqrt{\epsilon}}} dy D^+[q(t), \epsilon, \epsilon] \Psi_0(y) \Xi_2(q(t), \epsilon, \epsilon) \times \\ \displaystyle \int_{q(0)=\frac{y}{\sqrt{\epsilon}}} dy D^+[q(t), \epsilon, \epsilon] \Psi_0(y) \Xi_1(q(t), \epsilon, \epsilon) \times \end{array} \right] \quad (2.2.2.45.b)$$

$$\exp\left\{-\int_{0}^{T} dt \left[\{q(t)\}_{\eta}\right]^{2}\right\} \left(-\int_{0}^{T} \tilde{V}_{1}(q_{v(\epsilon, \epsilon)}(t)) dt\right)^{2n+1}].$$

Let us calculate now oscillatory path integral $\Theta_{1,\epsilon}(T, x, x_0, \epsilon, \eta)$
$$v_{1,\epsilon}\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) \le \Theta_{1,\epsilon}(T, x, x_0, \epsilon, \eta) +$$

$$\sum_{n=1}^{M} \frac{(-1)^n \epsilon^n}{(2n)!} \left[ \begin{array}{l} \displaystyle \int_{q(T)=\frac{x}{\sqrt{\epsilon}}} dy D^+[q(t), \epsilon, \epsilon] \Psi_0(y) \Xi_1(q(t), \epsilon, \epsilon) \times \\ \displaystyle \int_{q(0)=\frac{y}{\sqrt{\epsilon}}} dy D^+[q(t), \epsilon, \epsilon] \Psi_0(y) \Xi_2(q(t), \epsilon, \epsilon) \times \end{array} \right. \quad (2.2.2.45.a)$$

$$\exp\left\{-\int_{0}^{T} dt \left[\{q(t)\}_{\eta}\right]^{2}\right\} \left(\int_{0}^{T} \tilde{V}_{1}(q_{v(\epsilon, \epsilon)}(t)) dt\right)^{2n}.$$

and

$$v_{2,\epsilon}\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) \le \Theta_{2,\epsilon}(T, x, x_0, \epsilon, \eta) +$$

$$\sum_{n=1}^{M} \frac{(-1)^n \epsilon^{n+1/2}}{(2n+1)!} \left[ \begin{array}{l} \displaystyle \int_{q(T)=\frac{x}{\sqrt{\epsilon}}} dy D^+[q(t), \epsilon, \epsilon] \Psi_0(y) \Xi_2(q(t), \epsilon, \epsilon) \times \\ \displaystyle \int_{q(0)=\frac{y}{\sqrt{\epsilon}}} dy D^+[q(t), \epsilon, \epsilon] \Psi_0(y) \Xi_1(q(t), \epsilon, \epsilon) \times \end{array} \right] \quad (2.2.2.45.b)$$

$$\exp\left\{-\int_{0}^{T} dt \left[\{q(t)\}_{\eta}\right]^{2}\right\} \left(-\int_{0}^{T} \tilde{V}_{1}(q_{v(\epsilon, \epsilon)}(t)) dt\right)^{2n+1}.$$

Let us calculate now oscillatory path integral $\Theta_{1,\epsilon}(T, x, x_0, \epsilon, \eta)$
$$ \Theta_{1,\epsilon}(T, x, x_0, \epsilon, \eta) = $$

$$ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[q(t), \epsilon\epsilon] \Psi_{0}(\sqrt{\epsilon}q(0)) \times} \qquad (2.2.2.46) $$

$$ \cos\left(\int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right). $$

and oscillatory path integral $\Theta_{2,\epsilon}(T, x, x_0, \epsilon, \eta)$

$$ \Theta_{2,\epsilon}(T, x, x_0, \epsilon, \eta) = $$

$$ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[q(t), \epsilon\epsilon] \Psi_{0}(\sqrt{\epsilon}q(0)) \times} \qquad (2.2.2.47) $$

$$ \sin\left[\left(\int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right)\right]. $$

From Eqs.(2.2.2.46,47) we obtain
$$ \Theta_{1,\epsilon}(T, x, x_0, \epsilon, \eta) = $$

$$ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[q(t), \epsilon\epsilon] \Psi_{0}(\sqrt{\epsilon}q(0)) \times} \qquad (2.2.2.46) $$

$$ \cos\left(\int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right). $$

and oscillatory path integral $\Theta_{2,\epsilon}(T, x, x_0, \epsilon, \eta)$

$$ \Theta_{2,\epsilon}(T, x, x_0, \epsilon, \eta) = $$

$$ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[q(t), \epsilon\epsilon] \Psi_{0}(\sqrt{\epsilon}q(0)) \times} \qquad (2.2.2.47) $$

$$ \sin\left[\left(\int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t) \frac{q_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right)\right]. $$

From Eqs.(2.2.2.46,47) we obtain
$$ \Theta_{1,\epsilon}(T, x, x_0, \epsilon, \eta) = $$

$$ \frac{1}{2} \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[q(t), \epsilon, \epsilon]} \Psi_{0}(\sqrt{\epsilon} q(0)) \times \quad (2.2.2.48.a) $$

$$ \left\{ \left[ \exp[i\tilde{S}_0(T, x, x_0, q(t), \epsilon, \epsilon)] + \exp[-i\tilde{S}_0(T, x, x_0, q(t), \epsilon, \epsilon)] \right] \right\} = \\ \frac{1}{2}\tilde{\Theta}_{1,\epsilon}(T, x, x_0, \epsilon, \eta) + \frac{1}{2}\tilde{\Theta}_{2,\epsilon}(T, x, x_0, \epsilon, \eta) $$

and

$$ \Theta_{2,\epsilon}(T, x, x_0, \epsilon, \eta) = $$

$$ \frac{1}{2i} \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[q(t), \epsilon, \epsilon]} \Psi_{0}(\sqrt{\epsilon} q(0)) \times \quad (2.2.2.48.b) $$

$$ \left\{ \left[ \exp[i\tilde{S}_0(T, x, x_0, q(t), \epsilon, \epsilon)] - \exp[-i\tilde{S}_0(T, x, x_0, q(t), \epsilon, \epsilon)] \right] \right\} = \\ \frac{1}{2i}\Theta_{1,\epsilon}(T, x, x_0, \epsilon, \eta) - \frac{1}{2i}\tilde{\Theta}_{2,\epsilon}(T, x, x_0, \epsilon, \eta). $$

Here

$$ \tilde{S}_0(T, x, x_0, q_\epsilon(t), \epsilon, \epsilon) = \int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{d\tau} \right)^2 - a_\epsilon(t) q_{\epsilon}^2(t) + b_\epsilon(t) \frac{q_\epsilon(t)}{\sqrt{\epsilon}} \right] \quad (2.2.2.49) $$

and
$$ \Theta_{1,\epsilon}(T, x, x_0, \epsilon, \eta) = $$

$$ \frac{1}{2} \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[q(t), \epsilon, \epsilon]} \Psi_{0}(\sqrt{\epsilon} q(0)) \times \qquad (2.2.2.48.a) $$

$$ \left\{ \left[ \exp[i\tilde{S}_0(T, x, x_0, q(t), \epsilon, \epsilon)] + \exp[-i\tilde{S}_0(T, x, x_0, q(t), \epsilon, \epsilon)] \right] \right\} = \\ \frac{1}{2}\tilde{\Theta}_{1,\epsilon}(T, x, x_0, \epsilon, \eta) + \frac{1}{2}\tilde{\Theta}_{2,\epsilon}(T, x, x_0, \epsilon, \eta) $$

and

$$ \Theta_{2,\epsilon}(T, x, x_0, \epsilon, \eta) = $$

$$ \frac{1}{2i} \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[q(t), \epsilon, \epsilon]} \Psi_{0}(\sqrt{\epsilon} q(0)) \times \qquad (2.2.2.48.b) $$

$$ \left\{ \left[ \exp[i\tilde{S}_0(T, x, x_0, q(t), \epsilon, \epsilon)] - \exp[-i\tilde{S}_0(T, x, x_0, q(t), \epsilon, \epsilon)] \right] \right\} = \\ \frac{1}{2i}\Theta_{1,\epsilon}(T, x, x_0, \epsilon, \eta) - \frac{1}{2i}\tilde{\Theta}_{2,\epsilon}(T, x, x_0, \epsilon, \eta). $$

Here

$$ \tilde{S}_0(T, x, x_0, q_\epsilon(t), \epsilon, \epsilon) = \int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon\epsilon}(t)}{d\tau} \right)^2 - a_\epsilon(t) q_{\epsilon}^2(t) + b_\epsilon(t) \frac{q_\epsilon(t)}{\sqrt{\epsilon}} \right] \quad (2.2.2.49) $$

and
$$
\tilde{\Theta}_{1,\epsilon}(T, x, x_0, \epsilon, \eta) =
$$

$$
\overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[q(t), \epsilon\epsilon] \Psi_{0}(\sqrt{\epsilon}q(0)) \exp[i\tilde{S}_{0}(T, x, x_{0}, q(t), \epsilon, \epsilon)]}, \qquad (2.2.2.50)
$$

$$
\tilde{\Theta}_{2,\epsilon}(T,x,x_0,\epsilon,\eta) =
$$

$$
\overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[q(t), \epsilon\epsilon] \Psi_{0}(\sqrt{\epsilon}q(0)) \exp[-i\tilde{S}_{0}(T, x, x_{0}, q(t), \epsilon, \epsilon)]}.
$$

Let's calculate now path integral $\tilde{\Theta}_{1,\epsilon}(T,x,x_0,\epsilon)$ by using stationary phase method.

By using replacement $q(t) := \frac{q_{\epsilon}(t)}{\sqrt{\epsilon}}$ we rewrite (2.2.2.50) in the next equivalent form

$$
\tilde{\Theta}_{1,\epsilon}(T,x,x_0,\epsilon,\eta) = \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[q(t),\epsilon] \Psi_{0}(\sqrt{\epsilon}q(0)) \times} \qquad (2.2.2.51)
$$

$$
\frac{1}{\epsilon} \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon}(t)}{d\tau} \right)^{2} - a_{\epsilon}(t) q_{\epsilon}^{2}(t) + b_{\epsilon}(t) q_{\epsilon}(t) \right].
$$

We assume now that

$$
\Psi(x, 0, x_0) = \frac{1}{\sqrt{2\pi\epsilon/\delta}} \exp\left(-\frac{x^2}{\epsilon/\delta}\right), \qquad (2.2.2.52)
$$

$0 < \epsilon \ll \delta \ll 1.$

Substitution Eq.(2.2.2.52) into Eq.(2.2.2.51) gives
$$
\tilde{\Theta}_{1,\epsilon}(T, x, x_0, \epsilon, \eta) =
$$

$$
\overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[q(t), \epsilon\epsilon] \Psi_{0}(\sqrt{\epsilon}q(0)) \exp[i\tilde{S}_{0}(T, x, x_{0}, q(t), \epsilon, \epsilon)]}, \quad (2.2.2.50)
$$

$$
\tilde{\Theta}_{2,\epsilon}(T, x, x_0, \epsilon, \eta) =
$$

$$
\overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[q(t), \epsilon\epsilon] \Psi_{0}(\sqrt{\epsilon}q(0)) \exp[-i\tilde{S}_{0}(T, x, x_{0}, q(t), \epsilon, \epsilon)]}.
$$

Let's calculate now path integral $\tilde{\Theta}_{1,\epsilon}(T,x,x_0,\epsilon)$ by using stationary phase method.

By using replacement $q(t) := \frac{q_{\epsilon}(t)}{\sqrt{\epsilon}}$ we rewrite (2.2.2.50) in the next equivalent form

$$
\tilde{\Theta}_{1,\epsilon}(T,x,x_0,\epsilon,\eta) = \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[q(t),\epsilon] \Psi_{0}(\sqrt{\epsilon}q(0)) \times} \tag{2.2.2.51}
$$

$$
\frac{1}{\epsilon} \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon}(t)}{d\tau} \right)^2 - a_{\epsilon}(t) q_{\epsilon}^2(t) + b_{\epsilon}(t) q_{\epsilon}(t) \right].
$$

We assume now that

$$
\Psi(x, 0, x_0) = \frac{1}{\sqrt{2\pi\epsilon/\delta}} \exp\left(-\frac{x^2}{\epsilon/\delta}\right), \quad (2.2.2.52)
$$

$0 < \epsilon \ll \delta \ll 1.$

Substitution Eq.(2.2.2.52) into Eq.(2.2.2.51) gives
$$
\tilde{\Theta}_{1,\epsilon}(T,x,x_0,\epsilon,\eta) =
$$

$$
\frac{1}{\sqrt[4]{2\pi\epsilon/\delta}} \int_{\substack{q(T)=x \\ q(0)=y}} dy \tilde{D}_{\eta}^{+}[q(t), \epsilon] \exp\left(-\frac{y^2}{\epsilon/\delta}\right) \times
$$

$$
\exp\left\{\frac{i}{\epsilon} \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t) q_{\epsilon}^2(t) + b_{\epsilon}(t) q_{\epsilon}(t) \right] \right\} = \tag{2.2.2.53}
$$

$$
\frac{1}{\sqrt{2\pi\epsilon/\delta}} \int_{\substack{q(T)=x \\ q(0)=y}} dy \tilde{D}_{\eta}^{+}[q(t), \epsilon] \exp\left(-\frac{y^2}{\epsilon/\delta}\right) \times
$$

$$
\exp\left\{\frac{i}{\epsilon} \int_{0}^{T} \mathcal{L}(\dot{q}(t), q_{\epsilon}(t)) dt\right\},
$$

$$
\mathcal{L}\left(\frac{d\tilde{q}_{\epsilon}(t)}{dt}, q_{\epsilon}(t)\right) = \frac{m}{2}\left(\frac{d\tilde{q}_{\epsilon}(t)}{dt}\right)^2 - a_{\epsilon}(t)q_{\epsilon}^2(t) + b_{\epsilon}(t)q_{\epsilon}(t).
$$

We assume that $a_\epsilon(t) > 0$ and rewrite $\mathcal{L}(\dot{q}(t), q_\epsilon(t))$ in the next form

$$
\mathcal{L}\left(\frac{d\tilde{q}_{\epsilon}(t)}{dt}, q_{\epsilon}(t)\right) = \frac{m}{2}\left(\frac{d\tilde{q}_{\epsilon}(t)}{dt}\right)^2 - \frac{m\omega_{\epsilon}^2}{2}q_{\epsilon}^2(t) + b_{\epsilon}(t)q_{\epsilon}(t), \quad (2.2.2.54)
$$

$$
\frac{m\omega_{\epsilon}^2}{2} = a_{\epsilon}(t).
$$

From Eq.(2.2.2.1) and Eq.(2.2.2.54) for $\epsilon \approx 0$ we obtain
$$
\tilde{\Theta}_{1,\epsilon}(T,x,x_0,\epsilon,\eta) =
$$

$$
\frac{1}{\sqrt[4]{2\pi\epsilon/\delta}} \int_{\substack{q(T)=x \\ q(0)=y}} dy \tilde{D}_{\eta}^{+}[q(t), \epsilon] \exp\left(-\frac{y^2}{\epsilon/\delta}\right) \times
$$

$$
\exp\left\{\frac{i}{\epsilon} \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon}(t)}{dt} \right)^{2} - a_{\epsilon}(t) q_{\epsilon}^{2}(t) + b_{\epsilon}(t) q_{\epsilon}(t) \right] \right\} = \tag{2.2.2.53}
$$

$$
\frac{1}{\sqrt{2\pi\epsilon/\delta}} \int_{\substack{q(T)=x \\ q(0)=y}} dy \tilde{D}_{\eta}^{+}[q(t), \epsilon] \exp\left(-\frac{y^2}{\epsilon/\delta}\right) \times
$$

$$
\exp\left\{\frac{i}{\epsilon} \int_{0}^{T} \mathcal{L}(\dot{q}(t), q_{\epsilon}(t)) dt\right\},
$$

$$
\mathcal{L}\left(\frac{d\tilde{q}_{\epsilon}(t)}{dt}, q_{\epsilon}(t)\right) = \frac{m}{2}\left(\frac{d\tilde{q}_{\epsilon}(t)}{dt}\right)^2 - a_{\epsilon}(t)q_{\epsilon}^2(t) + b_{\epsilon}(t)q_{\epsilon}(t).
$$

We assume that $a_\epsilon(t) > 0$ and rewrite $\mathcal{L}(\dot{q}(t), q_\epsilon(t))$ in the next form

$$
\mathcal{L}\left(\frac{d\tilde{q}_{\epsilon}(t)}{dt}, q_{\epsilon}(t)\right) = \frac{m}{2}\left(\frac{d\tilde{q}_{\epsilon}(t)}{dt}\right)^2 - \frac{m\omega_{\epsilon}^2}{2}q_{\epsilon}^2(t) + b_{\epsilon}(t)q_{\epsilon}(t), \qquad (2.2.2.54)
$$

$$
\frac{m\omega_{\epsilon}^2}{2} = a_{\epsilon}(t).
$$

From Eq.(2.2.2.1) and Eq.(2.2.2.54) for $\epsilon \approx 0$ we obtain
$$
\begin{align}
\mathcal{L}\left(\frac{d\tilde{q}_\epsilon(t)}{dt}, q_\epsilon(t)\right) &= \frac{m}{2}\left(\frac{dq(t)}{dt}\right)^2 - \frac{m\omega_0^2}{2}q^2(t) + b_0(t)q(t) + o(\epsilon) = \tag{2.2.2.55} \\
&\qquad \mathcal{L}(\dot{q}, q) + o(\epsilon). \nonumber
\end{align}
$$

We assume now that $\omega_0(t) = \omega = \text{const.}$ The Euler-Lagrange equation of $\mathcal{L}(\dot{q}, q)$
with corresponding boundary conditions is

$$
\begin{equation}
\begin{gathered}
\frac{d}{dt} \left( \frac{\partial \mathcal{L}}{\partial \dot{q}} \right) - \frac{\partial \mathcal{L}}{\partial q} = 0, \\
q(0) = y, q(T) = x.
\end{gathered}
\tag{2.2.2.56}
\end{equation}
$$

Corresponding classic action $S_{\text{cl}}$ is [13]:

$$
\begin{equation}
\begin{aligned}
S_{\text{cl}}(y, x, T) ={}& \frac{m\omega}{2\sin\omega T} \left[ \right. \\
& \left. \quad (\cos\omega T)(y^2 + x^2) - 2xy + \frac{2x}{m\omega} \int_0^T b(t) \sin(\omega t) dt + \right. \\
& \left. \quad + \frac{2y}{m\omega} \int_0^T b(t) \sin\omega(T-t) dt - \right. \\
& \left. \quad - \frac{2}{m^2\omega^2} \int_0^T \int_0^T b(t)b(s) \sin\omega(T-t) \sin(\omega s) ds dt \left].
\end{aligned}
\tag{2.2.2.57}
\end{equation}
$$

Substitution Eq.(2.2.2.57) into Eq.(2.2.2.53) gives
$$ \mathcal{L}\left(\frac{d\tilde{q}_{\varepsilon}(t)}{dt}, q_{\varepsilon}(t)\right) = \frac{m}{2}\left(\frac{dq(t)}{dt}\right)^2 - \frac{m\omega_0^2}{2}q^2(t) + b_0(t)q(t) + o(\varepsilon) = \\ \mathcal{L}(\dot{q}, q) + o(\varepsilon). \qquad (2.2.2.55) $$

We assume now that $\omega_0(t) = \omega = \text{const.}$ The Euler-Lagrange equation of $\mathcal{L}(\dot{q}, q)$ with corresponding boundary conditions is

$$ \frac{d}{dt}\left(\frac{\partial \mathcal{L}}{\partial \dot{q}}\right) - \frac{\partial \mathcal{L}}{\partial q} = 0, \qquad (2.2.2.56) \\ q(0) = y, q(T) = x. $$

Corresponding classic action $S_{\text{cl}}$ is [13]:

$$ S_{\text{cl}}(y, x, T) = \frac{m\omega}{2\sin\omega T} \left[ \begin{aligned} & (\cos\omega T)(y^2 + x^2) - 2xy + \frac{2x}{m\omega} \int_0^T b(t)\sin(\omega t)dt + \\ & \qquad + \frac{2y}{m\omega} \int_0^T b(t)\sin\omega(T-t)dt - \\ & \qquad - \frac{2}{m^2\omega^2} \int_0^T \int_0^T b(t)b(s)\sin\omega(T-t)\sin(\omega s)dsdt \end{aligned} \right]. \qquad (2.2.2.57) $$

Substitution Eq.(2.2.2.57) into Eq.(2.2.2.53) gives
$$
\begin{align}
\tilde{\Theta}_{1,\epsilon}(T, x, 0, \epsilon, \eta) &= \nonumber \\
&\quad \frac{1}{\sqrt[4]{2\pi\epsilon/\delta}} \int_{\substack{q(T)=x \\ q(0)=y}} dy \tilde{D}_{\eta}[q(t), \epsilon] \exp\left(-\frac{y^2}{\epsilon/\delta}\right) \times \tag{2.2.2.58} \\
&\quad \exp\left\{\frac{i}{\epsilon} \int_{0}^{T} \mathcal{L}(\dot{q}(t), q_{\epsilon}(t))dt\right\} = \nonumber \\
&\quad = \hat{\Theta}_{1,\epsilon}(T, x, x_0, \epsilon, \eta). \nonumber
\end{align}
$$

Here

$$
\begin{align}
\hat{\Theta}_{1,\epsilon}(T,x,0,\epsilon) = & \nonumber \\
& \frac{1}{\sqrt[4]{2\pi\epsilon/\delta}} \sqrt{\frac{m\omega}{2\pi i(\epsilon/\delta)\sin\omega T}} \int dy \exp\left(-\frac{y^2}{\epsilon/\delta}\right) \times \tag{2.2.2.59} \\
& \exp\left[\frac{i}{\epsilon}S_{\text{cl}}(y,x,T) + \frac{i\pi\gamma}{2}\right], \nonumber \\
& \sin\omega T \neq 0. \nonumber
\end{align}
$$

Let's calculate now oscillatory integral $\hat{\Theta}_{1,\epsilon}(T, x, 0, \epsilon)$ by using stationary phase method.

$$
\frac{\partial S_{\text{cl}}}{\partial y} = 2y \cos \omega T - 2x + \frac{2}{m\bar{\omega}} \int_{0}^{T} b(t) \sin \omega (T-t) dt = 0. \quad (2.2.2.60)
$$

Therefore critical point $y_{cr}(x, T)$ is
$$
\begin{align}
\tilde{\Theta}_{1,\epsilon}(T, x, 0, \epsilon, \eta) &= \nonumber \\
&\quad \frac{1}{\sqrt[4]{2\pi\epsilon/\delta}} \int_{\substack{q(T)=x \\ q(0)=y}} dy \tilde{D}_{\eta}[q(t), \epsilon] \exp\left(-\frac{y^2}{\epsilon/\delta}\right) \times \tag{2.2.2.58} \\
&\quad \exp\left\{\frac{i}{\epsilon} \int_{0}^{T} \mathcal{L}(\dot{q}(t), q_{\epsilon}(t))dt\right\} = \nonumber \\
&\quad = \hat{\Theta}_{1,\epsilon}(T, x, x_0, \epsilon, \eta). \nonumber
\end{align}
$$

Here

$$
\begin{align}
\hat{\Theta}_{1,\epsilon}(T,x,0,\epsilon) = & \nonumber \\
& \frac{1}{\sqrt[4]{2\pi\epsilon/\delta}} \sqrt{\frac{m\omega}{2\pi i(\epsilon/\delta)\sin\omega T}} \int dy \exp\left(-\frac{y^2}{\epsilon/\delta}\right) \times \tag{2.2.2.59} \\
& \exp\left[\frac{i}{\epsilon}S_{\text{cl}}(y,x,T) + \frac{i\pi\gamma}{2}\right], \nonumber \\
& \sin\omega T \neq 0. \nonumber
\end{align}
$$

Let's calculate now oscillatory integral $\hat{\Theta}_{1,\epsilon}(T, x, 0, \epsilon)$ by using stationary phase method.

$$
\frac{\partial S_{\text{cl}}}{\partial y} = 2y \cos \omega T - 2x + \frac{2}{m\bar{\omega}} \int_{0}^{T} b(t) \sin \omega (T-t) dt = 0. \quad (2.2.2.60)
$$

Therefore critical point $y_{cr}(x, T)$ is
$$
\begin{align}
y_{cr}(x, T) &= \frac{x}{\cos \omega T} - \frac{1}{m\omega \cos \omega T} \int_0^T b(t) \sin \omega (T-t) dt = \nonumber \\
&\qquad \frac{1}{\cos \omega T} \left( x - \frac{1}{m\omega} \int_0^T b(t) \sin \omega (T-t) dt \right), \tag{2.2.2.61} \\
&\qquad \cos \omega T \neq 0. \nonumber
\end{align}
$$

From Eq.(2.2.2.59) and Eq.(2.2.2.61) we obtain

$$
\begin{align}
\hat{\Theta}_{1,\epsilon}(T,x,0,\epsilon) &= \frac{1}{\sqrt[4]{2\pi\epsilon/\delta} |\cos\omega T|^{1/2}} \exp\left[-\frac{y_{cr}^2(x,T)}{\epsilon/\delta}\right] \times \nonumber \\
&\quad \exp\left[\frac{i}{\epsilon} \tilde{S}_{cl}(x,y_{cr},T) + \frac{i\pi\gamma}{2} + \frac{i\pi\gamma'}{4}\right] + O(\epsilon/m) = \nonumber \\
&= \frac{1}{\sqrt[4]{2\pi\epsilon/\delta} |\cos\omega T|^{1/2}} \times \tag{2.2.2.62} \\
&\quad \times \exp\left[-\frac{1}{\epsilon/\delta} \left(\frac{x}{\cos\omega T} - \frac{1}{m\omega \cos\omega T} \int_0^T b(t) \sin\omega(T-t)dt\right)^2\right] \times \nonumber \\
&\quad \exp\left[\frac{i}{\epsilon} \tilde{S}_{cl}(x,y_{cr},\lambda,T) + \frac{i\pi\gamma}{2} + \frac{i\pi\gamma'}{4}\right] + O(\epsilon). \nonumber
\end{align}
$$

Let's calculate now Laplace integral
$$
\begin{align*}
y_{cr}(x, T) &= \frac{x}{\cos \omega T} - \frac{1}{m\omega \cos \omega T} \int_{0}^{T} b(t) \sin \omega (T-t) dt = \\
&\qquad \frac{1}{\cos \omega T} \left( x - \frac{1}{m\omega} \int_{0}^{T} b(t) \sin \omega (T-t) dt \right), \tag{2.2.2.61} \\
&\qquad \cos \omega T \neq 0.
\end{align*}
$$

From Eq.(2.2.2.59) and Eq.(2.2.2.61) we obtain

$$
\begin{align}
\hat{\Theta}_{1,\epsilon}(T,x,0,\epsilon) &= \frac{1}{\sqrt[4]{2\pi\epsilon/\delta} |\cos\omega T|^{1/2}} \exp\left[-\frac{y_{cr}^2(x,T)}{\epsilon/\delta}\right] \times \nonumber \\
&\quad \exp\left[\frac{i}{\epsilon} \tilde{S}_{cl}(x,y_{cr},T) + \frac{i\pi\gamma}{2} + \frac{i\pi\gamma'}{4}\right] + O(\epsilon/m) = \nonumber \\
&\quad \frac{1}{\sqrt[4]{2\pi\epsilon/\delta} |\cos\omega T|^{1/2}} \times \tag{2.2.2.62} \\
&\quad \times \exp\left[-\frac{1}{\epsilon/\delta} \left(\frac{x}{\cos\omega T} - \frac{1}{m\omega \cos\omega T} \int_0^T b(t) \sin\omega(T-t)dt\right)^2\right] \times \nonumber \\
&\quad \exp\left[\frac{i}{\epsilon} \tilde{S}_{cl}(x,y_{cr},\lambda,T) + \frac{i\pi\gamma}{2} + \frac{i\pi\gamma'}{4}\right] + O(\epsilon). \nonumber
\end{align}
$$

Let's calculate now Laplace integral
$$
\begin{align*}
U(T) &= \int dx|x| |\hat{\Theta}_{1,\epsilon}(T,x,0,\epsilon)|^2 = \\
&\qquad \int dx \left\{ \frac{|x|}{\sqrt{2\pi\epsilon/\delta} |\cos \omega T|} \times \right. \\
&\qquad \left. \exp\left[-\frac{2}{(\epsilon/\delta)\cos \omega T} \left(x - \frac{1}{m\omega} \int_0^T b(t) \sin \omega(T-t)dt\right)^2\right] \right\} \tag{2.2.2.63}
\end{align*}
$$

using Laplace method. The maximum point is

$$
x_{\max} = \frac{1}{m\bar{\omega}} \int_{0}^{T} b(t) \sin \omega (T-t) dt. \qquad (2.2.2.64)
$$

Finally we obtain

$$
U(T) \approx \left| \frac{1}{m\bar{\omega}} \int_{0}^{T} b(t) \sin \omega(T-t) dt \right|. \qquad (2.2.2.65)
$$

From Eq.(2.2.2.45.a,b) we obtain

$$
\begin{align}
v_{1,\epsilon}\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) &\le \Theta_{1,\epsilon}(T,x,x_0,\epsilon,\eta) + \mathfrak{R}_{1,\epsilon}(T,x,x_0,\epsilon,\eta), \tag{2.2.2.66} \\
v_{2,\epsilon}\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) &\le \Theta_{2,\epsilon}(T,x,x_0,\epsilon,\eta) + \mathfrak{R}_{2,\epsilon}(T,x,x_0,\epsilon,\eta), \nonumber
\end{align}
$$

Here
$$
\begin{align*}
U(T) &= \int dx|x| |\hat{\Theta}_{1,\epsilon}(T,x,0,\epsilon)|^2 = \\
&\quad \int dx \left\{ \frac{|x|}{\sqrt{2\pi\epsilon/\delta} |\cos \omega T|} \times \right. \\
&\qquad \left. \exp\left[-\frac{2}{(\epsilon/\delta)\cos \omega T} \left(x - \frac{1}{m\omega} \int_0^T b(t) \sin \omega(T-t)dt\right)^2\right] \right\} \tag{2.2.2.63}
\end{align*}
$$

using Laplace method. The maximum point is

$$
x_{\max} = \frac{1}{m\bar{\omega}} \int_{0}^{T} b(t) \sin \omega (T-t) dt. \qquad (2.2.2.64)
$$

Finally we obtain

$$
U(T) \approx \left| \frac{1}{m\bar{\omega}} \int_{0}^{T} b(t) \sin \omega(T-t) dt \right|. \quad (2.2.2.65)
$$

From Eq.(2.2.2.45.a,b) we obtain

$$
\begin{align}
v_{1,\epsilon}\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) &\le \Theta_{1,\epsilon}(T,x,x_0,\epsilon,\eta) + \mathfrak{R}_{1,\epsilon}(T,x,x_0,\epsilon,\eta), \tag{2.2.2.66} \\
v_{2,\epsilon}\left(T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta\right) &\le \Theta_{2,\epsilon}(T,x,x_0,\epsilon,\eta) + \mathfrak{R}_{2,\epsilon}(T,x,x_0,\epsilon,\eta), \nonumber
\end{align}
$$

Here
$$
\mathfrak{R}_{1,\epsilon}(T, x, x_0, \epsilon, \eta) =
$$

$$
\sum_{n=1}^{M} \frac{(-1)^n \epsilon^n}{(2n)!} \left[ \frac{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} dy D^+[q(t),\epsilon,\epsilon] \Psi_0(y) \Xi_1(q(t),\epsilon,\epsilon) \times \times}{q(0)=\frac{y}{\sqrt{\epsilon}}} \right] \quad (2.2.2.67.a)
$$

$$
\exp\left\{-\int_{0}^{T} dt \left[\{q(t)\}_{\eta}\right]^{2}\right\} \left(\int_{0}^{T} \tilde{V}_{1}(q_{v(\epsilon,\epsilon)}(t)) dt\right)^{2n}].
$$

and

$$
\mathfrak{R}_{2,\epsilon}(T, x, x_0, \epsilon, \eta) =
$$

$$
\sum_{n=1}^{M} \frac{(-1)^n \epsilon^{n+1/2}}{(2n+1)!} \left[ \frac{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} dy D^+[q(t),\epsilon,\epsilon] \Psi_0(y) \Xi_2(q(t),\epsilon,\epsilon) \times \times}{q(0)=\frac{y}{\sqrt{\epsilon}}} \right] \quad (2.2.2.67.b)
$$

$$
\exp\left\{-\int_{0}^{T} dt \left[ \{q(t)\}_{\eta} \right]^2 \right\} \left( -\int_{0}^{T} \tilde{V}_{1}(q_{v(\epsilon,\epsilon)}(t)) dt \right)^{2n+1}].
$$

From (2.2.2.66) we obtain
$$
\Re_{1,\varepsilon}(T, x, x_0, \varepsilon, \eta) =
$$

$$
\sum_{n=1}^{M} \frac{(-1)^n \epsilon^n}{(2n)!} \left[ \begin{array}{l} \displaystyle \int_{\substack{q(T)=\frac{x}{\sqrt{\epsilon}} \\ q(0)=\frac{y}{\sqrt{\epsilon}}}} dy D^+[q(t), \epsilon, \epsilon] \Psi_0(y) \Xi_1(q(t), \epsilon, \epsilon) \times \\ \times \exp\left\{ -\int_0^T dt \left[ \{q(t)\}_\eta \right]^2 \right\} \left( \int_0^T \tilde{V}_1(q_{v(\epsilon,\epsilon)}(t)) dt \right)^{2n} \end{array} \right]. \quad (2.2.2.67.a)
$$

$$
\exp\left\{-\int_{0}^{T} dt \left[\{q(t)\}_{\eta}\right]^{2}\right\} \left(\int_{0}^{T} \tilde{V}_{1}(q_{v(\epsilon, \epsilon)}(t)) dt\right)^{2n}].
$$

and

$$
\mathfrak{R}_{2,\varepsilon}(T, x, x_0, \varepsilon, \eta) =
$$

$$
\sum_{n=1}^{M} \frac{(-1)^n \epsilon^{n+1/2}}{(2n+1)!} \left[ \begin{array}{l} \displaystyle \int_{\substack{q(T)=\frac{x}{\sqrt{\epsilon}} \\ q(0)=\frac{y}{\sqrt{\epsilon}}}} dy D^+[q(t), \epsilon, \epsilon] \Psi_0(y) \Xi_2(q(t), \epsilon, \epsilon) \times \\ \times \exp\left\{ -\int_0^T dt \left[ \{q(t)\}_\eta \right]^2 \right\} \left( -\int_0^T \tilde{V}_1(q_{v(\epsilon,\epsilon)}(t)) dt \right)^{2n+1} \end{array} \right]. \quad (2.2.2.67.b)
$$

$$
\exp\left\{-\int_{0}^{T} dt \left[\{q(t)\}_{\eta}\right]^{2}\right\} \left(-\int_{0}^{T} \tilde{V}_{1}(q_{v(\epsilon, \epsilon)}(t)) dt\right)^{2n+1}].
$$

From (2.2.2.66) we obtain
$$
\begin{align}
v_{1,\epsilon}^2 \left( T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta \right) &\le \nonumber \\
&[\Theta_{1,\epsilon}(T,x,x_0,\epsilon,\eta) + \mathfrak{R}_{1,\epsilon}(T,x,x_0,\epsilon,\eta)]^2 \le \tag{2.2.2.68.a} \\
&2[\Theta_{1,\epsilon}^2(T,x,x_0,\epsilon,\eta) + \mathfrak{R}_{1,\epsilon}^2(T,x,x_0,\epsilon,\eta)] \nonumber
\end{align}
$$

and

$$
\begin{align}
v_{2,\epsilon}^2 \left( T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta \right) &\le \nonumber \\
&[\Theta_{2,\epsilon}(T,x,x_0,\epsilon,\eta) + \mathfrak{R}_{2,\epsilon}(T,x,x_0,\epsilon,\eta)]^2 \le \tag{2.2.2.68.b} \\
&2[\Theta_{2,\epsilon}^2(T,x,x_0,\epsilon,\eta) + \mathfrak{R}_{2,\epsilon}^2(T,x,x_0,\epsilon,\eta)]. \nonumber
\end{align}
$$

From Eq.(2.2.2.12) and (2.2.2.68.a,b) we obtain
$$
\begin{align}
v_{1,\epsilon}^2 \left( T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta \right) &\le \nonumber \\
&[\Theta_{1,\epsilon}(T,x,x_0,\epsilon,\eta) + \mathfrak{R}_{1,\epsilon}(T,x,x_0,\epsilon,\eta)]^2 \le \tag{2.2.2.68.a} \\
&2[\Theta_{1,\epsilon}^2(T,x,x_0,\epsilon,\eta) + \mathfrak{R}_{1,\epsilon}^2(T,x,x_0,\epsilon,\eta)] \nonumber
\end{align}
$$

and

$$
\begin{align}
v_{2,\epsilon}^2 \left( T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta \right) &\le \nonumber \\
&[\Theta_{2,\epsilon}(T,x,x_0,\epsilon,\eta) + \mathfrak{R}_{2,\epsilon}(T,x,x_0,\epsilon,\eta)]^2 \le \tag{2.2.2.68.b} \\
&2[\Theta_{2,\epsilon}^2(T,x,x_0,\epsilon,\eta) + \mathfrak{R}_{2,\epsilon}^2(T,x,x_0,\epsilon,\eta)]. \nonumber
\end{align}
$$

From Eq.(2.2.2.12) and (2.2.2.68.a,b) we obtain
$$
\begin{align*}
\chi_1(T, \epsilon, \epsilon, \eta) &= \int dx|x| \left[ v_{1,\epsilon} \left( T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta \right) + v_{2,\epsilon} \left( T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta \right) \right]^2 \le \\
&= 2 \int dx|x| \left[ v_{1,\epsilon}^2 \left( T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta \right) + v_{2,\epsilon}^2 \left( T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta \right) \right] = \\
&= 2 \left[ \int dx|x| v_{1,\epsilon}^2 \left( T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta \right) + \int dx|x| v_{2,\epsilon}^2 \left( T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta \right) \right] \le \\
&= 4 \left[ \int dx|x| \Theta_{1,\epsilon}^2 (T, x, x_0, \epsilon, \eta) + \int dx|x| \Theta_{2,\epsilon}^2 (T, x, x_0, \epsilon, \eta) + \right. \\
&\qquad \left. \int dx|x| \mathfrak{R}_{1,\epsilon}^2 (T, x, x_0, \epsilon, \eta) + \int dx|x| \mathfrak{R}_{2,\epsilon}^2 (T, x, x_0, \epsilon, \eta) \right] =
\end{align*}
$$

$$4U(T) + 4\mathcal{F}_{1,\epsilon}(T, x_0, \epsilon, \eta) + 4\mathcal{F}_{2,\epsilon}(T, x_0, \epsilon, \eta).$$

$$\mathcal{F}_{1,\epsilon}(T, x_0, \epsilon, \eta) = \int dx|x|\mathfrak{R}_{1,\epsilon}^2(T, x, x_0, \epsilon, \eta),$$

$$\mathcal{F}_{2,\epsilon}(T, x_0, \epsilon, \eta) = \int dx|x|\mathfrak{R}_{2,\epsilon}^2(T, x, x_0, \epsilon, \eta).$$

Let us evaluate now path integral $\mathcal{F}_{1,\epsilon}(T, x_0, \epsilon, \eta)$ and $\mathcal{F}_{2,\epsilon}(T, x_0, \epsilon, \eta)$. From Eq.(2.2.2.67.a,b) we obtain
$$
\begin{align*}
\chi_1(T, \epsilon, \epsilon, \eta) &= \int dx|x| \left[ v_{1,\epsilon} \left( T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta \right) + v_{2,\epsilon} \left( T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta \right) \right]^2 \le \\
&\quad 2 \int dx|x| \left[ v_{1,\epsilon}^2 \left( T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta \right) + v_{2,\epsilon}^2 \left( T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta \right) \right] = \\
&\quad 2 \left[ \int dx|x| v_{1,\epsilon}^2 \left( T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta \right) + \int dx|x| v_{2,\epsilon}^2 \left( T, \frac{x}{\sqrt{\epsilon}}, x_0, \epsilon, \eta \right) \right] \le \\
&\quad 4 \left[ \int dx|x| \Theta_{1,\epsilon}^2 (T, x, x_0, \epsilon, \eta) + \int dx|x| \Theta_{2,\epsilon}^2 (T, x, x_0, \epsilon, \eta) + \right. \\
&\qquad \left. \int dx|x| \mathfrak{R}_{1,\epsilon}^2 (T, x, x_0, \epsilon, \eta) + \int dx|x| \mathfrak{R}_{2,\epsilon}^2 (T, x, x_0, \epsilon, \eta) \right] = \\
&\quad 4U(T) + 4\mathcal{F}_{1,\epsilon}(T, x_0, \epsilon, \eta) + 4\mathcal{F}_{2,\epsilon}(T, x_0, \epsilon, \eta).
\end{align*}
$$

$$
\mathcal{F}_{1,\epsilon}(T, x_0, \epsilon, \eta) = \int dx|x| \mathfrak{R}_{1,\epsilon}^2(T, x, x_0, \epsilon, \eta),
$$

$$
\mathcal{F}_{2,\epsilon}(T, x_0, \epsilon, \eta) = \int dx|x| \mathfrak{R}_{2,\epsilon}^2(T, x, x_0, \epsilon, \eta).
$$

Let us evaluate now path integral $\mathcal{F}_{1,\epsilon}(T, x_0, \epsilon, \eta)$ and $\mathcal{F}_{2,\epsilon}(T, x_0, \epsilon, \eta)$. From Eq.(2.2.2.67.a,b) we obtain
$$
\mathcal{F}_{1,\epsilon}(T, x_0, \epsilon, \eta) = \int dx|x| \mathfrak{R}_{1,\epsilon}^2(T, x, x_0, \epsilon, \eta) \le
$$

$$
2 \sum_{n=1}^{M} \frac{\epsilon^{2n}}{[(2n)!]^2} \times
$$

$$
\int dx|x| \left[ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \int_{q(0)=\frac{y}{\sqrt{\epsilon}}} dy D^+[q(t),\epsilon,\epsilon] \Psi_0(y) \exp\left\{-\int_0^T dt \left[\{q(t)\}_\eta\right]^2\right\}} \right] \\
\left(\int_0^T \tilde{V}_1(q_{v(\epsilon,\epsilon)}(t)) dt\right)^n = \tag{2.2.2.70.a}
$$

$$
2 \sum_{n=1}^{M} \frac{\epsilon^{2n}}{[(2n)!]^2} \int dx|x| \mathfrak{I}_{1,n}^2(T, x, x_0, \epsilon, \eta) =
$$

$$
2 \sum_{n=1}^{M} \frac{\epsilon^{2n}}{[(2n)!]^2} \mathcal{L}_{1,n}(T, x_0, \epsilon, \eta).
$$

$$
\mathfrak{I}_{1,n}(T,x,x_0,\epsilon,\eta) =
$$

$$
\overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} dy D^+[q(t),\epsilon,\epsilon]\Psi_0(y) \exp\left\{-\int_0^T dt \left[\{q(t)\}_\eta\right]^2\right\} \left(\int_0^T \tilde{V}_1(q_{v(\epsilon,\epsilon)}(t))dt\right)^n,}
$$

$$
\mathcal{L}_{1,n}(T, x_0, \epsilon, \eta) = \int dx |x| \mathfrak{I}_{1,n}^2(T, x, x_0, \epsilon, \eta).
$$

and
$$
\mathcal{F}_{1,\epsilon}(T, x_0, \epsilon, \eta) = \int dx|x| \mathfrak{R}_{1,\epsilon}^2(T, x, x_0, \epsilon, \eta) \le
$$

$$
2 \sum_{n=1}^{M} \frac{\epsilon^{2n}}{[(2n)!]^2} \times
$$

$$
\int dx|x| \left[ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} \int_{q(0)=\frac{y}{\sqrt{\epsilon}}} dy D^+[q(t),\epsilon,\epsilon] \Psi_0(y) \exp\left\{-\int_0^T dt \left[\{q(t)\}_\eta\right]^2\right\}} \right] \\
\left(\int_0^T \tilde{V}_1(q_{v(\epsilon,\epsilon)}(t)) dt\right)^n = \tag{2.2.2.70.a}
$$

$$
2 \sum_{n=1}^{M} \frac{\epsilon^{2n}}{[(2n)!]^2} \int dx|x| \mathfrak{I}_{1,n}^2(T, x, x_0, \epsilon, \eta) =
$$

$$
2 \sum_{n=1}^{M} \frac{\epsilon^{2n}}{[(2n)!]^2} \mathcal{L}_{1,n}(T, x_0, \epsilon, \eta).
$$

$$
\mathfrak{I}_{1,n}(T, x, x_0, \epsilon, \eta) =
$$

$$
\overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} dy D^+[q(t),\epsilon,\epsilon]\Psi_0(y) \exp\left\{-\int_0^T dt \left[\{q(t)\}_\eta\right]^2\right\} \left(\int_0^T \tilde{V}_1(q_{v(\epsilon,\epsilon)}(t))dt\right)^n,}
$$

$$
\mathcal{L}_{1,n}(T, x_0, \epsilon, \eta) = \int dx|x| \mathfrak{I}_{1,n}^2(T, x, x_0, \epsilon, \eta).
$$

and
$$
\mathcal{F}_{2,\varepsilon}(T, x_0, \varepsilon, \eta) = \int dx|x| \mathfrak{R}_{2,\varepsilon}^2(T, x, x_0, \varepsilon, \eta) \le
$$

$$
2 \sum_{n=1}^{M} \frac{\epsilon^{2n+1}}{[(2n+1)!]^2} \times
$$

$$
\int dx|x| \left[ \begin{array}{l} \displaystyle \int \frac{dyD^+[q(t), \epsilon, \epsilon]\Psi_0(y) \exp\left\{-\int_0^T dt \left[\{q(t)\}_\eta\right]^2\right\}}{q(T)=\frac{x}{\sqrt{\epsilon}}} \\ \\ \displaystyle \int \frac{dyD^+[q(t), \epsilon, \epsilon]\Psi_0(y) \exp\left\{-\int_0^T dt \left[\{q(t)\}_\eta\right]^2\right\}}{q(0)=\frac{y}{\sqrt{\epsilon}}} \end{array} \right]
$$

$$
\left( -\int_{0}^{T} \tilde{V}_{1}(q_{v(\epsilon, \epsilon)}(t)) dt \right)^{2n+1} \right]^2 =
$$

$$
2 \sum_{n=1}^{M} \frac{\epsilon^{2n+1}}{[(2n+1)!]^2} \int dx|x| \mathfrak{I}_{2,n}^2(T, x, x_0, \epsilon, \eta) = \qquad (2.2.2.70.b)
$$

$$
2 \sum_{n=1}^{M} \frac{\epsilon^{2n+1}}{[(2n+1)!]^2} \mathcal{L}_{2,n}(T, x_0, \epsilon, \eta).
$$

$$
\mathfrak{I}_{2,n}(T, x, x_0, \epsilon, \eta) =
$$

$$
\overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} dy D^+[q(t), \epsilon, \epsilon] \Psi_0(y) \Xi_2(q(t), \epsilon, \epsilon) \exp\left\{-\int_0^T dt \left[\{q(t)\}_\eta\right]^2\right\} \times}
$$

$$
\left( -\int_{0}^{T} \tilde{V}_{1}(q_{v(\epsilon, \epsilon)}(t)) dt \right)^{2n+1} .
$$

$$
\mathcal{L}_{2,n}(T, x_0, \epsilon, \eta) = \int dx|x| \mathfrak{I}_{2,n}^2(T, x, x_0, \epsilon, \eta)
$$
$$
\mathcal{F}_{2,\varepsilon}(T, x_0, \varepsilon, \eta) = \int dx|x| \mathfrak{R}_{2,\varepsilon}^2(T, x, x_0, \varepsilon, \eta) \le
$$

$$
2 \sum_{n=1}^{M} \frac{\epsilon^{2n+1}}{[(2n+1)!]^2} \times
$$

$$
\int dx|x| \left[
\begin{array}{@{}l@{}}
  \displaystyle \int \frac{dyD^+[q(t), \epsilon, \epsilon]\Psi_0(y) \exp\left\{-\int_0^T dt \left[\{q(t)\}_\eta\right]^2\right\}}{q(T)=\frac{x}{\sqrt{\epsilon}}} \\[2ex]
  \displaystyle q(0)=\frac{y}{\sqrt{\epsilon}}
\end{array}
\right]
$$

$$
\left( -\int_{0}^{T} \tilde{V}_{1}(q_{v(\epsilon, \epsilon)}(t)) dt \right)^{2n+1} \right]^2 =
$$

$$
2 \sum_{n=1}^{M} \frac{\epsilon^{2n+1}}{[(2n+1)!]^2} \int dx|x| \mathfrak{I}_{2,n}^2(T, x, x_0, \epsilon, \eta) = \qquad (2.2.2.70.b)
$$

$$
2 \sum_{n=1}^{M} \frac{\epsilon^{2n+1}}{[(2n+1)!]^2} \mathcal{L}_{2,n}(T, x_0, \epsilon, \eta).
$$

$$
\mathfrak{I}_{2,n}(T, x, x_0, \epsilon, \eta) =
$$

$$
\overline{\int}_{q(T)=\frac{x}{\sqrt{\epsilon}}} dy D^+[q(t), \epsilon, \epsilon] \Psi_0(y) \Xi_2(q(t), \epsilon, \epsilon) \exp \left\{ -\int_0^T dt \left[ \{q(t)\}_\eta \right]^2 \right\} \times \\
q(0)=\frac{y}{\sqrt{\epsilon}}
$$

$$
\left( -\int_{0}^{T} \tilde{V}_{1}(q_{v(\epsilon, \epsilon)}(t)) dt \right)^{2n+1} .
$$

$$
\mathcal{L}_{2,n}(T, x_0, \epsilon, \eta) = \int dx|x| \mathfrak{I}_{2,n}^2(T, x, x_0, \epsilon, \eta)
$$
Let us evaluate now path integral $\mathcal{L}_{1,n}(T, x_0, \epsilon, \eta)$ and $\mathcal{L}_{2,n}(T, x_0, \epsilon, \eta)$. From Eqs.(2.2.2.70.a,b) we obtain

$$ \mathcal{L}_{1,n}(T, x_0, \epsilon, \eta) = $$

$$ \int dx|x| \left[ \begin{array}{l} \displaystyle \overline{\displaystyle\int_{q(T)=\frac{x}{\sqrt{\epsilon}}}^{\int dyD^{+}[q(t),\epsilon,\epsilon]\Psi_0(y) \exp \times} q(0)=\frac{y}{\sqrt{\epsilon}}} \end{array} \right. \quad (2.2.2.71.a) $$

$$ \left\{ -\int_{0}^{T} dt \left[ \{q(t)\}_{\eta}^{2} \right] \right\} \left( \int_{0}^{T} \tilde{V}_{1}(q_{v(\epsilon,\epsilon)}(t)) dt \right)^{n} $$

and

$$ \mathcal{L}_{2,n}(T, x_0, \epsilon, \eta) = $$

$$ \int dx|x| \left[ \begin{array}{l} \displaystyle \overline{\displaystyle\int_{q(T)=\frac{x}{\sqrt{\epsilon}}}^{\int dyD^{+}[q(t),\epsilon,\epsilon]\Psi_0(y) \exp \times} q(0)=\frac{y}{\sqrt{\epsilon}}} \end{array} \right. \quad (2.2.2.71.b) $$

$$ \left\{ -\int_{0}^{T} dt \left[ \{q(t)\}_{\eta}^{2} \right] \right\} \left( \int_{0}^{T} \tilde{V}_{1}(q_{v(\epsilon,\epsilon)}(t)) dt \right)^{n+1} $$

Substitution into Eq.(2.2.2.71.a,b) $q(t) := \delta(t) + z(t)$, here
Let us evaluate now path integral $\mathcal{L}_{1,n}(T, x_0, \epsilon, \eta)$ and $\mathcal{L}_{2,n}(T, x_0, \epsilon, \eta)$. From Eqs.(2.2.2.70.a,b) we obtain

$$ \mathcal{L}_{1,n}(T, x_0, \epsilon, \eta) = $$

$$ \int dx|x| \left[ \begin{array}{l} \displaystyle \overline{\displaystyle \int_{q(T)=\frac{x}{\sqrt{\epsilon}}}^{\int_{q(0)=\frac{y}{\sqrt{\epsilon}}}^{T} dyD^{+}[q(t), \epsilon, \epsilon]\Psi_{0}(y) \exp \times} \end{array} \right. \quad (2.2.2.71.a) $$

$$ \left\{ -\int_{0}^{T} dt \left[ \{q(t)\}_{\eta}^{2} \right] \right\} \left( \int_{0}^{T} \tilde{V}_{1}(q_{v(\epsilon, \epsilon)}(t)) dt \right)^{n} $$

and

$$ \mathcal{L}_{2,n}(T, x_0, \epsilon, \eta) = $$

$$ \int dx|x| \left[ \begin{array}{l} \displaystyle \overline{\displaystyle \int_{q(T)=\frac{x}{\sqrt{\epsilon}}}^{\int_{q(0)=\frac{y}{\sqrt{\epsilon}}}^{T} dyD^{+}[q(t), \epsilon, \epsilon]\Psi_{0}(y) \exp \times} \end{array} \right. \quad (2.2.2.71.b) $$

$$ \left\{ -\int_{0}^{T} dt \left[ \{q(t)\}_{\eta}^{2} \right] \right\} \left( \int_{0}^{T} \tilde{V}_{1}(q_{v(\epsilon, \epsilon)}(t)) dt \right)^{n+1} $$

Substitution into Eq.(2.2.2.71.a,b) $q(t) := \delta(t) + z(t)$, here
$$z(t) = \frac{x-y}{T\sqrt{\epsilon}}t + \frac{y}{\sqrt{\epsilon}}, \qquad (2.2.2.72)$$

$$\delta(t) = \sum_{n=1}^{\infty} a_n \sin\left(\frac{n\pi t}{T}\right)$$

gives

$$\mathcal{L}_{1,n}(T, x_0, \epsilon, \eta) =$$

$$\frac{1}{\sqrt{2\pi\epsilon/\delta}} \int dx|x| \left[ \overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}}^{q(0)=\frac{y}{\sqrt{\epsilon}}} dy D^{+}[q(t), \epsilon, \epsilon] \exp\left(-\frac{y^2}{\epsilon/\delta}\right) \exp \times \left\{ -\int_{0}^{T} dt \left[ \{q(t)\}_{\eta} \right]^2 \right\} \left( \int_{0}^{T} \tilde{V}_1(q_{v(\epsilon,\epsilon)}(t)) dt \right)^n \right]^2 = \qquad (2.2.2.73.a)$$

$$\frac{1}{\sqrt{2\pi\epsilon/\delta}} \int dx|x| \left[ \overline{\int_{q(T)=0}^{q(0)=0} dy D^{+}[\delta(t), \epsilon, \epsilon] \exp\left(-\frac{y^2}{\epsilon/\delta}\right) \exp \times \left\{ -\int_{0}^{T} dt \left[ \{\delta(t)\}_{\eta} + \eta \left( \frac{x-y}{T\sqrt{\epsilon}}t + \frac{y}{\sqrt{\epsilon}} \right) \right]^2 \right\} \times \left( \int_{0}^{T} \tilde{V}_1 \left( \delta(t) + \left( \frac{x-y}{T\sqrt{\epsilon}}t + \frac{y}{\sqrt{\epsilon}} \right) \right) dt + o(\epsilon\epsilon) \right)^n \right]^2$$
$$z(t) = \frac{x-y}{T\sqrt{\epsilon}}t + \frac{y}{\sqrt{\epsilon}}, \qquad (2.2.2.72)$$

$$\delta(t) = \sum_{n=1}^{\infty} a_n \sin\left(\frac{n\pi t}{T}\right)$$

gives

$$\mathcal{L}_{1,n}(T, x_0, \epsilon, \eta) =$$

$$\frac{1}{\sqrt{2\pi\epsilon/\delta}} \int dx|x| \left\{ \overline{\int_{q(0)=\frac{y}{\sqrt{\epsilon}}}^{\overline{\int_{q(T)=\frac{x}{\sqrt{\epsilon}}} dyD^{+}[q(t),\epsilon,\epsilon] \exp\left(-\frac{y^2}{\epsilon/\delta}\right) \exp\left(-\frac{\epsilon^2}{\delta}\right)}} \right\} \times \\ \left\{ -\int_{0}^{T} dt \left[ \{q(t)\}_{\eta} \right]^2 \right\} \left( \int_{0}^{T} \tilde{V}_1(q_{v(\epsilon,\epsilon)}(t)) dt \right)^n = \qquad (2.2.2.73.a)$$

$$\frac{1}{\sqrt{2\pi\epsilon/\delta}} \int dx|x| \left[ \overline{\int_{q(0)=0}^{\overline{\int_{q(T)=0} dyD^{+}[\delta(t),\epsilon,\epsilon] \exp\left(-\frac{y^2}{\epsilon/\delta}\right) \exp\left(-\frac{\epsilon^2}{\delta}\right)}} \right. \\ \left. \times \left\{ -\int_{0}^{T} dt \left[ \{\delta(t)\}_{\eta} + \eta \left( \frac{x-y}{T\sqrt{\epsilon}}t + \frac{y}{\sqrt{\epsilon}} \right) \right]^2 \right\} \times \left( \int_{0}^{T} \tilde{V}_1 \left( \delta(t) + \left( \frac{x-y}{T\sqrt{\epsilon}}t + \frac{y}{\sqrt{\epsilon}} \right) \right) dt + o(\epsilon\epsilon) \right)^n \right]^2$$
and

$$ \mathcal{L}_{2,n}(T, x_0, \epsilon, \eta) = $$

$$ \frac{1}{\sqrt{2\pi\epsilon/\delta}} \int dx|x| \left[ \overline{\int_{q(T)=\frac{y}{\sqrt{\epsilon}}}^{T} dy D^{+}[q(t), \epsilon, \epsilon]} \exp\left(-\frac{y^2}{\epsilon/\delta}\right) \exp \times \right. \\ \left. \left\{ -\int_{0}^{T} dt \left[ \{q(t)\}_{\eta} \right]^2 \right\} \left( \int_{0}^{T} \tilde{V}_1(q_{v(\epsilon,\epsilon)}(t)) dt \right)^{n+1} \right]^2 = \tag{2.2.2.73.b} $$

$$ \frac{1}{\sqrt{2\pi\epsilon/\delta}} \int dx|x| \left[ \overline{\int_{q(T)=0}^{T} dy D^{+}[\delta(t), \epsilon, \epsilon]} \exp\left(-\frac{y^2}{\epsilon/\delta}\right) \exp \times \right. \\ \left. \left\{ -\int_{0}^{T} dt \left[ \{\delta(t)\}_{\eta} + \eta \left( \frac{x-y}{T\sqrt{\epsilon}}t + \frac{y}{\sqrt{\epsilon}} \right) \right]^2 \right\} \times \right. \\ \left. \left( \int_{0}^{T} \tilde{V}_1 \left( \delta(t) + \left( \frac{x-y}{T\sqrt{\epsilon}}t + \frac{y}{\sqrt{\epsilon}} \right) \right) dt + o(\epsilon\epsilon) \right)^{n+1} \right]^2 $$

Substitution $x := x\sqrt{\epsilon}$, $y := y\sqrt{\epsilon}$, into Eq.(2.2.2.73.a,b) gives
and

$$ \mathcal{L}_{2,n}(T, x_0, \epsilon, \eta) = $$

$$ \frac{1}{\sqrt{2\pi\epsilon/\delta}} \int dx|x| \left[ \overline{\int_{q(T)=\frac{y}{\sqrt{\epsilon}}}^{T} dy D^{+}[q(t), \epsilon, \epsilon]} \exp\left(-\frac{y^2}{\epsilon/\delta}\right) \exp \times \right. \\ \left. \left\{ -\int_{0}^{T} dt \left[ \{q(t)\}_{\eta} \right]^2 \right\} \left( \int_{0}^{T} \tilde{V}_1(q_{v(\epsilon,\epsilon)}(t)) dt \right)^{n+1} \right]^2 = \qquad (2.2.2.73.b) $$

$$ \frac{1}{\sqrt{2\pi\epsilon/\delta}} \int dx|x| \left[ \overline{\int_{q(T)=0}^{T} dy D^{+}[\delta(t), \epsilon, \epsilon]} \exp\left(-\frac{y^2}{\epsilon/\delta}\right) \exp \times \right. \\ \left. \left\{ -\int_{0}^{T} dt \left[ \{\delta(t)\}_{\eta} + \eta \left( \frac{x-y}{T\sqrt{\epsilon}}t + \frac{y}{\sqrt{\epsilon}} \right) \right]^2 \right\} \times \right. \\ \left. \left( \int_{0}^{T} \tilde{V}_1 \left( \delta(t) + \left( \frac{x-y}{T\sqrt{\epsilon}}t + \frac{y}{\sqrt{\epsilon}} \right) \right) dt + o(\epsilon\epsilon) \right)^{n+1} \right]^2 . $$

Substitution $x := x\sqrt{\epsilon}$, $y := y\sqrt{\epsilon}$, into Eq.(2.2.2.73.a,b) gives
$$ \mathcal{L}_{1,n}(T, x_0, \epsilon, \eta) = $$

$$
\begin{align*}
& \frac{\epsilon(\sqrt{\epsilon})}{\sqrt{2\pi\epsilon/\delta}} \int dx|x| \left[ \overline{\int_{q(0)=0} dy D^+[\delta(t), \epsilon, \epsilon]} \exp(-\delta y^2) \exp \times \right. \\
& \qquad \left. \left\{ -\int_0^T dt \left[ \{\delta(t)\}_\eta + \eta \left( \frac{x-y}{T} t + y \right) \right]^2 \right\} \times \right. \\
& \qquad \left. \left( \int_0^T \tilde{V}_1 \left( \delta(t) + \left( \frac{x-y}{T} t + y \right) \right) dt + o(\epsilon\epsilon) \right)^n \right] = \\
& \qquad \left( \sqrt{\delta} \epsilon \right)^{\eta^{-N(\epsilon)}} O(\gamma_{1,n}(T, \delta)).
\end{align*}
$$

and
$$ \mathcal{L}_{1,n}(T, x_0, \epsilon, \eta) = $$

$$ \frac{\epsilon(\sqrt{\epsilon})}{\sqrt{2\pi\epsilon/\delta}} \int dx|x| \left[ \int_{\substack{q(T)=0 \\ q(0)=0}} dy D^+[\delta(t), \epsilon, \epsilon] \exp(-\delta y^2) \exp \times \right. \\ \left. \left\{ -\int_0^T dt \left[ \{\delta(t)\}_\eta + \eta \left( \frac{x-y}{T}t + y \right) \right]^2 \right\} \times \right. \\ \left. \left( \int_0^T \tilde{V}_1 \left( \delta(t) + \left( \frac{x-y}{T}t + y \right) \right) dt + o(\epsilon\epsilon) \right)^n \right] = \tag{2.2.2.74.a} $$

$$ (\sqrt{\delta} \epsilon) \eta^{-N(\epsilon)} O(\gamma_{1,n}(T, \delta)). $$

and
$$ \mathcal{L}_{2,n}(T, x_0, \epsilon, \eta) = $$

$$
\begin{aligned}
& \frac{\epsilon(\sqrt{\epsilon})}{\sqrt{2\pi\epsilon/\delta}} \int dx|x| \left[ \overline{\int_{q(0)=0} dy D^+[\delta(t), \epsilon, \epsilon] \exp(-\delta y^2) \exp \times} \right. \\
& \qquad \left. \left\{ -\int_0^T dt \left[ \{\delta(t)\}_\eta + \eta \left( \frac{x-y}{T} t + y \right) \right]^2 \right\} \times \right. \\
& \qquad \left. \left( \int_0^T \tilde{V}_1 \left( \delta(t) + \left( \frac{x-y}{T} t + y \right) \right) dt + o(\epsilon\epsilon) \right)^{n+1} \right]^2 \\
& = (\sqrt{\delta}\epsilon) \eta^{-N(\epsilon)} O(\gamma_{2,n}(T, \delta)).
\end{aligned}
\qquad (2.2.2.74.b)
$$

We note that $\gamma_{1,n}(T, \delta) \approx \sqrt{\delta}, \gamma_{2,n}(T, \delta) \approx \sqrt{\delta}$ if $\delta \to 0$.

Substitution Eq.(2.2.2.74.a,b) into Eq.(2.2.2.70.a,b) gives

$$
\begin{aligned}
\mathcal{F}_{1,\epsilon}(T, x_0, \epsilon, \eta) &= \int dx|x| \mathfrak{R}_{1,\epsilon}^2(T, x, x_0, \epsilon, \eta) \le \\
& 2 \sum_{n=1}^{M} \frac{\epsilon^{2n}}{[(2n)!]^2} \mathcal{L}_{1,n}(T, x_0, \epsilon, \eta) = \\
& 2 \sum_{n=1}^{M} \frac{\epsilon^{2n}}{[(2n)!]^2} \epsilon \eta^{-N(\epsilon)} O(\gamma_{1,n}(T, \delta)) = \\
& 2 \eta^{-N(\epsilon)} \sum_{n=1}^{M} \frac{\epsilon^{2n+1}}{[(2n)!]^2} O(\tilde{\gamma}_{1,n}(T))
\end{aligned}
\qquad (2.2.2.75.a)
$$
$$ \mathcal{L}_{2,n}(T, x_0, \epsilon, \eta) = $$

$$
\begin{aligned}
& \frac{\epsilon(\sqrt{\epsilon})}{\sqrt{2\pi\epsilon/\delta}} \int dx|x| \left[ \overline{\int_{q(0)=0} dy} \int_{q(T)=0} dyD^+[\delta(t), \epsilon, \epsilon] \exp(-\delta y^2) \exp \times \right. \\
& \left. \left\{ -\int_0^T dt \left[ \{\delta(t)\}_\eta + \eta \left( \frac{x-y}{T}t + y \right) \right]^2 \right\} \times \right. \\
& \left. \left( \int_0^T \tilde{V}_1 \left( \delta(t) + \left( \frac{x-y}{T}t + y \right) \right) dt + o(\epsilon\epsilon) \right)^{n+1} \right]^2 \\
& = (\sqrt{\delta}\epsilon)\eta^{-N(\epsilon)}O(\gamma_{2,n}(T,\delta)).
\end{aligned}
\qquad (2.2.2.74.b)
$$

We note that $\gamma_{1,n}(T,\delta) \approx \sqrt{\delta}, \gamma_{2,n}(T,\delta) \approx \sqrt{\delta}$ if $\delta \to 0$.

Substitution Eq.(2.2.2.74.a,b) into Eq.(2.2.2.70.a,b) gives

$$
\begin{aligned}
\mathcal{F}_{1,\epsilon}(T, x_0, \epsilon, \eta) &= \int dx|x| \mathfrak{R}_{1,\epsilon}^2(T, x, x_0, \epsilon, \eta) \le \\
&2 \sum_{n=1}^{M} \frac{\epsilon^{2n}}{[(2n)!]^2} \mathcal{L}_{1,n}(T, x_0, \epsilon, \eta) = \\
&2 \sum_{n=1}^{M} \frac{\epsilon^{2n}}{[(2n)!]^2} \epsilon \eta^{-N(\epsilon)} O(\gamma_{1,n}(T, \delta)) = \\
&2 \eta^{-N(\epsilon)} \sum_{n=1}^{M} \frac{\epsilon^{2n+1}}{[(2n)!]^2} O(\tilde{\gamma}_{1,n}(T))
\end{aligned}
\qquad (2.2.2.75.a)
$$
and

$$
\begin{align*}
F_{2,\epsilon}(T, x_0, \epsilon, \eta) &= \int dx |x| \mathfrak{R}_{2,\epsilon}^2(T, x, x_0, \epsilon, \eta) \le \\
&\qquad 2 \sum_{n=1}^{M} \frac{\epsilon^{2n+1}}{[(2n+1)!]^2} \mathcal{L}_{2,n}(T, x_0, \epsilon, \eta) = \\
&\qquad 2 \sum_{n=1}^{M} \frac{\epsilon^{2n+1}}{[(2n+1)!]^2} \epsilon \eta^{-N(\epsilon)} O(\gamma_n(T, \delta)) = \\
&\qquad 2 \eta^{-N(\epsilon)} \sum_{n=1}^{M} \frac{\epsilon^{2n+2}}{[(2n+1)!]^2} O(\tilde{\gamma}_n(T)).
\end{align*}
$$

Finally from (2.2.2.69) and Eqs.(2.2.2.75.a,b) we obtain

$$
\begin{align*}
\chi_1(T, \epsilon, \varepsilon, \eta) &\le \\
&4U(T) + 4\mathcal{F}_{1,\varepsilon}(T, x_0, \varepsilon, \eta) + 4\mathcal{F}_{2,\varepsilon}(T, x_0, \varepsilon, \eta) = \\
&\qquad 4U(T) + \\
&\qquad 8\eta^{-N(\varepsilon)} \sum_{n=1}^{M} \frac{\epsilon^{2n+1}}{[(2n)!]^2} O(\tilde{\gamma}_{1,n}(T)) + 8\eta^{-N(\varepsilon)} \sum_{n=1}^{M} \frac{\epsilon^{2n+2}}{[(2n+1)!]^2} O(\tilde{\gamma}_{n}(T)) = \\
&\qquad 4U(T) + 8\eta^{-N(\varepsilon)} \sum_{n=1}^{M} \left[ \frac{\epsilon^{2n+1}}{[(2n)!]^2} O(\tilde{\gamma}_{1,n}(T)) + \sum_{n=1}^{M} \frac{\epsilon^{2n+2}}{[(2n+1)!]^2} O(\tilde{\gamma}_{n}(T)) \right].
\end{align*}
$$

Finally we obtain the master inequality:
and

$$
\begin{align*}
\mathcal{F}_{2,\epsilon}(T, x_0, \epsilon, \eta) &= \int dx |x| \mathfrak{R}_{2,\epsilon}^2(T, x, x_0, \epsilon, \eta) \le \\
&\qquad 2 \sum_{n=1}^{M} \frac{\epsilon^{2n+1}}{[(2n+1)!]^2} \mathcal{L}_{2,n}(T, x_0, \epsilon, \eta) = \\
&\qquad 2 \sum_{n=1}^{M} \frac{\epsilon^{2n+1}}{[(2n+1)!]^2} \epsilon \eta^{-N(\epsilon)} O(\gamma_n(T, \delta)) = \\
&\qquad 2 \eta^{-N(\epsilon)} \sum_{n=1}^{M} \frac{\epsilon^{2n+2}}{[(2n+1)!]^2} O(\tilde{\gamma}_n(T)).
\end{align*}
$$

Finally from (2.2.2.69) and Eqs.(2.2.2.75.a,b) we obtain

$$
\begin{align*}
\chi_1(T, \epsilon, \varepsilon, \eta) &\le \\
&4U(T) + 4\mathcal{F}_{1,\varepsilon}(T, x_0, \varepsilon, \eta) + 4\mathcal{F}_{2,\varepsilon}(T, x_0, \varepsilon, \eta) = \\
&\qquad 4U(T) + \\
&\qquad 8\eta^{-N(\varepsilon)} \sum_{n=1}^{M} \frac{\epsilon^{2n+1}}{[(2n)!]^2} O(\tilde{\gamma}_{1,n}(T)) + 8\eta^{-N(\varepsilon)} \sum_{n=1}^{M} \frac{\epsilon^{2n+2}}{[(2n+1)!]^2} O(\tilde{\gamma}_{n}(T)) = \\
&\qquad 4U(T) + 8\eta^{-N(\varepsilon)} \sum_{n=1}^{M} \left[ \frac{\epsilon^{2n+1}}{[(2n)!]^2} O(\tilde{\gamma}_{1,n}(T)) + \sum_{n=1}^{M} \frac{\epsilon^{2n+2}}{[(2n+1)!]^2} O(\tilde{\gamma}_{n}(T)) \right].
\end{align*}
$$

Finally we obtain the master inequality:
**Theorem 2.2.2.1.**

$$|\langle x, t, 0; \epsilon, \delta \rangle| \leq U(T)(C + O(\delta)). \qquad (2.2.2.77)$$

Here

$$U(T) = \left| \frac{1}{m\omega} \int_{0}^{T} b(t) \sin \omega(T-t) dt \right|. \qquad (2.2.2.78)$$

Let us considered the quantity $\langle x_f, t_f, x_0; \epsilon \rangle - \lambda, \lambda \in \mathbb{R}$. We assume that $\int dx_f |\mathbf{K}(x_f, t_f|x_0, 0)|^2 = 1$. Then from Eq.(2.2.2.2) we obtain

$$ \begin{aligned} \langle T, x_0, \lambda; \epsilon \rangle &= \langle T, x_0; \epsilon \rangle - \lambda = \\ &= \int dx_f x_f |\mathbf{K}(x_f, T|x_0, 0)|^2 - \lambda \int dx_f |\mathbf{K}(x_f, T|x_0, 0)|^2 \\ &= \int dx_f (x_f - \lambda) |\mathbf{K}(x_f, T|x_0, 0)|^2. \end{aligned} \qquad (2.2.2.79) $$

$$T = t_f.$$

From Eq.(2.2.2.79) we obtain
**Theorem 2.2.2.1.**

$$|\langle x, t, 0; \epsilon, \delta \rangle| \leq U(T)(C + O(\delta)). \qquad (2.2.2.77)$$

Here

$$U(T) = \left| \frac{1}{m\omega} \int_{0}^{T} b(t) \sin \omega(T-t) dt \right|. \qquad (2.2.2.78)$$

Let us considered the quantity $\langle x_f, t_f, x_0; \epsilon \rangle - \lambda, \lambda \in \mathbb{R}$. We assume that $\int dx_f |\mathbf{K}(x_f, t_f|x_0, 0)|^2 = 1$. Then from Eq.(2.2.2.2) we obtain

$$ \begin{align} \langle T, x_0, \lambda; \epsilon \rangle &= \langle T, x_0; \epsilon \rangle - \lambda = \nonumber \\ &= \int dx_f x_f |\mathbf{K}(x_f, T|x_0, 0)|^2 - \lambda \int dx_f |\mathbf{K}(x_f, T|x_0, 0)|^2 \tag{2.2.2.79} \\ &= \int dx_f (x_f - \lambda) |\mathbf{K}(x_f, T|x_0, 0)|^2. \nonumber \end{align} $$

$$T = t_f.$$

From Eq.(2.2.2.79) we obtain
$$|\langle T, x_0, \lambda; \varepsilon, \epsilon \rangle| \le \int dx |x - \lambda| \times$$

$$\left| \int_{q(T)=x} \tilde{D}_{\eta}^{+}[q(t), \varepsilon] \Psi_{0}(q(0)) \exp \left[ \frac{i}{\varepsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\varepsilon}(t)}{dt} \right)^{2} - V(q_{\varepsilon}(t)) \right] \right) \right] \right|^{2} =$$

$$\int dx \left\{ \left| \int_{q(T)=x} \tilde{D}_{\eta}^{+}[q(t), \varepsilon] \Psi_{0}(q(0)) \sqrt{|x - \lambda|} \right. \right. $$

$$\left. \exp \left[ \left| \frac{i}{\epsilon} \left( \int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_\epsilon(t)}{dt} \right)^2 - V(q_\epsilon(t)) \right] \right) \right| \right]^2 \right\} = \qquad (2.2.2.80)$$

$$\int dx \left\{ \left| \int_{q(T)=x} D^{+}[q(t), \epsilon] \Psi_{0}(q(0)) \sqrt{|q(T) - \lambda|} \times \right. \right. $$

$$\exp\left\{-\frac{m}{\epsilon} \int_{0}^{T} dt \left[ \{q(t)\}_{\eta} + \eta g(t, q(T), q(0), \lambda) \right]^2 \right\} \times$$

$$\exp\left[ \left| \frac{i}{\epsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon}(t)}{dt} \right)^{2} - V(q_{\epsilon}(t)) \right] \right) \right|^{2} \right\}.$$

Here the function $V(q_{\epsilon}(t), t)$ is given via formula

$$V(q_{\epsilon}(t), t) = a_{1,\epsilon}(t)q_{\epsilon}(t) + a_{2,\epsilon}(t)q_{\epsilon}^{2}(t) + a_{3,\epsilon}q_{\epsilon}^{3}(t) + \dots + a_{k,\epsilon}q_{\epsilon}^{k}(t),$$

$$q_{\epsilon}(t) = \frac{q(t)}{1 + \epsilon^k q^m(t)}, \qquad (2.2.2.80')$$

$k \ge 1, m \ge 2$

From Eq.(2.2.2.80) we obtain
$$|\langle T, x_0, \lambda; \varepsilon, \epsilon \rangle| \le \int dx |x - \lambda| \times$$

$$\left| \int_{q(T)=x} \tilde{D}_{\eta}^{+}[q(t), \varepsilon] \Psi_{0}(q(0)) \exp \left[ \frac{i}{\varepsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\varepsilon}(t)}{dt} \right)^2 - V(q_{\varepsilon}(t)) \right] \right) \right] \right|^2 =$$

$$\int dx \left\{ \left| \int_{q(T)=x} \tilde{D}_{\eta}^{+}[q(t), \varepsilon] \Psi_{0}(q(0)) \sqrt{|x - \lambda|} \right. \right.$$

$$\left. \exp \left[ \left| \frac{i}{\epsilon} \left( \int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon}(t)}{dt} \right)^2 - V(q_{\epsilon}(t)) \right] \right) \right| \right|^2 \right\} = \qquad (2.2.2.80)$$

$$\int dx \left\{ \left| \int_{q(T)=x} D^{+}[q(t), \epsilon] \Psi_{0}(q(0)) \sqrt{|q(T) - \lambda|} \right. \times \right.$$

$$\exp \left\{ -\frac{m}{\epsilon} \int_{0}^{T} dt \left[ \{q(t)\}_{\eta} + \eta g(t, q(T), q(0), \lambda) \right]^2 \right\} \times$$

$$\exp \left[ \left| \frac{i}{\epsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon}(t)}{dt} \right)^2 - V(q_{\epsilon}(t)) \right] \right) \right|^2 \right].$$

Here the function $V(q_{\epsilon}(t), t)$ is given via formula

$$V(q_{\epsilon}(t), t) = a_{1,\epsilon}(t)q_{\epsilon}(t) + a_{2,\epsilon}(t)q_{\epsilon}^{2}(t) + a_{3,\epsilon}q_{\epsilon}^{3}(t) + \dots + a_{k,\epsilon}q_{\epsilon}^{k}(t),$$

$$q_{\epsilon}(t) = \frac{q(t)}{1 + \epsilon^k q^m(t)}, \qquad (2.2.2.80')$$

$$k \ge 1, m \ge 2$$

From Eq.(2.2.2.80) we obtain
$$|\langle T, x_0, \lambda; \varepsilon, \epsilon \rangle| \le$$

$$
\begin{aligned}
& \int dx \left\{ \left[ \int_{q(T)=x} \tilde{D}_{\eta}^{+}[q(t), \varepsilon] \Psi_{0}(q(0)) \sqrt{|q(T) - \lambda|} \times \right. \right. \\
& \qquad \left. \left. \cos \left[ \frac{1}{\varepsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\varepsilon}(t)}{dt} \right)^2 - V(q_{\varepsilon}(t)) \right] \right) \right] \right] \right. \\
& \qquad \left. + \int_{q(T)=x} \tilde{D}_{\eta}^{+}[q(t), \varepsilon] \Psi_{0}(q(0)) \sqrt{|q(T) - \lambda|} \times \right. \\
& \qquad \left. \left. \sin \left[ \frac{1}{\varepsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\varepsilon}(t)}{dt} \right)^2 - V(q_{\varepsilon}(t)) \right] \right) \right] \right] \right\} = \\
& \qquad \qquad (2.2.2.81)
\end{aligned}
$$

$$
\begin{aligned}
& \int dx \left\{ \left[ \int_{q(T)=x} \tilde{D}_{\eta}^{+}[q(t), \varepsilon] \Psi_{0}(q(0)) \sqrt{|q(T) - \lambda|} \times \right. \right. \\
& \qquad \left. \left. \cos \left[ \frac{1}{\varepsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\varepsilon}(t)}{dt} \right)^2 - V(q_{\varepsilon}(t)) \right] \right) \right] \right] \right\} + \\
& \qquad \int dx \left\{ \left[ \int_{q(T)=x} \tilde{D}_{\eta}^{+}[q(t), \varepsilon] \Psi_{0}(q(0)) \sqrt{|q(T) - \lambda|} \times \right. \right. \\
& \qquad \left. \left. \sin \left[ \frac{1}{\varepsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\varepsilon}(t)}{dt} \right)^2 - V(q_{\varepsilon}(t)) \right] \right) \right] \right] \right\} = \\
& \qquad \qquad \chi_{1}(T, x_{0}, \lambda, \varepsilon, \varepsilon, \eta) + \chi_{2,\varepsilon}(T, x_{0}, \lambda, \varepsilon, \varepsilon, \eta).
\end{aligned}
$$

Here
$$|\langle T, x_0, \lambda; \varepsilon, \varepsilon \rangle| \le$$

$$ \int dx \left\{ \left[ \int_{q(T)=x} \tilde{D}_{\eta}^{+}[q(t), \varepsilon] \Psi_{0}(q(0)) \sqrt{|q(T) - \lambda|} \times \right. \\ \cos \left[ \frac{1}{\varepsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\varepsilon}(t)}{dt} \right)^2 - V(q_{\varepsilon}(t)) \right] \right) \right] + \\ \left. \int_{q(T)=x} \tilde{D}_{\eta}^{+}[q(t), \varepsilon] \Psi_{0}(q(0)) \sqrt{|q(T) - \lambda|} \times \right. \\ \sin \left[ \frac{1}{\varepsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\varepsilon}(t)}{dt} \right)^2 - V(q_{\varepsilon}(t)) \right] \right) \right] \right\} = \tag{2.2.2.81} $$

$$ \int dx \left\{ \left[ \int_{q(T)=x} \tilde{D}_{\eta}^{+}[q(t), \varepsilon] \Psi_{0}(q(0)) \sqrt{|q(T) - \lambda|} \times \right. \\ \cos \left[ \frac{1}{\varepsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\varepsilon}(t)}{dt} \right)^2 - V(q_{\varepsilon}(t)) \right] \right) \right] \right\} + $$

$$ \int dx \left\{ \left[ \int_{q(T)=x} \tilde{D}_{\eta}^{+}[q(t), \varepsilon] \Psi_{0}(q(0)) \sqrt{|q(T) - \lambda|} \times \right. \\ \left. \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\varepsilon}(t)}{dt} \right)^2 - V(q_{\varepsilon}(t)) \right] \right] \right\} = $$

$$ \chi_1(T, x_0, \lambda, \varepsilon, \varepsilon, \eta) + \chi_{2,\varepsilon}(T, x_0, \lambda, \varepsilon, \varepsilon, \eta). $$

Here
$$
\chi_1(T, x_0, \lambda, \varepsilon, \epsilon, \eta) =
$$

$$
\begin{equation}
\begin{aligned}
& \int dx \left\{ \left[ \int_{q(T)=x} \tilde{D}_{\eta}^{+}[q(t),\varepsilon]\Psi_{0}(q(0)) \sqrt{|q(T)-\lambda|} \right] \times \right. \\
& \qquad \left. \exp\left\{ -\frac{m}{\epsilon} \int_{0}^{T} dt \left[ \{q(t)\}_{\eta} + \eta g(t,q(T),q(0),\lambda) \right]^2 \right\} \right. \\
& \qquad \left. \cos\left[ \frac{1}{\epsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\varepsilon}(t)}{dt} \right)^2 - V(q_{\varepsilon}(t)) \right] \right) \right] \right\},
\end{aligned}
\tag{2.2.2.82.a}
\end{equation}
$$

and

$$
\chi_2(T, x_0, \lambda, \epsilon, \epsilon, \eta) =
$$

$$
\begin{equation}
\begin{aligned}
& \int dx \left\{ \left[ \int_{q(T)=x} \tilde{D}_{\eta}^{+}[q(t), \varepsilon] \Psi_{0}(q(0)) \sqrt{|q(T)-\lambda|} \right] \times \right. \\
& \qquad \left. \exp\left\{ -\frac{m}{\epsilon} \int_{0}^{T} dt \left[ \{q(t)\}_{\eta} + \eta g(t, q(T), q(0), \lambda) \right]^2 \right\} \right. \\
& \qquad \left. \sin\left[ \frac{1}{\epsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon}(t)}{dt} \right)^2 - V(q_{\epsilon}(t)) \right] \right) \right] \right\}.
\end{aligned}
\tag{2.2.2.82.b}
\end{equation}
$$
$$
\chi_1(T, x_0, \lambda, \varepsilon, \epsilon, \eta) =
$$

$$
\int dx \left\{ \left[ \int_{q(T)=x} \tilde{D}_{\eta}^{+}[q(t), \varepsilon] \Psi_{0}(q(0)) \sqrt{|q(T) - \lambda|} \times \right. \right. \\
\left. \left. \exp\left\{ -\frac{m}{\epsilon} \int_{0}^{T} dt \left[ \{q(t)\}_{\eta} + \eta g(t, q(T), q(0), \lambda) \right]^{2} \right\} \times \right] \right. \\
\left. \left. \cos\left[ \frac{1}{\epsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\varepsilon}(t)}{dt} \right)^{2} - V(q_{\varepsilon}(t)) \right] \right) \right] \right] \right\}, \quad (2.2.2.82.a)
$$

$$
\cos \left[ \frac{1}{\epsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon}(t)}{dt} \right)^{2} - V(q_{\epsilon}(t)) \right] \right) \right]^{2}
$$

and

$$
\chi_2(T, x_0, \lambda, \epsilon, \epsilon, \eta) =
$$

$$
\int dx \left\{ \left[ \int_{q(T)=x} \tilde{D}_{\eta}^{+}[q(t), \epsilon] \Psi_{0}(q(0)) \sqrt{|q(T) - \lambda|} \times \right. \right. \\
\left. \left. \exp\left\{ -\frac{m}{\epsilon} \int_{0}^{T} dt \left[ \{q(t)\}_{\eta} + \eta g(t, q(T), q(0), \lambda) \right]^2 \right\} \times \right] \right.
$$

$$
\sin \left[ \frac{1}{\epsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{q}_{\epsilon}(t)}{dt} \right)^{2} - V(q_{\epsilon}(t)) \right] \right) \right]^{2}
$$
By using replacement $q(t) - \lambda = u(t)$, from Eq.(2.2.2.80) we obtain

$$ \chi_1(T, x_0, \lambda, \varepsilon, \epsilon, \eta) = $$

$$ \begin{aligned} & \int dx \left\{ \left[ \int_{u(T)=x-\lambda} D^+[u(t),\varepsilon]\Psi_0(u(0)+\lambda)\sqrt{|u(T)|} \times \right. \\ & \qquad \left. \exp\left\{ -\frac{m}{\epsilon}\int_0^T dt \left[ \{u(t)+\lambda\}_\eta + \eta g(t, u(T) + \lambda, u(0) + \lambda, \lambda) \right]^2 \right\} \times \right. \\ & \qquad \left. \cos\left[ \frac{1}{\epsilon}\left(\int_0^T dt \left[ \frac{m}{2}\left(\frac{d\tilde{u}_\epsilon(t)}{dt}\right)^2 - V(u_\epsilon(t)+\lambda) \right] \right) \right] \right]^2, \end{aligned} \quad (2.2.2.83.a) $$

and

$$ \chi_2(T, x_0, \lambda, \varepsilon, \epsilon, \eta) = $$

$$ \begin{aligned} & \int dx \left\{ \left[ \int_{u(T)=x-\lambda} D^+[u(t),\varepsilon]\Psi_0(u(0)+\lambda)\sqrt{|u(T)|} \times \right. \\ & \qquad \left. \exp\left\{ -\frac{m}{\epsilon}\int_0^T dt \left[ \{u(t)+\lambda\}_\eta + \eta g(t, u(T) + \lambda, u(0) + \lambda, \lambda) \right]^2 \right\} \times \right. \\ & \qquad \left. \sin\left[ \frac{1}{\epsilon}\left(\int_0^T dt \left[ \frac{m}{2}\left(\frac{d\tilde{u}_\epsilon(t)}{dt}\right)^2 - V(u_\epsilon(t)+\lambda) \right] \right) \right] \right]^2, \end{aligned} \quad (2.2.2.83.b) $$
By using replacement $q(t) - \lambda = u(t)$, from Eq.(2.2.2.80) we obtain

$$ \chi_1(T, x_0, \lambda, \varepsilon, \epsilon, \eta) = $$

$$ \begin{aligned} & \int dx \left\{ \left[ \int_{u(T)=x-\lambda} D^+[u(t),\varepsilon]\Psi_0(u(0)+\lambda)\sqrt{|u(T)|} \times \right. \\ & \qquad \left. \exp\left\{ -\frac{m}{\epsilon}\int_0^T dt \left[ \{u(t)+\lambda\}_\eta + \eta g(t, u(T) + \lambda, u(0) + \lambda, \lambda) \right]^2 \right\} \times \right. \\ & \qquad \left. \cos\left[ \frac{1}{\epsilon}\left(\int_0^T dt \left[ \frac{m}{2}\left(\frac{d\tilde{u}_\epsilon(t)}{dt}\right)^2 - V(u_\epsilon(t)+\lambda) \right] \right) \right] \right]^2, \end{aligned} \quad (2.2.2.83.a) $$

and

$$ \chi_2(T, x_0, \lambda, \varepsilon, \epsilon, \eta) = $$

$$ \begin{aligned} & \int dx \left\{ \left[ \int_{u(T)=x-\lambda} D^+[u(t),\varepsilon]\Psi_0(u(0)+\lambda)\sqrt{|u(T)|} \times \right. \\ & \qquad \left. \exp\left\{ -\frac{m}{\epsilon}\int_0^T dt \left[ \{u(t)+\lambda\}_\eta + \eta g(t, u(T) + \lambda, u(0) + \lambda, \lambda) \right]^2 \right\} \times \right. \\ & \qquad \left. \sin\left[ \frac{1}{\epsilon}\left(\int_0^T dt \left[ \frac{m}{2}\left(\frac{d\tilde{u}_\epsilon(t)}{dt}\right)^2 - V(u_\epsilon(t)+\lambda) \right] \right) \right] \right]^2, \end{aligned} \quad (2.2.2.83.b) $$
Let us rewrite a function $V(u_{\epsilon}(t) + \lambda, t)$ in equivalent form:

$$V(u_{\epsilon}(t) + \lambda, t) = V_0(u_{\epsilon}(t), t, \lambda) + V_1(u_{\epsilon}(t), t, \lambda)$$

$$V_0(u_{\epsilon}(t), t, \lambda) = a_{\epsilon}(\lambda)u_{\epsilon}^2(t) + b_{\epsilon}(t, \lambda)(q_{\epsilon}(t)),$$

$$V_1(u_{\epsilon}(t), t, \lambda) = a_{3,\epsilon}(\lambda,t)u_{\epsilon}^3(t) + \dots + a_{k,\epsilon}(\lambda,t)u_{\epsilon}^k(t) + c(t,\lambda), \qquad (2.2.2.84)$$

$$a_{\epsilon}(\lambda, t) > 0,$$

$$u_{\epsilon}(t) = \frac{u(t)}{1 + \epsilon^k u^m(t)},$$

$$k \geq 1, m \geq 2$$

Let us now evaluate a quantities $(\chi_1(T, x_0, \lambda, \epsilon, \eta))_{\epsilon}$:
Let us rewrite a function $V(u_{\varepsilon}(t) + \lambda, t)$ in equivalent form:

$$V(u_{\varepsilon}(t) + \lambda, t) = V_0(u_{\varepsilon}(t), t, \lambda) + V_1(u_{\varepsilon}(t), t, \lambda)$$

$$V_0(u_{\varepsilon}(t), t, \lambda) = a_{\varepsilon}(\lambda)u_{\varepsilon}^2(t) + b_{\varepsilon}(t, \lambda)(q_{\varepsilon}(t)),$$

$$V_1(u_{\varepsilon}(t), t, \lambda) = a_{3,\varepsilon}(\lambda,t)u_{\varepsilon}^3(t) + \dots + a_{k,\varepsilon}(\lambda,t)u_{\varepsilon}^k(t) + c(t, \lambda), \quad (2.2.2.84)$$

$$a_{\varepsilon}(\lambda, t) > 0,$$

$$u_{\varepsilon}(t) = \frac{u(t)}{1 + \varepsilon^k u^m(t)},$$

$$k \geq 1, m \geq 2$$

Let us now evaluate a quantities $(\chi_1(T, x_0, \lambda, \epsilon, \eta))_{\varepsilon}$:
$$ \chi_1(T, x_0, \lambda, \epsilon, \epsilon, \eta) = $$

$$ \int dx \left\{ \left[ \int_{u(T)=x-\lambda} D^+[u(t),\epsilon]\Psi_0(u(0)+\lambda)\sqrt{|u(T)|} \times \right. \right. \\ \exp\left\{ -\frac{m}{\epsilon} \int_0^T dt \left[ \{u(t)+\lambda\}_\eta + \eta g(t, u(T)+\lambda, u(0)+\lambda, \lambda) \right]^2 \right\} \times \quad (2.2.2.85.a) \\ \left. \left. \cos\left[ \frac{1}{\epsilon} \left( \int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{u}_\epsilon(t)}{dt} \right)^2 - V(u_\epsilon(t)+\lambda) \right] \right) \right] \right]^2 = \right. \\ \left. \int dx \Pi_1^2(T, x_0, \lambda, \epsilon, \epsilon, \eta) \right\} $$

and $\chi_2(T, x_0, \lambda, \epsilon, \epsilon, \eta)$ :

$$ \chi_2(T, x_0, \lambda, \epsilon, \epsilon, \eta) = $$

$$ \int dx \left\{ \left[ \int_{u(T)=x-\lambda} D^+[u(t),\epsilon]\Psi_0(u(0)+\lambda)\sqrt{|u(T)|} \times \right. \right. \\ \exp\left\{ -\frac{m}{\epsilon} \int_0^T dt \left[ \{u(t)+\lambda\}_\eta + \eta g(t, u(T)+\lambda, u(0)+\lambda, \lambda) \right]^2 \right\} \times \quad (2.2.2.85.b) \\ \left. \left. \sin\left[ \frac{1}{\epsilon} \left( \int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{u}_\epsilon(t)}{dt} \right)^2 - V(u_\epsilon(t)+\lambda) \right] \right) \right] \right]^2 = \right. \\ \left. \int dx \Pi_2^2(T, x_0, \lambda, \epsilon, \epsilon, \eta). \right\} $$
$$ \chi_1(T, x_0, \lambda, \epsilon, \epsilon, \eta) = $$

$$ \int dx \left\{ \left[ \int_{u(T)=x-\lambda} D^+[u(t),\epsilon]\Psi_0(u(0)+\lambda)\sqrt{|u(T)|} \times \right. \right. \\ \exp\left\{ -\frac{m}{\epsilon} \int_0^T dt \left[ \{u(t)+\lambda\}_\eta + \eta g(t, u(T)+\lambda, u(0)+\lambda, \lambda) \right]^2 \right\} \times \quad (2.2.2.85.a) \\ \left. \left. \cos\left[ \frac{1}{\epsilon} \left( \int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{u}_\epsilon(t)}{dt} \right)^2 - V(u_\epsilon(t)+\lambda) \right] \right) \right] \right]^2 = \right. \\ \left. \int dx \Pi_1^2(T, x_0, \lambda, \epsilon, \epsilon, \eta) \right\} $$

and $\chi_2(T, x_0, \lambda, \epsilon, \epsilon, \eta)$ :

$$ \chi_2(T, x_0, \lambda, \epsilon, \epsilon, \eta) = $$

$$ \int dx \left\{ \left[ \int_{u(T)=x-\lambda} D^+[u(t),\epsilon]\Psi_0(u(0)+\lambda)\sqrt{|u(T)|} \times \right. \right. \\ \exp\left\{ -\frac{m}{\epsilon} \int_0^T dt \left[ \{u(t)+\lambda\}_\eta + \eta g(t, u(T)+\lambda, u(0)+\lambda, \lambda) \right]^2 \right\} \times \quad (2.2.2.85.b) \\ \left. \left. \sin\left[ \frac{1}{\epsilon} \left( \int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{u}_\epsilon(t)}{dt} \right)^2 - V(u_\epsilon(t)+\lambda) \right] \right) \right] \right]^2 = \right. \\ \left. \int dx \Pi_2^2(T, x_0, \lambda, \epsilon, \epsilon, \eta). \right\} $$
Here

$$ \Pi_1(T, x_0, \lambda, \epsilon, \varepsilon, \eta) = $$

$$ \overline{\int_{u(T)=x-\lambda} D^+[u(t), \varepsilon]\Psi_0(u(0) + \lambda) \sqrt{|u(T)|} \times} $$

$$ \exp\left\{-\frac{m}{\epsilon}\int_{0}^{T}dt\left[\{u(t)+\lambda\}_{\eta}+\eta g(t,u(T)+\lambda,u(0)+\lambda,\lambda)\right]^2\right\} \times \quad (2.2.2.86.a) $$

$$ \cos\left[\frac{1}{\epsilon}\left(\int_{0}^{T}dt\left[\frac{m}{2}\left(\frac{d\tilde{u}_{\varepsilon}(t)}{dt}\right)^{2}-V(u_{\varepsilon}(t)+\lambda)\right]\right)\right]. $$

and

$$ \Pi_2(T, x_0, \lambda, \epsilon, \varepsilon, \eta) = $$

$$ \overline{\int_{u(T)=x-\lambda} D^+[u(t), \varepsilon]\Psi_0(u(0) + \lambda) \sqrt{|u(T)|} \times} $$

$$ \exp\left\{-\frac{m}{\epsilon}\int_{0}^{T}dt\left[\{u(t)+\lambda\}_{\eta}+\eta g(t,u(T)+\lambda,u(0)+\lambda,\lambda)\right]^2\right\} \times \quad (2.2.2.86.b) $$

$$ \sin\left[\frac{1}{\epsilon}\left(\int_{0}^{T}dt\left[\frac{m}{2}\left(\frac{d\tilde{u}_{\epsilon}(t)}{dt}\right)^{2}-V(u_{\epsilon}(t)+\lambda)\right]\right)\right]. $$

Substitution Eq.(2.2.2.84) into Eq.(2.2.2.86.a,b) gives
Here

$$
\begin{align}
\Pi_1(T, x_0, \lambda, \epsilon, \epsilon, \eta) = \nonumber \\
& \quad \overline{\int_{u(T)=x-\lambda} D^+[u(t), \epsilon]\Psi_0(u(0) + \lambda) \sqrt{|u(T)|}} \times \nonumber \\
& \quad \exp\left\{-\frac{m}{\epsilon} \int_0^T dt \left[\{u(t) + \lambda\}_\eta + \eta g(t, u(T) + \lambda, u(0) + \lambda, \lambda)\right]^2\right\} \times \tag{2.2.2.86.a} \\
& \quad \cos\left[\frac{1}{\epsilon} \left(\int_0^T dt \left[\frac{m}{2}\left(\frac{d\tilde{u}_\epsilon(t)}{dt}\right)^2 - V(u_\epsilon(t) + \lambda)\right]\right)\right]. \nonumber
\end{align}
$$

and

$$
\begin{align}
\Pi_2(T, x_0, \lambda, \epsilon, \epsilon, \eta) = & \nonumber \\
& \overline{\int_{u(T)=x-\lambda} D^+[u(t), \epsilon]\Psi_0(u(0) + \lambda) \sqrt{|u(T)|}} \times \nonumber \\
& \exp\left\{-\frac{m}{\epsilon} \int_0^T dt \left[\{u(t) + \lambda\}_\eta + \eta g(t, u(T) + \lambda, u(0) + \lambda, \lambda)\right]^2\right\} \times \tag{2.2.2.86.b} \\
& \sin\left[\frac{1}{\epsilon} \left(\int_0^T dt \left[\frac{m}{2}\left(\frac{d\tilde{u}_\epsilon(t)}{dt}\right)^2 - V(u_\epsilon(t) + \lambda)\right]\right)\right]. \nonumber
\end{align}
$$

Substitution Eq.(2.2.2.84) into Eq.(2.2.2.86.a,b) gives
$$ (\Pi_1(T, x, x_0, \lambda, \epsilon, \eta))_\epsilon = $$

$$ \overline{\int_{u(T)=x-\lambda} D^+[u(t), \epsilon]\Psi_0(u(0) + \lambda) \sqrt{|u(T)|} \times} $$

$$ \exp\left\{-\frac{m}{\epsilon}\int_{0}^{T} dt \left[ \{u(t) + \lambda\}_{\eta} + \eta g(t, u(T) + \lambda, u(0) + \lambda, \lambda) \right]^2\right\} \times $$

$$ \cos\left[\frac{1}{\epsilon}\left(\int_{0}^{T} dt \left[\frac{m}{2}\left(\frac{d\tilde{u}_{\epsilon}(t)}{dt}\right)^2 - V_0(u_{\epsilon}(t), t, \lambda) - V_1(u_{\epsilon}(t), t, \lambda)\right]\right)\right] = $$

$$ \overline{\int_{u(T)=x-\lambda} \tilde{D}_{\eta}^{+}[u(t), \epsilon]\Psi_0(u(0) + \lambda) \sqrt{|u(T)|} \times} $$

$$ \cos\left[\frac{1}{\epsilon}\left(\int_{0}^{T} dt \left[\frac{m}{2}\left(\frac{d\tilde{u}_{\epsilon}(t)}{dt}\right)^2 - V_0(u_{\epsilon}(t), t, \lambda)\right]\right)\right] \times \quad (2.2.2.87.a) $$

$$ \cos\left[\frac{1}{\epsilon}\left(\int_{0}^{T} V_{1}(u_{\epsilon}(t), t, \lambda) dt\right)\right] + $$

$$ \overline{\int_{u(T)=x-\lambda} \tilde{D}_{\eta}^{+}[u(t), \epsilon]\Psi_0(q(0) + \lambda) \sqrt{|u(T)|} \times} $$

$$ \sin\left[\frac{1}{\epsilon}\left(\int_{0}^{T} dt \left[\frac{m}{2}\left(\frac{d\tilde{u}_{\epsilon}(t)}{dt}\right)^2 - V_0(u_{\epsilon}(t), t, \lambda)\right]\right)\right] \times $$

$$ \sin\left[-\frac{1}{\epsilon}\left(\int_{0}^{T} V_{1}(u_{\epsilon}(t), t, \lambda) dt\right)\right] = $$

$$ v_{1,\epsilon}(T, x, x_0, \lambda, \epsilon, \eta) + v_{1,\epsilon}(T, x, x_0, \lambda, \epsilon, \eta). $$
$$ (\Pi_1(T, x, x_0, \lambda, \epsilon, \eta))_{\epsilon} = $$

$$ \overline{\int_{u(T)=x-\lambda} D^+[u(t), \epsilon]\Psi_0(u(0) + \lambda) \sqrt{|u(T)|} \times} $$

$$ \exp\left\{-\frac{m}{\epsilon}\int_{0}^{T} dt \left[ \{u(t) + \lambda\}_{\eta} + \eta g(t, u(T) + \lambda, u(0) + \lambda, \lambda) \right]^2\right\} \times $$

$$ \cos\left[ \frac{1}{\epsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{u}_{\epsilon}(t)}{dt} \right)^2 - V_0(u_{\epsilon}(t), t, \lambda) - V_1(u_{\epsilon}(t), t, \lambda) \right] \right) \right] = $$

$$ \overline{\int_{u(T)=x-\lambda} \tilde{D}_{\eta}^{+}[u(t), \epsilon]\Psi_0(u(0) + \lambda) \sqrt{|u(T)|} \times} $$

$$ \cos\left[ \frac{1}{\epsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{u}_{\epsilon}(t)}{dt} \right)^2 - V_0(u_{\epsilon}(t), t, \lambda) \right] \right) \right] \times \quad (2.2.2.87.a) $$

$$ \cos\left[ \frac{1}{\epsilon} \left( \int_{0}^{T} V_{1}(u_{\epsilon}(t), t, \lambda) dt \right) \right] + $$

$$ \overline{\int_{u(T)=x-\lambda} \tilde{D}_{\eta}^{+}[u(t), \epsilon]\Psi_0(q(0) + \lambda) \sqrt{|u(T)|} \times} $$

$$ \sin\left[ \frac{1}{\epsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{u}_{\epsilon}(t)}{dt} \right)^2 - V_0(u_{\epsilon}(t), t, \lambda) \right] \right) \right] \times $$

$$ \sin\left[ -\frac{1}{\epsilon} \left( \int_{0}^{T} V_{1}(u_{\epsilon}(t), t, \lambda) dt \right) \right] = $$

$$ v_{1,\epsilon}(T, x, x_0, \lambda, \epsilon, \eta) + v_{1,\epsilon}(T, x, x_0, \lambda, \epsilon, \eta). $$
Here
Here
$$v_{1,\epsilon}(T, x, x_0, \lambda, \epsilon, \eta) =$$

$$\overline{\int_{u(T)=x-\lambda} \tilde{D}_{\eta}^{+}[u(t), \epsilon] \Psi_{0}(u(0)+\lambda) \sqrt{|u(T)|} \times}$$

$$\cos\left[ \frac{1}{\epsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{u}_{\epsilon}(t)}{dt} \right)^2 - V_0(u_{\epsilon}(t), t, \lambda) \right] \right) \right] \times$$

$$\cos\left[ \frac{1}{\epsilon} \left( \int_{0}^{T} V_{1}(u_{\epsilon}(t), t, \lambda) dt \right) \right] =$$

$$\bar{\int} dy \left\{ \overline{\int_{u(T)=x-\lambda, u(0)=y-\lambda} \tilde{D}_{\eta}^{+}[u(t), \epsilon] \Psi_{0}(u(0)+\lambda) \sqrt{|u(T)|} \times} \right.$$

$$\sin\left[ \frac{1}{\epsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{u}_{\epsilon}(t)}{dt} \right)^2 - V_0(u_{\epsilon}(t), t, \lambda) \right] \right) \right] \times$$

(2.2.2.88.a)

$$\cos\left[ \frac{1}{\epsilon} \left( \int_{0}^{T} V_{1}(u_{\epsilon}(t), t, \lambda) dt \right) \right] =$$

$$\bar{\int} dy \left\{ \overline{\int_{u(T)=x-\lambda, u(0)=y-\lambda} D^{+}[u(t), \epsilon] \Psi_{0}(y) \sqrt{|u(T)|} \times} \right.$$

$$\exp\left\{ -\frac{m}{\epsilon} \int_{0}^{T} dt \left[ \{u(t) + \lambda\}_{\eta} + \eta g(t, u(T) + \lambda, u(0) + \lambda, \lambda) \right]^2 \right\} \times$$

$$\cos\left[ \frac{1}{\epsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{u}_{\epsilon}(t)}{dt} \right)^2 - V_0(u_{\epsilon}(t), t, \lambda) \right] \right) \right] \times$$

$$\cos\left[ \frac{1}{\epsilon} \left( \int_{0}^{T} V_{1}(u_{\epsilon}(t), t, \lambda) dt \right) \right]$$
$$v_{1,\epsilon}(T, x, x_0, \lambda, \epsilon, \eta) =$$

$$\overline{\int_{u(T)=x-\lambda} \tilde{D}_{\eta}^{+}[u(t), \epsilon] \Psi_{0}(u(0)+\lambda) \sqrt{|u(T)|} \times}$$

$$\cos\left[ \frac{1}{\epsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{u}_{\epsilon}(t)}{dt} \right)^2 - V_0(u_{\epsilon}(t), t, \lambda) \right] \right) \right] \times$$

$$\cos\left[ \frac{1}{\epsilon} \left( \int_{0}^{T} V_{1}(u_{\epsilon}(t), t, \lambda) dt \right) \right] =$$

$$\bar{\int} dy \left\{ \overline{\int_{u(T)=x-\lambda, u(0)=y-\lambda} \tilde{D}_{\eta}^{+}[u(t), \epsilon] \Psi_{0}(u(0)+\lambda) \sqrt{|u(T)|} \times} \right.$

$$\sin\left[ \frac{1}{\epsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{u}_{\epsilon}(t)}{dt} \right)^2 - V_0(u_{\epsilon}(t), t, \lambda) \right] \right) \right] \times$$

(2.2.2.88.a)

$$\cos\left[ \frac{1}{\epsilon} \left( \int_{0}^{T} V_{1}(u_{\epsilon}(t), t, \lambda) dt \right) \right] =$$

$$\bar{\int} dy \left\{ \overline{\int_{u(T)=x-\lambda, u(0)=y-\lambda} D^{+}[u(t), \epsilon] \Psi_{0}(y) \sqrt{|u(T)|} \times} \right.$

$$\exp\left\{ -\frac{m}{\epsilon} \int_{0}^{T} dt \left[ \{u(t) + \lambda\}_{\eta} + \eta g(t, u(T) + \lambda, u(0) + \lambda, \lambda) \right]^2 \right\} \times$$

$$\cos\left[ \frac{1}{\epsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{u}_{\epsilon}(t)}{dt} \right)^2 - V_0(u_{\epsilon}(t), t, \lambda) \right] \right) \right] \times$$

$$\cos\left[ \frac{1}{\epsilon} \left( \int_{0}^{T} V_{1}(u_{\epsilon}(t), t, \lambda) dt \right) \right]$$
and
and
$$v_{2,\varepsilon}(T, x, x_0, \lambda, \varepsilon, \eta) =$$

$$\overline{\int_{u(T)=x-\lambda} \tilde{D}_{\eta}^{+}[u(t), \varepsilon] \Psi_{0}(u(0)+\lambda) \sqrt{|u(T)|} \times}$$

$$\sin\left[\frac{1}{\epsilon}\left(\int_{0}^{T} dt\left[\frac{m}{2}\left(\frac{d\tilde{u}_{\varepsilon}(t)}{dt}\right)^{2}-V_{0}(u_{\varepsilon}(t),t,\lambda)\right]\right)\right] \times$$

$$\sin\left[-\frac{1}{\epsilon}\left(\int_{0}^{T} V_{1}(u_{\varepsilon}(t),t,\lambda)dt\right)\right] =$$

$$\overline{\int dy \left\{ \int_{u(T)=x-\lambda, u(0)=y-\lambda} \tilde{D}_{\eta}^{+}[u(t), \varepsilon] \Psi_{0}(u(0)+\lambda) \sqrt{|u(T)|} \times \right.}$$

$$\sin\left[\frac{1}{\epsilon}\left(\int_{0}^{T} dt\left[\frac{m}{2}\left(\frac{d\tilde{u}_{\varepsilon}(t)}{dt}\right)^{2}-V_{0}(u_{\varepsilon}(t),t,\lambda)\right]\right)\right] \times \qquad (2.2.2.88.b)$$

$$\left.\sin\left[-\frac{1}{\epsilon}\left(\int_{0}^{T} V_{1}(u_{\varepsilon}(t),t,\lambda)dt\right)\right]\right\} =$$

$$\overline{\int dy \left\{ \int_{u(T)=x-\lambda, u(0)=y-\lambda} D^{+}[u(t), \varepsilon] \Psi_{0}(y) \sqrt{|u(T)|} \times \right.}$$

$$\exp\left\{-\frac{m}{\epsilon}\int_{0}^{T} dt\left[\{u(t)+\lambda\}_{\eta}+\eta g(t, u(T)+\lambda, u(0)+\lambda, \lambda)\right]^{2}\right\} \times$$

$$\sin\left[\frac{1}{\epsilon}\left(\int_{0}^{T} dt\left[\frac{m}{2}\left(\frac{d\tilde{u}_{\varepsilon}(t)}{dt}\right)^{2}-V_{0}(u_{\varepsilon}(t),t,\lambda)\right]\right)\right] \times$$

$$\left.\sin\left[-\frac{1}{\epsilon}\left(\int_{0}^{T} V_{1}(u_{\varepsilon}(t),t,\lambda)dt\right)\right]\right\}.$$
$$v_{2,\epsilon}(T, x, x_0, \lambda, \epsilon, \eta) =$$

$$
\begin{align*}
& \overline{\int_{u(T)=x-\lambda} \tilde{D}_{\eta}^{+}[u(t), \epsilon] \Psi_{0}(u(0) + \lambda) \sqrt{|u(T)|} \times} \\
& \sin\left[ \frac{1}{\epsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{u}_{\epsilon}(t)}{dt} \right)^2 - V_0(u_{\epsilon}(t), t, \lambda) \right] \right) \right] \times \\
& \sin\left[ -\frac{1}{\epsilon} \left( \int_{0}^{T} V_1(u_{\epsilon}(t), t, \lambda) dt \right) \right] = \\
& \overline{\int dy \left\{ \int_{u(T)=x-\lambda, u(0)=y-\lambda} \tilde{D}_{\eta}^{+}[u(t), \epsilon] \Psi_{0}(u(0) + \lambda) \sqrt{|u(T)|} \times} \\
& \sin\left[ \frac{1}{\epsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{u}_{\epsilon}(t)}{dt} \right)^2 - V_0(u_{\epsilon}(t), t, \lambda) \right] \right) \right] \times \tag{2.2.2.88.b} \\
& \sin\left[ -\frac{1}{\epsilon} \left( \int_{0}^{T} V_1(u_{\epsilon}(t), t, \lambda) dt \right) \right] = \\
& \overline{\int dy \left\{ \int_{u(T)=x-\lambda, u(0)=y-\lambda} D^{+}[u(t), \epsilon] \Psi_{0}(y) \sqrt{|u(T)|} \times} \\
& \exp\left\{ -\frac{m}{\epsilon} \int_{0}^{T} dt \left[ \{u(t) + \lambda\}_{\eta} + \eta g(t, u(T) + \lambda, u(0) + \lambda, \lambda) \right]^2 \right\} \times \\
& \sin\left[ \frac{1}{\epsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{d\tilde{u}_{\epsilon}(t)}{dt} \right)^2 - V_0(u_{\epsilon}(t), t, \lambda) \right] \right) \right] \times \\
& \sin\left[ -\frac{1}{\epsilon} \left( \int_{0}^{T} V_1(u_{\epsilon}(t), t, \lambda) dt \right) \right].
\end{align*}
$$
Let us now evaluate a quantities $v_{1,\epsilon}(T, x_0, \lambda, \epsilon, \epsilon, \eta)$ and $v_{2,\epsilon}(T, x_0, \lambda, \epsilon, \epsilon, \eta)$. By using replacement $u(t) \rightarrow \sqrt{\epsilon} u(t)$, from Eq.(2.2.2.88.a,b) we obtain
Let us now evaluate a quantity $v_{1,\epsilon}(T, x_0, \lambda, \epsilon, \epsilon, \eta)$ and $v_{2,\epsilon}(T, x_0, \lambda, \epsilon, \epsilon, \eta)$. By using replacement $u(t) \rightarrow \sqrt{\epsilon} u(t)$, from Eq.(2.2.2.88.a,b) we obtain
$$v_{1,\epsilon}(T, x, x_0, \lambda, \epsilon, \eta) =$$

$$ \begin{aligned} & \int dy \left\{ \overline{\int_{u(T)=\frac{x-\lambda}{\sqrt{\epsilon}}, u(0)=\frac{y-\lambda}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[u(t), \epsilon] \Psi_{0}(y) \sqrt{\sqrt{\epsilon} |u(T)|} \times \right. \\ & \cos \left[ \frac{1}{\epsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{\sqrt{\epsilon} d\tilde{u}_{\epsilon}(t)}{dt} \right)^2 - V_0(\sqrt{\epsilon} u_{\epsilon}(t), t, \lambda) \right] \right) \times \right. \\ & \left. \cos \left[ \frac{1}{\epsilon} \left( \int_{0}^{T} \hat{V}_1(\sqrt{\epsilon} u_{\epsilon}(t), t, \lambda) dt \right) \right] \right\} = \\ & \int dy \left\{ \overline{\int_{u(T)=\frac{x-\lambda}{\sqrt{\epsilon}}, u(0)=\frac{y-\lambda}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[u(t), \epsilon] \Psi_{0}(y) \sqrt{\sqrt{\epsilon} |u(T)|} \times \right. \\ & \cos \left[ \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{\sqrt{\epsilon} d\tilde{u}_{\epsilon}(t)}{dt} \right)^2 - \frac{1}{\epsilon} V_0(\sqrt{\epsilon} u_{\epsilon}(t), t, \lambda) \right) \right] \times \right. \\ & \left. \cos \left[ \frac{1}{\epsilon} \left( \int_{0}^{T} \hat{V}_1(\sqrt{\epsilon} u_{\epsilon}(t), t, \lambda) dt \right) \right] \right\} = \\ & \int dy \left\{ \overline{\int_{u(T)=\frac{x-\lambda}{\sqrt{\epsilon}}, u(0)=\frac{y-\lambda}{\sqrt{\epsilon}}} D^{+}[u(t), \epsilon] \Psi_{0}(y) \sqrt{\sqrt{\epsilon} |u(T)|} \times \right. \\ & \exp \left\{ -\frac{m}{\epsilon} \int_{0}^{T} dt \left[ \{u(t) + \lambda\}_{\eta} + \eta g(t, \sqrt{\epsilon} u(T) + \lambda, \sqrt{\epsilon} u(0) + \lambda, \lambda) \right]^2 \right\} \times \\ & \cos \left[ \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{\sqrt{\epsilon} d\tilde{u}_{\epsilon}(t)}{dt} \right)^2 - \frac{1}{\epsilon} V_0(\sqrt{\epsilon} u_{\epsilon}(t), t, \lambda) \right) \right] \times \end{aligned} \quad (2.2.2.89.a) $$

$$ \int dy \left\{ \overline{\int_{u(T)=\frac{x-\lambda}{\sqrt{\epsilon}}, u(0)=\frac{y-\lambda}{\sqrt{\epsilon}}} D^{+}[u(t), \epsilon] \Psi_{0}(y) \sqrt{\sqrt{\epsilon} |u(T)|} \times \right. \\ \exp \left\{ -\frac{m}{\epsilon} \int_{0}^{T} dt \left[ \{u(t) + \lambda\}_{\eta} + \eta g(t, \sqrt{\epsilon} u(T) + \lambda, \sqrt{\epsilon} u(0) + \lambda, \lambda) \right]^2 \right\} \times \\ \cos \left[ \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{\sqrt{\epsilon} d\tilde{u}_{\epsilon}(t)}{dt} \right)^2 - \frac{1}{\epsilon} V_0(\sqrt{\epsilon} u_{\epsilon}(t), t, \lambda) \right) \right] \times $$
$$v_{1,\epsilon}(T, x, x_0, \lambda, \epsilon, \eta) =$$

$$ \begin{aligned} & \int dy \left\{ \overline{\int_{u(T)=\frac{x-\lambda}{\sqrt{\epsilon}}, u(0)=\frac{y-\lambda}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[u(t), \epsilon] \Psi_{0}(y) \sqrt{\sqrt{\epsilon}|u(T)|} \times \right. \\ & \cos \left[ \frac{1}{\epsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{\sqrt{\epsilon} d\tilde{u}_{\epsilon}(t)}{dt} \right)^2 - V_0(\sqrt{\epsilon} u_{\epsilon}(t), t, \lambda) \right] \right) \times \right. \\ & \left. \cos \left[ \frac{1}{\epsilon} \left( \int_{0}^{T} \hat{V}_1(\sqrt{\epsilon} u_{\epsilon}(t), t, \lambda) dt \right) \right] \right\} = \\ & \int dy \left\{ \overline{\int_{u(T)=\frac{x-\lambda}{\sqrt{\epsilon}}, u(0)=\frac{y-\lambda}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[u(t), \epsilon] \Psi_{0}(y) \sqrt{\sqrt{\epsilon}|u(T)|} \times \right. \\ & \cos \left[ \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{\sqrt{\epsilon} d\tilde{u}_{\epsilon}(t)}{dt} \right)^2 - \frac{1}{\epsilon} V_0(\sqrt{\epsilon} u_{\epsilon}(t), t, \lambda) \right) \right] \times \right. \\ & \left. \cos \left[ \frac{1}{\epsilon} \left( \int_{0}^{T} \hat{V}_1(\sqrt{\epsilon} u_{\epsilon}(t), t, \lambda) dt \right) \right] \right\} = \\ & \int dy \left\{ \overline{\int_{u(T)=\frac{x-\lambda}{\sqrt{\epsilon}}, u(0)=\frac{y-\lambda}{\sqrt{\epsilon}}} D^{+}[u(t), \epsilon] \Psi_{0}(y) \sqrt{\sqrt{\epsilon}|u(T)|} \times \right. \\ & \exp \left\{ -\frac{m}{\epsilon} \int_{0}^{T} dt \left[ \{u(t) + \lambda\}_{\eta} + \eta g(t, \sqrt{\epsilon} u(T) + \lambda, \sqrt{\epsilon} u(0) + \lambda, \lambda) \right]^2 \right\} \times \\ & \cos \left[ \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{\sqrt{\epsilon} d\tilde{u}_{\epsilon}(t)}{dt} \right)^2 - \frac{1}{\epsilon} V_0(\sqrt{\epsilon} u_{\epsilon}(t), t, \lambda) \right) \right] \times \right. \end{aligned} \quad (2.2.2.89.a) $$

$$ \int dy \left\{ \overline{\int_{u(T)=\frac{x-\lambda}{\sqrt{\epsilon}}, u(0)=\frac{y-\lambda}{\sqrt{\epsilon}}} D^{+}[u(t), \epsilon] \Psi_{0}(y) \sqrt{\sqrt{\epsilon}|u(T)|} \times \right. \\ \exp \left\{ -\frac{m}{\epsilon} \int_{0}^{T} dt \left[ \{u(t) + \lambda\}_{\eta} + \eta g(t, \sqrt{\epsilon} u(T) + \lambda, \sqrt{\epsilon} u(0) + \lambda, \lambda) \right]^2 \right\} \times \\ \cos \left[ \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{\sqrt{\epsilon} d\tilde{u}_{\epsilon}(t)}{dt} \right)^2 - \frac{1}{\epsilon} V_0(\sqrt{\epsilon} u_{\epsilon}(t), t, \lambda) \right) \right] \times $$
and
and
$v_{2,\epsilon}(T, x, x_0, \lambda, \epsilon, \eta) =$

$$
\begin{aligned}
& \int dy \left\{ \overline{\int_{u(T)=\frac{x-\lambda}{\sqrt{\epsilon}}, u(0)=\frac{y-\lambda}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[u(t), \epsilon] \Psi_{0}(y) \sqrt{\sqrt{\epsilon} |u(T)|} \times \right. \\
& \qquad \left. \sin \left[ \frac{1}{\epsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{\sqrt{\epsilon} d\tilde{u}_{\epsilon}(t)}{dt} \right)^2 - V_0(\sqrt{\epsilon} u_{\epsilon}(t), t, \lambda) \right] \right) \right] \times \right. \\
& \qquad \left. \sin \left[ -\frac{1}{\epsilon} \left( \int_{0}^{T} \hat{V}_1(\sqrt{\epsilon} u_{\epsilon}(t), t, \lambda) dt \right) \right] \right\} = \\
& \int dy \left\{ \overline{\int_{u(T)=\frac{x-\lambda}{\sqrt{\epsilon}}, u(0)=\frac{y-\lambda}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[u(t), \epsilon] \Psi_{0}(y) \sqrt{\sqrt{\epsilon} |u(T)|} \times \right. \\
& \qquad \left. \sin \left[ \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{\sqrt{\epsilon} d\tilde{u}_{\epsilon}(t)}{dt} \right)^2 - \frac{1}{\epsilon} V_0(\sqrt{\epsilon} u_{\epsilon}(t), t, \lambda) \right) \right] \right] \right\} \times \quad (2.2.2.89.b) \\
& \sin \left[ \left( -\int_{0}^{T} \hat{V}_1(\sqrt{\epsilon} u_{\epsilon}(t), t, \lambda) dt \right) \right] = \\
& \int dy \left\{ \overline{\int_{u(T)=\frac{x-\lambda}{\sqrt{\epsilon}}, u(0)=\frac{y-\lambda}{\sqrt{\epsilon}}} D^{+}[u(t), \epsilon] \Psi_{0}(y) \sqrt{\sqrt{\epsilon} |u(T)|} \times \right. \\
& \qquad \left. \exp \left\{ -\frac{m}{\epsilon} \int_{0}^{T} dt \left[ \{u(t) + \lambda\}_{\eta} + \eta g(t, \sqrt{\epsilon} u(T) + \lambda, \sqrt{\epsilon} u(0) + \lambda, \lambda) \right]^2 \right\} \times \right. \\
& \qquad \left. \sin \left[ \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{\sqrt{\epsilon} d\tilde{u}_{\epsilon}(t)}{dt} \right)^2 - \frac{1}{\epsilon} V_0(\sqrt{\epsilon} u_{\epsilon}(t), t, \lambda) \right) \right] \right] \right\} \times
\end{aligned}
$$

$$
\sin\left[\left(-\int_{0}^{T} \hat{V}_{1}(\sqrt{\epsilon} u_{v(\epsilon, \epsilon)}(t), t, \lambda) dt\right)\right] =
$$
$$v_{2,\epsilon}(T, x, x_0, \lambda, \epsilon, \eta) =$$

$$ \begin{aligned} & \int dy \left\{ \overline{\int_{u(T)=\frac{x-\lambda}{\sqrt{\epsilon}}, u(0)=\frac{y-\lambda}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[u(t), \epsilon] \Psi_{0}(y) \sqrt{\sqrt{\epsilon} |u(T)|} \times \right. \\ & \left. \sin \left[ \frac{1}{\epsilon} \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{\sqrt{\epsilon} d\tilde{u}_{\epsilon}(t)}{dt} \right)^2 - V_0(\sqrt{\epsilon} u_{\epsilon}(t), t, \lambda) \right] \right) \right] \times \right. \\ & \left. \sin \left[ -\frac{1}{\epsilon} \left( \int_{0}^{T} \hat{V}_1(\sqrt{\epsilon} u_{\epsilon}(t), t, \lambda) dt \right) \right] \right\} = \\ & \int dy \left\{ \overline{\int_{u(T)=\frac{x-\lambda}{\sqrt{\epsilon}}, u(0)=\frac{y-\lambda}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[u(t), \epsilon] \Psi_{0}(y) \sqrt{\sqrt{\epsilon} |u(T)|} \times \right. \\ & \left. \sin \left[ \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{\sqrt{\epsilon} d\tilde{u}_{\epsilon}(t)}{dt} \right)^2 - \frac{1}{\epsilon} V_0(\sqrt{\epsilon} u_{\epsilon}(t), t, \lambda) \right) \right] \right] \right\} \times \right. \\ & \left. \sin \left[ \left( -\int_{0}^{T} \hat{V}_1(\sqrt{\epsilon} u_{\epsilon}(t), t, \lambda) dt \right) \right] \right\} = \\ & \int dy \left\{ \overline{\int_{u(T)=\frac{x-\lambda}{\sqrt{\epsilon}}, u(0)=\frac{y-\lambda}{\sqrt{\epsilon}}} D^{+}[u(t), \epsilon] \Psi_{0}(y) \sqrt{\sqrt{\epsilon} |u(T)|} \times \right. \\ & \left. \exp \left\{ -\frac{m}{\epsilon} \int_{0}^{T} dt \left[ \{u(t) + \lambda\}_{\eta} + \eta g(t, \sqrt{\epsilon} u(T) + \lambda, \sqrt{\epsilon} u(0) + \lambda, \lambda) \right]^2 \right\} \times \right. \\ & \left. \sin \left[ \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{\sqrt{\epsilon} d\tilde{u}_{\epsilon}(t)}{dt} \right)^2 - \frac{1}{\epsilon} V_0(\sqrt{\epsilon} u_{\epsilon}(t), t, \lambda) \right) \right] \right] \right\} \times \end{aligned} \quad (2.2.2.89.b) $$

$$ \begin{aligned} & \sin \left[ \left( -\int_{0}^{T} \hat{V}_{1}(\sqrt{\epsilon} u_{\epsilon}(t), t, \lambda) dt \right) \right] = \\ & \int dy \left\{ \overline{\int_{u(T)=\frac{x-\lambda}{\sqrt{\epsilon}}, u(0)=\frac{y-\lambda}{\sqrt{\epsilon}}} D^{+}[u(t), \epsilon] \Psi_{0}(y) \sqrt{\sqrt{\epsilon} |u(T)|} \times \right. \\ & \left. \exp \left\{ -\frac{m}{\epsilon} \int_{0}^{T} dt \left[ \{u(t) + \lambda\}_{\eta} + \eta g(t, \sqrt{\epsilon} u(T) + \lambda, \sqrt{\epsilon} u(0) + \lambda, \lambda) \right]^2 \right\} \times \right. \\ & \left. \sin \left[ \left( \int_{0}^{T} dt \left[ \frac{m}{2} \left( \frac{\sqrt{\epsilon} d\tilde{u}_{\epsilon}(t)}{dt} \right)^2 - \frac{1}{\epsilon} V_{0}(\sqrt{\epsilon} u_{\epsilon}(t), t, \lambda) \right) \right] \right] \right\} \times \end{aligned} $$
From Eqs.(2.2.2.84) and Eqs.(2.2.2.89.a,b) we obtain

$$
\begin{align*}
& v_{1,\epsilon}(T, x, x_0, \lambda, \epsilon, \eta) = \\
& \int dy \left\{ \overline{\int_{u(T)=\frac{x-\lambda}{\sqrt{\epsilon}}, u(0)=\frac{y-\lambda}{\sqrt{\epsilon}}} \tilde{D}_{\eta}^{+}[u(t), \epsilon] \Psi_0(y) \sqrt{\sqrt{\epsilon} |u(T)|} \times \right. \\
& \cos \left[ \left( \int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{u}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t, \lambda) u_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t, \lambda) \frac{u_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \times \right. \\
& \left. \cos \left[ \left( \sqrt{\epsilon} \int_0^T \tilde{V}_1(u_{v(\epsilon,\epsilon)}(t), t, \lambda) dt \right) \right] \right\} = \\
& \int dy \left\{ \overline{\int_{u(T)=\frac{x-\lambda}{\sqrt{\epsilon}}, u(0)=\frac{y-\lambda}{\sqrt{\epsilon}}} D^+[u(t), \epsilon] \Psi_0(y) \sqrt{\sqrt{\epsilon} |u(T)|} \times \right. \\
& \exp \left\{ -\frac{m}{\epsilon} \int_0^T dt \left[ \{u(t) + \lambda\}_\eta + \eta g(t, \sqrt{\epsilon} u(T) + \lambda, \sqrt{\epsilon} u(0) + \lambda, \lambda) \right]^2 \right\} \times \\
& \cos \left[ \left( \int_0^T dt \left[ \frac{m}{2} \left( \frac{d\tilde{u}_{\epsilon\epsilon}(t)}{dt} \right)^2 - a_{\epsilon}(t, \lambda) u_{v(\epsilon,\epsilon)}^2(t) + b_{\epsilon}(t, \lambda) \frac{u_{v(\epsilon,\epsilon)}(t)}{\sqrt{\epsilon}} \right] \right) \times \right. \\
& \left. \cos \left[ \left( \sqrt{\epsilon} \int_0^T \tilde{V}_1(u_{v(\epsilon,\epsilon)}(t), t, \lambda) dt \right) \right] \right\} \tag{2.2.2.90.a}
\end{align*}
$$

and