# Exact quasiclassical asymptotics beyond Maslov canonical operator

jaykov foukzon, Stanislav Podosenov, Alexander Potapov

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jaykov foukzon, Stanislav Podosenov, Alexander Potapov. Exact quasiclassical asymptotics beyond Maslov canonical operator. 2024. hal-04584026

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# Exact quasiclassical asymptotics beyond Maslov canonical operator.

J.Foukzon<fnref n="1" />, A.A.Potapov<fnref n="2" />, S.A. Podosenov<fnref n="3" />, .

**Abstract:** The main purpose of this paper is to calculate exact quasiclassical asymptotic of the quantum averages without any reference to the corresponding quasiclassical asymptotic of the Schrödinger wave function $\Psi(x,t,x_0;\hbar)$ given via Maslov canonical operator $K_{\Lambda^n(t)}^{1/\hbar}[\cdot]$. We suggest a new asymptotic representation for the quantum averages with position variable with localized initial data.

## Content

### I.0.Introduction.

**I.0.1.** Colombeau solutions of the Schrödinger equation using Maslov canonical operator.

**I.0.2.** The main result.

**I.0.3.** Colombeau-Wiener measure $(\tilde{D}_x^\mathbb{R}[\omega(\tau), v_\varepsilon])_\varepsilon$.

**I.0.4.** Regularized Colombeau-Feynman Path Integral $(J_{\varepsilon, \nu}^{x,t}[\mathcal{L}_\varepsilon(\dot{\omega}_\varepsilon(\tau), \omega(\tau)), \psi_\varepsilon(\omega(\tau))])_\varepsilon$ based on Colombeau-Wiener measure $(\tilde{D}_x^\mathbb{R}[\omega(\tau), v_\varepsilon])_\varepsilon$.

**I.0.5.** Regularized Colombeau-Feynman Path Integral $(J_{\varepsilon, \nu}^{x,t}[\mathcal{L}_\varepsilon(\dot{\omega}_\varepsilon(\tau), \omega(\tau)), \psi_\varepsilon(\omega(\tau))])_\varepsilon$ based on Colombeau-Wiener measure $(\tilde{D}_x^\mathbb{R}[\omega(\tau), \varepsilon])_\varepsilon$.

**I.0.6.** Colombeau-Feynman Path Integral based on complex

<footnote n="1">¹Israel Institute of Technology.</footnote>

<footnote n="2">²IRE RAS,</footnote>

<footnote n="3">³All-Russian Scientific-Research Institute
for Optical and Physical Measurements.</footnote>
# Exact quasiclassical asymptotics beyond Maslov canonical operator.

J.Foukzon<fnref n="1" />, A.A.Potapov<fnref n="2" />, S.A. Podosenov<fnref n="3" />, .

**Abstract:** The main purpose of this paper is to calculate exact quasiclassical asymptotic of the quantum averages without any reference to the corresponding quasiclassical asymptotic of the Schrödinger wave function $\Psi(x,t,x_0;\hbar)$ given via Maslov canonical operator $K_{\Lambda^n(t)}^{1/\hbar}[\cdot]$. We suggest a new asymptotic representation for the quantum averages with position variable with localized initial data.

## Content

### I.0.Introduction.

**I.0.1.** Colombeau solutions of the Schrödinger equation using Maslov canonical operator.

**I.0.2.** The main result.

**I.0.3.** Colombeau-Wiener measure $(\tilde{D}_x^\mathbb{R}[\omega(\tau), v_\varepsilon])_\varepsilon$.

**I.0.4.** Regularized Colombeau-Feynman Path Integral $(J_{\varepsilon, \nu}^{x,t}[\mathcal{L}_\varepsilon(\dot{\omega}_\varepsilon(\tau), \omega(\tau)), \psi_\varepsilon(\omega(\tau))])_\varepsilon$ based on Colombeau-Wiener measure $(\tilde{D}_x^\mathbb{R}[\omega(\tau), v_\varepsilon])_\varepsilon$.

**I.0.5.** Regularized Colombeau-Feynman Path Integral $(J_{\varepsilon, \nu}^{x,t}[\mathcal{L}_\varepsilon(\dot{\omega}_\varepsilon(\tau), \omega(\tau)), \psi_\varepsilon(\omega(\tau))])_\varepsilon$ based on Colombeau-Wiener measure $(\tilde{D}_x^\mathbb{R}[\omega(\tau), \varepsilon])_\varepsilon$.

**I.0.6.** Colombeau-Feynman Path Integral based on complex

<footnote n="1">¹Israel Institute of Technology.</footnote>

<footnote n="2">²IRE RAS,</footnote>

<footnote n="3">³All-Russian Scientific-Research Institute
for Optical and Physical Measurements.</footnote>
Colombeau-Wiener measure $\tilde{D}^C[x(\tau), v_\epsilon]$.

I.0.7. Colombeau-Wiener Path Integral and generalized Haba theorem.

I.1. Colombeau generalized functions.

I.1.1. Point Values of Generalized Functions on M and Generalized Numbers.

I.1.2. Colombeau algebra of compactly supported generalized functions and Colombeau algebra of tempered generalized functions.

I.1.3. Algebra of generalized functions based on $\mathcal{L}(\mathbb{R}^n)$. Colombeau algebra $G_{\tau, \mathcal{L}}(\mathbb{R}^n)$.

I.1.4. Regular Colombeau algebras.

I.1.5. Integration in Colombeau theory.

I.1.6. Fourier transform in Colombeau theory.

I.1.7. Topological $\tilde{\mathbb{C}}$-modules.

I.1.8. Completeness and duality theory for topological $\tilde{\mathbb{C}}$-modules.

I.1.9. Definition and basic properties of $G_E$ and $\tilde{G}_B$.

I.1.10. Definition and basic properties of $G_B$ and $\tilde{G}_B$.

I.1.11. Colombeau Sobolev spaces. Definition and basic properties of $(W_p^{l,k,\epsilon}(\Omega))_{\epsilon\in(0,1]}$.

I.2. Pseudodifferential operators with coefficients are distributions.

I.3. The generalized oscillatory integral.

I.4. Pseudodifferential operators with generalized symbol.

I.5. Formal series and symbols.

I.6. Oscillatory integrals with generalized phase function on $\mathbb{R}^n$.

I.7. Generalized symbols and amplitudes of weighted type.

I.8. Pseudodifferential operators acting on $G_{\tau, \mathcal{L}}(\mathbb{R}^n)$.

I.9. Oscillatory integrals of weighted type.

II.1.1. Feynman-Colombeau path integral.

II.1.2. Generalized Maslov theorem.

II.2. Master Equation for the Feynman-Colombeau path integral.

II.2.1. Master Equation for the Feynman-Colombeau path integral corresponding with a Schrödinger equation with a non-Hermitian Hamiltonian.

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

II.3. Quantum anharmonic oscillator with a cubic potential.

II.4. Quantum anharmonic oscillator with a double well potential.

III. Comparizon with a perturbation theory.

III.1. Path-integrals calculation by using saddle-point approximation.

IV. Wave function collapse and weak Colombeau solutions of the Colombeau-Schrödinger equation.

IV.1. Weak Colombeau solutions of the Colombeau-Schrödinger equation.
Colombeau-Wiener measure $\tilde{D}^C[x(\tau), v_\epsilon]$.

I.0.7. Colombeau-Wiener Path Integral and generalized Haba theorem.

I.1. Colombeau generalized functions.

I.1.1. Point Values of Generalized Functions on M and Generalized Numbers.

I.1.2. Colombeau algebra of compactly supported generalized functions and Colombeau algebra of tempered generalized functions.

I.1.3. Algebra of generalized functions based on $\mathcal{L}(\mathbb{R}^n)$. Colombeau algebra $G_{\tau, \mathcal{L}}(\mathbb{R}^n)$.

I.1.4. Regular Colombeau algebras.

I.1.5. Integration in Colombeau theory.

I.1.6. Fourier transform in Colombeau theory.

I.1.7. Topological $\tilde{\mathbb{C}}$-modules.

I.1.8. Completeness and duality theory for topological $\tilde{\mathbb{C}}$-modules.

I.1.9. Definition and basic properties of $G_E$ and $\tilde{G}_B$.

I.1.10. Definition and basic properties of $G_B$ and $\tilde{G}_B$.

I.1.11. Colombeau Sobolev spaces. Definition and basic properties of $(W_p^{l,k,\epsilon}(\Omega))_{\epsilon\in(0,1]}$.

I.2. Pseudodifferential operators with coefficients are distributions.

I.3. The generalized oscillatory integral.

I.4. Pseudodifferential operators with generalized symbol.

I.5. Formal series and symbols.

I.6. Oscillatory integrals with generalized phase function on $\mathbb{R}^n$.

I.7. Generalized symbols and amplitudes of weighted type.

I.8. Pseudodifferential operators acting on $G_{\tau, \mathcal{L}}(\mathbb{R}^n)$.

I.9. Oscillatory integrals of weighted type.

II.1.1. Feynman-Colombeau path integral.

II.1.2. Generalized Maslov theorem.

II.2. Master Equation for the Feynman-Colombeau path integral.

II.2.1. Master Equation for the Feynman-Colombeau path integral corresponding with a Schrödinger equation with a non-Hermitian Hamiltonian.

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

II.3. Quantum anharmonic oscillator with a cubic potential.

II.4. Quantum anharmonic oscillator with a double well potential.

III. Comparizon with a perturbation theory.

III.1. Path-integrals calculation by using saddle-point approximation.

IV. Wave function collapse and weak Colombeau solutions of the Colombeau-Schrödinger equation.

IV.1. Weak Colombeau solutions of the Colombeau-Schrödinger equation.
**V. Schrödinger's cat paradox resolution using GRW collapse model.**

**Appendix.** CAT-A.

**Appendix.** CAT-B.

**Appendix.** CAT-C.

**Apendix I.0. Generalized Colombeau Propagator.**

**Apendix I.1. Derivation of the Feynman-Colombeau path integral for the non-Hermitian generalized Hamiltonian.**

**Apendix I.2.** Colombeau-Schrödinger Equation from Feynman-Colombeau path integral.

**Apendix I.3.** Colombeau-Feynman oscillatory path integral.

**Apendix II.1.1.** Upper and lower infinite-dimensional Feynman subintegrals.

**Apendix II.1.2.** Upper and lower infinite-dimensional Colombeau-Feynman subintegrals.

**Apendix II.2.1.** Colombeau Signed Measures.

**Apendix II.2.2.** Colombeau Submeasures.

**Apendix II.2.3.** Infinite-Dimensional Colombeau-Feynman Signed Measures.

**Apendix III.** Colombeau-Albeverio normed path integral.

**Apendix IV.** The Schrödinger equation with singular potential $(V_\epsilon)_\epsilon$ in Colombeau module $E_M[\mathcal{F}(\mathbb{R}^d)]$.

# I.0.Introduction

## I.0.1. Colombeau solutions of the Schrödinger equation using Maslov canonical operator.

Let $\mathcal{H}$ be a complex infinite dimensional separable Hilbert space, with inner product $\langle \cdot, \cdot \rangle$ and norm $|\cdot|$. Let us consider Schrödinger equation:
**V. Schrödinger's cat paradox resolution using GRW collapse model.**

**Appendix.** CAT-A.

**Appendix.** CAT-B.

**Appendix.** CAT-C.

**Apendix I.0. Generalized Colombeau Propagator.**

**Apendix I.1. Derivation of the Feynman-Colombeau path integral for the non-Hermitian generalized Hamiltonian.**

**Apendix I.2.** Colombeau-Schrödinger Equation from Feynman-Colombeau path integral.

**Apendix I.3.** Colombeau-Feynman oscillatory path integral.

**Apendix II.1.1.** Upper and lower infinite-dimensional Feynman subintegrals.

**Apendix II.1.2.** Upper and lower infinite-dimensional Colombeau-Feynman subintegrals.

**Apendix II.2.1.** Colombeau Signed Measures.

**Apendix II.2.2.** Colombeau Submeasures.

**Apendix II.2.3.** Infinite-Dimensional Colombeau-Feynman Signed Measures.

**Apendix III.** Colombeau-Albeverio normed path integral.

**Apendix IV.** The Schrödinger equation with singular potential $(V_\epsilon)_\epsilon$ in Colombeau module $E_M[\mathcal{F}(\mathbb{R}^d)]$.

# I.0.Introduction

## I.0.1. Colombeau solutions of the Schrödinger equation using Maslov canonical operator.

Let $\mathcal{H}$ be a complex infinite dimensional separable Hilbert space, with inner product $\langle \cdot, \cdot \rangle$ and norm $|\cdot|$. Let us consider Schrödinger equation:
$$ -i\hbar \frac{\partial \Psi(t)}{\partial t} + \hat{H}(t)\Psi(t) = 0, \qquad (I.0.1.1) $$

$$ \Psi(0) = \Psi_0(x), x \in \mathbb{R}^n, $$

where operator $H(t)$ given via formula

$$ H(t) = -\frac{\hbar^2}{2m} \Delta + V(x,t) \qquad (I.0.1.2) $$

and $H$ is essentially self-adjoint, $\hat{H}$ is the closure of $H, \Psi(t) : \mathbb{R} \to \mathbf{H}$ and $\hat{H}(t) : \mathbb{R} \times \mathbf{H} \to \mathbf{H}$.

Maslov local quasiclassical asymptotic of the Green function of the Schrödinger equation (I.0.1.1) with potential $V(x,t)$ is [26]:

$$ G(t,x,y;\hbar) = (2\pi i\hbar)^{-n/2} \sqrt{|\frac{\partial^2 S}{\partial x \partial y}|} \exp\left[\frac{i}{\hbar}S(t,x,y)\right] + \hbar\zeta(t,x,y;\hbar), \qquad (I.0.1.3) $$

$$ S(t,x,y) = \int_{0}^{t} \left[ \frac{m}{2} \left( \frac{\partial X(\tau; t, x, y)}{\partial \tau} \right)^2 - V(X(\tau; t, x, y), \tau) \right] d\tau. $$

Here $t < T[V(x,t)], x,y \in \mathbb{R}^n, \eta(t,y) = \|\zeta(t,x,y;\hbar)\|_{L_2(\mathbb{R}^n)} < C$ and

$$ m\ddot{X} = -\frac{\partial V(X,t)}{\partial X}, \qquad (I.0.1.4) $$

$$ X(0; t, x, y) = y, X(t; t, x, y) = x. $$

By using convolution one obtain [25]-[27]:

$$ G(t_1 + t_2, x, y; \hbar) = \int_{\mathbb{R}^n} G(t_1, \tilde{y}, y; \hbar) G(t_2, x, \tilde{y}; \hbar) d\tilde{y}. \qquad (I.0.1.5) $$
$$ -i\hbar \frac{\partial \Psi(t)}{\partial t} + \hat{H}(t) \Psi(t) = 0, \qquad (I.0.1.1) $$

$$ \Psi(0) = \Psi_0(x), x \in \mathbb{R}^n, $$

where operator $H(t)$ given via formula

$$ H(t) = -\frac{\hbar^2}{2m} \Delta + V(x, t) \qquad (I.0.1.2) $$

and $H$ is essentially self-adjoint, $\hat{H}$ is the closure of $H, \Psi(t) : \mathbb{R} \to \mathbf{H}$ and $\hat{H}(t) : \mathbb{R} \times \mathbf{H} \to \mathbf{H}$.

Maslov local quasiclassical asymptotic of the Green function of the Schrödinger equation (I.0.1.1) with potential $V(x, t)$ is [26]:

$$ G(t,x,y;\hbar) = (2\pi i\hbar)^{-n/2} \sqrt{|\frac{\partial^2 S}{\partial x \partial y}|} \exp\left[\frac{i}{\hbar}S(t,x,y)\right] + \hbar\zeta(t,x,y;\hbar), \qquad (I.0.1.3) $$

$$ S(t,x,y) = \int_{0}^{t} \left[ \frac{m}{2} \left( \frac{\partial X(\tau; t, x, y)}{\partial \tau} \right)^2 - V(X(\tau; t, x, y), \tau) \right] d\tau. $$

Here $t < T[V(x,t)]$, $x,y \in \mathbb{R}^n$, $\eta(t,y) = \|\zeta(t,x,y;\hbar)\|_{L_2(\mathbb{R}^n)} < C$ and

$$ m\ddot{X} = -\frac{\partial V(X,t)}{\partial X}, \qquad (I.0.1.4) $$

$$ X(0; t, x, y) = y, X(t; t, x, y) = x. $$

By using convolution one obtain [25]-[27]:

$$ G(t_1 + t_2, x, y; \hbar) = \int_{\mathbb{R}^n} G(t_1, \tilde{y}, y; \hbar) G(t_2, x, \tilde{y}; \hbar) d\tilde{y}. \qquad (I.0.1.5) $$
We assume now that [25]-[27]:

(i) $V(x) \in C^{\infty}(\mathbb{R}^n)$ and

$$ 
\begin{array}{l}
\displaystyle \sup_{x \in \mathbb{R}^n} |\mathrm{grad}_x V(x)| < \infty, \\
\\
\displaystyle \inf_{x \in \mathbb{R}^n} V(x) \geq -\infty,
\end{array}
\qquad (0.1.6) $$

**Remark 0.1.1.** We remind that ander conditions (i) H is essentially self-adjoint.

(ii) for a fixed $t$ and $y$ boundary problem is a system, (0.1.4) has a finite number of solutions $X_k(\tau; t, x, y), k = 1, \dots, N$,

(iii) for a given $t, x, y$ the set $\{X_k(t; t, x, y) | k = 1, \dots, N\}$ does not contain focal points.

Then Maslov global quasiclassical asymptotic of the Green function obtained from Eq.(0.1.6) is [27]:

$$ G(t, x, y; \hbar) = $$

$$ (2\pi i\hbar)^{-n/2} \left[ \sum_{k=1}^{N} \sqrt{\left| \frac{\partial^2 S_k(t,x,y)}{\partial x \partial y} \right|} \exp \left[ \frac{i}{\hbar} S_k(t,x,y) - \frac{i\pi\gamma_k}{2} \right] + O(\hbar) \right], \qquad (0.1.7) $$

$$ S_k(t,x,y) = \int_0^t \left[ \frac{m}{2} \left( \frac{\partial X_k(\tau, t, x, y)}{\partial \tau} \right)^2 - V(X_k(\tau, t, x, y)) \right] d\tau. $$

Here the number $\gamma_k$ is the Morse index of the trajectory $X_k(\tau; t, x, y)$.

Unfortunately for real physical problems assumption (i) does not satisfies and consequently Maslov global quasiclassical asymptotic of the Green function (0.1.7) does not valid. We improve this difficulties using Colombeau approach [1]-[12],[42], [43].

Let us consider Colombeau-Schrödinger equation [42],[43]:
We assume now that [25]-[27]:

(i) $V(x) \in C^{\infty}(\mathbb{R}^n)$ and

$$ \sup_{x \in \mathbb{R}^n} |\mathrm{grad}_x V(x)| < \infty, \qquad (0.1.6) $$

$$ \inf_{x \in \mathbb{R}^n} V(x) \geq -\infty, $$

**Remark 0.1.1.** We remind that ander conditions (i) $H$ is essentially self-adjoint.

(ii) for a fixed $t$ and $y$ boundary problem is a system, (0.1.4) has a finite number of solutions $X_k(\tau; t, x, y), k = 1, \dots, N$,

(iii) for a given $t, x, y$ the set $\{X_k(t; t, x, y) | k = 1, \dots, N\}$ does not contain focal points.

Then Maslov global quasiclassical asymptotic of the Green function obtained from Eq.(0.1.6) is [27]:

$$ G(t, x, y; \hbar) = $$

$$ (2\pi i\hbar)^{-n/2} \left[ \sum_{k=1}^{N} \sqrt{\left| \frac{\partial^2 S_k(t,x,y)}{\partial x \partial y} \right|} \exp \left[ \frac{i}{\hbar} S_k(t,x,y) - \frac{i\pi\gamma_k}{2} \right] + O(\hbar) \right], \qquad (0.1.7) $$

$$ S_k(t,x,y) = \int_0^t \left[ \frac{m}{2} \left( \frac{\partial X_k(\tau, t, x, y)}{\partial \tau} \right)^2 - V(X_k(\tau, t, x, y)) \right] d\tau. $$

Here the number $\gamma_k$ is the Morse index of the trajectory $X_k(\tau; t, x, y)$.

Unfortunately for real physical problems assumption (i) does not satisfies and consequently Maslov global quasiclassical asymptotic of the Green function (0.1.7) does not valid. We improve this difficulties using Colombeau approach [1]-[12],[42], [43].

Let us consider Colombeau-Schrödinger equation [42],[43]:
$$ -i((h_{\varepsilon})_{\varepsilon}) \left( \frac{\partial \Psi_{\varepsilon}(t)}{\partial t} \right)_{\varepsilon} + (\hat{H}_{\varepsilon}(t) \Psi_{\varepsilon}(t))_{\varepsilon} = 0, \qquad (0.1.8) $$

$$ (\Psi_{\varepsilon}(0))_{\varepsilon} = (\Psi_{0,\varepsilon}(x))_{\varepsilon}, x \in \mathbb{R}^n, $$

where $\forall \varepsilon \in (0,1]$ operator $H_{\varepsilon}(t)$ given via formula

$$ H_{\varepsilon}(t) = -\frac{h_{\varepsilon}^2}{2m}\Delta + V_{\varepsilon}(x), $$
$$ (V_{\varepsilon}(x))_{\varepsilon} \in G(\mathbb{R}^n), \qquad (0.1.9) $$
$$ \inf_{x \in \mathbb{R}^n} (V_{\varepsilon}(x))_{\varepsilon} \geq -\infty $$

and $V_0(x) = V(x)$.

**Remark 0.1.2.** We note that $(V_{\varepsilon}(x))_{\varepsilon} \in G(\mathbb{R}^n) \Rightarrow$

$$ \forall \varepsilon(\varepsilon \in (0,1]) \left[ \sup_{x \in \mathbb{R}^n} |\mathbf{grad}_x V_{\varepsilon}(x)| < \infty \right]. \qquad (0.1.10) $$

Maslov local quasiclassical asymptotic of the Green function of the Colombeau-Schrödinger equation (0.1.8) with potential $(V_{\varepsilon}(x))_{\varepsilon} \in G(\mathbb{R}^n)$ is:
$$ -i((h_{\varepsilon})_{\varepsilon}) \left( \frac{\partial \Psi_{\varepsilon}(t)}{\partial t} \right)_{\varepsilon} + (\hat{H}_{\varepsilon}(t) \Psi_{\varepsilon}(t))_{\varepsilon} = 0, \qquad (0.1.8) $$

$$ (\Psi_{\varepsilon}(0))_{\varepsilon} = (\Psi_{0,\varepsilon}(x))_{\varepsilon}, x \in \mathbb{R}^{n}, $$

where $\forall \varepsilon \in (0,1]$ operator $H_{\varepsilon}(t)$ given via formula

$$ H_{\varepsilon}(t) = -\frac{h_{\varepsilon}^2}{2m}\Delta + V_{\varepsilon}(x), $$ 
$$ (V_{\varepsilon}(x))_{\varepsilon} \in G(\mathbb{R}^n), \qquad (0.1.9) $$ 
$$ \inf_{x \in \mathbb{R}^n} (V_{\varepsilon}(x))_{\varepsilon} \geq -\infty $$

and $V_0(x) = V(x)$.

**Remark 0.1.2.** We note that $(V_{\varepsilon}(x))_{\varepsilon} \in G(\mathbb{R}^n) \Rightarrow$

$$ \forall \varepsilon(\varepsilon \in (0,1]) \left[ \sup_{x \in \mathbb{R}^n} |\mathbf{grad}_x V_{\varepsilon}(x)| < \infty \right]. \qquad (0.1.10) $$

Maslov local quasiclassical asymptotic of the Green function of the Colombeau-Schrödinger equation (0.1.8) with potential $(V_{\varepsilon}(x))_{\varepsilon} \in G(\mathbb{R}^n)$ is:
$$ (G_{\varepsilon}(t_{\varepsilon}, x, y; \hbar_{\varepsilon}))_{\varepsilon} = (2\pi i ((\hbar_{\varepsilon})_{\varepsilon}))^{-n/2} \sqrt{|\left(\frac{\partial^2 S_{\varepsilon}(t_{\varepsilon}, x, y)}{\partial x \partial y}\right)_{\varepsilon}|} \exp\left[\frac{i}{(\hbar_{\varepsilon})_{\varepsilon}}(S_{\varepsilon}(t_{\varepsilon}, x, y))_{\varepsilon}\right] + ((\hbar_{\varepsilon})_{\varepsilon})((\zeta_{\varepsilon}(t_{\varepsilon}, x, y; \hbar))_{\varepsilon}), \quad (0.1.11) $$

$$ (S_{\varepsilon}(t_{\varepsilon}, x, y))_{\varepsilon} = \left( \int_{0}^{t_{\varepsilon}} \left[ \frac{m}{2} \left( \frac{\partial X_{\varepsilon}(\tau; t_{\varepsilon}, x, y)}{\partial \tau} \right)^2 - V_{\varepsilon}(X_{\varepsilon}(\tau; t_{\varepsilon}, x, y), \tau) \right] d\tau \right)_{\varepsilon}. $$

Here $(t_{\varepsilon})_{\varepsilon} < (T[V_{\varepsilon}(x, t_{\varepsilon})])_{\varepsilon}, x, y \in \mathbb{R}^n, (\|\zeta(t_{\varepsilon}, x, y; \hbar)\|_{L^2(\mathbb{R}^n)})_{\varepsilon} < (C_{\varepsilon})_{\varepsilon} \in \tilde{\mathbb{R}}$ and

$$ m((\ddot{X}_{\varepsilon})_{\varepsilon}) = -\left(\frac{\partial V_{\varepsilon}(X_{\varepsilon}, t)}{\partial X_{\varepsilon}}\right)_{\varepsilon}, \quad (0.1.12) $$
$$ (X_{\varepsilon}(0, t, x, y))_{\varepsilon} = y, (X_{\varepsilon}(t, t, x, y))_{\varepsilon} = x. $$

By using convolution we obtain

$$ (G_{\varepsilon}(t_1 + t_2, x, y; \hbar_{\varepsilon}))_{\varepsilon} = \left( \int_{\mathbb{R}^n} G_{\varepsilon}(t_1, \tilde{y}, y; \hbar) G_{\varepsilon}(t_2, x, \tilde{y}; \hbar) d\tilde{y} \right)_{\varepsilon}. \quad (0.1.13) $$

We assume now that:

(i) for a fixed $\varepsilon \in (0, 1]$, $t$ and $y$ boundary problem is a system, (0.1.12) has a finite number of solutions $X_{k,\varepsilon}(\tau; t, x, y)$, $k = 1, \dots, N(\varepsilon)$,

(ii) for a given $t, x, y: \forall \varepsilon \in (0, 1]$ the set $\{X_{k,\varepsilon}(t; t, x, y)|k = 1, \dots, N(\varepsilon)\}$ does not contain focal points.

Then Maslov global quasiclassical asymptotic of the Green function obtained from Eq.(0.1.13) is:
$$ (G_{\varepsilon}(t_{\varepsilon}, x, y; \hbar_{\varepsilon}))_{\varepsilon} = (2\pi i ((\hbar_{\varepsilon})_{\varepsilon}))^{-n/2} \sqrt{|\left(\frac{\partial^2 S_{\varepsilon}(t_{\varepsilon}, x, y)}{\partial x \partial y}\right)_{\varepsilon}|} \exp\left[\frac{i}{(\hbar_{\varepsilon})_{\varepsilon}}(S_{\varepsilon}(t_{\varepsilon}, x, y))_{\varepsilon}\right] + ((\hbar_{\varepsilon})_{\varepsilon})((\zeta_{\varepsilon}(t_{\varepsilon}, x, y; \hbar))_{\varepsilon}), \quad (0.1.11) $$

$$ (S_{\varepsilon}(t_{\varepsilon}, x, y))_{\varepsilon} = \left( \int_{0}^{t_{\varepsilon}} \left[ \frac{m}{2} \left( \frac{\partial X_{\varepsilon}(\tau; t_{\varepsilon}, x, y)}{\partial \tau} \right)^2 - V_{\varepsilon}(X_{\varepsilon}(\tau; t_{\varepsilon}, x, y), \tau) \right] d\tau \right)_{\varepsilon}. $$

Here $(t_{\varepsilon})_{\varepsilon} < (T[V_{\varepsilon}(x, t_{\varepsilon})])_{\varepsilon}, x, y \in \mathbb{R}^n, (\|\zeta(t_{\varepsilon}, x, y; \hbar)\|_{L^2(\mathbb{R}^n)})_{\varepsilon} < (C_{\varepsilon})_{\varepsilon} \in \tilde{\mathbb{R}} \text{ and}$

$$ m((\ddot{X}_{\varepsilon})_{\varepsilon}) = -\left(\frac{\partial V_{\varepsilon}(X_{\varepsilon}, t)}{\partial X_{\varepsilon}}\right)_{\varepsilon}, \quad (0.1.12) $$
$$ (X_{\varepsilon}(0, t, x, y))_{\varepsilon} = y, (X_{\varepsilon}(t, t, x, y))_{\varepsilon} = x. $$

By using convolution we obtain

$$ (G_{\varepsilon}(t_1 + t_2, x, y; \hbar_{\varepsilon}))_{\varepsilon} = \left( \int_{\mathbb{R}^n} G_{\varepsilon}(t_1, \tilde{y}, y; \hbar) G_{\varepsilon}(t_2, x, \tilde{y}; \hbar) d\tilde{y} \right)_{\varepsilon}. \quad (0.1.13) $$

We assume now that:

(i) for a fixed $\varepsilon \in (0, 1]$, $t$ and $y$ boundary problem is a system, (0.1.12) has a finite number of solutions $X_{k,\varepsilon}(\tau; t, x, y)$, $k = 1, \dots, N(\varepsilon)$,

(ii) for a given $t, x, y: \forall \varepsilon \in (0, 1]$ the set $\{X_{k,\varepsilon}(t; t, x, y)|k = 1, \dots, N(\varepsilon)\}$ does not contain focal points.

Then Maslov global quasiclassical asymptotic of the Green function obtained from Eq.(0.1.13) is:
$$ (G_{\varepsilon}(t, x, y; \hbar_{\varepsilon}))_{\varepsilon} = $$

$$ (2\pi i(\hbar_{\varepsilon}))_{\varepsilon}^{-n/2} \times $$

$$ \left[ \left( \sum_{k=1}^{N(\varepsilon)} \sqrt{\left| \left( \frac{\partial^2 S_{k,\varepsilon}(t,x,y)}{\partial x \partial y} \right) \right|} \right)_{\varepsilon} \times \right. \tag{0.1.14} $$

$$ \exp\left[ \frac{i}{(\hbar_{\varepsilon})_{\varepsilon}} \left( S_{k,\varepsilon}(t,x,y)_{\varepsilon} - \frac{i\pi(\gamma_{k,\varepsilon})_{\varepsilon}}{2} \right) \right] + (O(\hbar_{\varepsilon}))_{\varepsilon}, $$

$$ (S_{k,\varepsilon}(t,x,y))_{\varepsilon} = \left( \int_{0}^{t} \left[ \frac{m}{2} \left( \frac{\partial X_k(\tau, t, x, y)}{\partial \tau} \right)^2 - V_{\varepsilon}(X_{k,\varepsilon}(\tau, t, x, y)) \right] d\tau \right)_{\varepsilon}. $$

Here the number $\gamma_{k,\varepsilon}$ is the Morse index of the trajectory $X_{k,\varepsilon}(\tau; t, x, y)$.

### I.0.2. The Main result.

The main purpose of this paper is:

(I) To calculate exact quasiclassical asymptotic of the complex $\tilde{\mathbb{C}}$-valued generalized quantum averages

$$ \left\langle \left\langle s, t, \mathbf{x}_0; \hbar_{\varepsilon} \right\rangle_{\varepsilon} \right\rangle_{\varepsilon} = \frac{\left( \int x_s \Psi_{\varepsilon}(\mathbf{x}, t, \mathbf{x}_0; \hbar_{\varepsilon}) d^n \mathbf{x} \right)_{\varepsilon}}{\left( \int \Psi_{\varepsilon}(\mathbf{x}, t, \mathbf{x}_0; \hbar_{\varepsilon}) d^n \mathbf{x} \right)_{\varepsilon}}, \tag{0.2.1} $$

$$ (\Psi_{\varepsilon}(\mathbf{x}, 0, \mathbf{x}_0; \hbar))_{\varepsilon} = \delta(\mathbf{x} - \mathbf{x}_0), $$

$$ s = 1, \dots, n $$
$$ (G_{\varepsilon}(t,x,y;\hbar_{\varepsilon}))_{\varepsilon} = $$

$$ (2\pi i(\hbar_{\varepsilon}))_{\varepsilon}^{-n/2} \times $$

$$ \left[ \left( \sum_{k=1}^{N(\varepsilon)} \sqrt{\left| \left( \frac{\partial^2 S_{k,\varepsilon}(t,x,y)}{\partial x \partial y} \right) \right|} \right)_{\varepsilon} \times \right. \tag{0.1.14} $$

$$ \exp \left[ \frac{i}{(\hbar_{\varepsilon})_{\varepsilon}} \left( S_{k,\varepsilon}(t,x,y)_{\varepsilon} - \frac{i\pi(\gamma_{k,\varepsilon})_{\varepsilon}}{2} \right) \right] + (O(\hbar_{\varepsilon}))_{\varepsilon}, $$

$$ (S_{k,\varepsilon}(t,x,y))_{\varepsilon} = \left( \int_{0}^{t} \left[ \frac{m}{2} \left( \frac{\partial X_k(\tau, t, x, y)}{\partial \tau} \right)^2 - V_{\varepsilon}(X_{k,\varepsilon}(\tau, t, x, y)) \right] d\tau \right)_{\varepsilon}. $$

Here the number $\gamma_{k,\varepsilon}$ is the Morse index of the trajectory $X_{k,\varepsilon}(\tau; t, x, y)$.

## 1.0.2. The Main result.

The main purpose of this paper is:

(I) To calculate exact quasiclassical asymptotic of the complex $\tilde{\mathbb{C}}$-valued generalized quantum averages

$$ \left\langle \left\langle s, t, \mathbf{x}_0; \hbar_{\varepsilon} \right\rangle_{\varepsilon} \right\rangle_{\varepsilon} = \frac{\left( \int x_s \Psi_{\varepsilon}(\mathbf{x}, t, \mathbf{x}_0; \hbar_{\varepsilon}) d^n \mathbf{x} \right)_{\varepsilon}}{\left( \int \Psi_{\varepsilon}(\mathbf{x}, t, \mathbf{x}_0; \hbar_{\varepsilon}) d^n \mathbf{x} \right)_{\varepsilon}}, \tag{0.2.1} $$

$$ (\Psi_{\varepsilon}(\mathbf{x}, 0, \mathbf{x}_0; \hbar))_{\varepsilon} = \delta(\mathbf{x} - \mathbf{x}_0), $$

$$ s = 1, \dots, n $$
without any reference to the corresponding exact quasiclassical asymptotic of the Schrödinger generalized wave function $(\Psi_{\varepsilon}(\mathbf{x},t,\mathbf{x}_0;\hbar))_{\varepsilon}$ given from Colombeau-Schrödinger Eq.(0.1.8)-(0.1.9)

(II) To calculate exact quasiclassical asymptotic of the real $\tilde{\mathbb{R}}$-valued generalized quantum averages

$$ (\langle s, t, \mathbf{x}_0; \hbar_{\varepsilon} \rangle)_{\varepsilon} = \frac{\left( \int_{\mathbb{R}^n} x_s |\Psi_{\varepsilon}(\mathbf{x}, t, \mathbf{x}_0; \hbar_{\varepsilon})|^2 d^n x \right)_{\varepsilon}}{\left( \int_{\mathbb{R}^n} |\Psi_{\varepsilon}(\mathbf{x}, t, \mathbf{x}_0; \hbar_{\varepsilon})|^2 d^n x \right)_{\varepsilon}}, \quad (0.2.2) $$

$$ s = 1, \dots, n $$

without any reference to the corresponding exact quasiclassical asymptotic of the Schrödinger wave function $(\Psi_{\varepsilon}(x,t,x_0;\hbar))_{\varepsilon}$ given from Colombeau-Schrödinger Eq.(0.1.8)-(0.1.9).

**Remark.** Note that such quasiclassical asymptotic very important from point of view of the "quantum jumps" problem, well known in modern quantum mechanics. The existence of such jumps was required by Bohr in his theory of the atom. He assumed that an atom remained in an atomic eigenstate until it made an instantaneous jump to another state with the emission or absorption of a photon. Since these jumps do not appear to occur in solutions of the Schrödinger equation, something similar to Bohr's idea has been added as an extra postulate in modern quantum mechanics. The question arises whether an explanation of these jumps can be found to result from a solution $\Psi(\mathbf{x}, t, \mathbf{x}_0; \hbar)$ of the Schrödinger equation (0.1.1)-(0.1.2) alone without additional postulates.

We suggest a new asymptotic representation for the quantum averages (0.2.2) with position variable with well localized initial data, i.e.

$$ (\langle i, 0, \mathbf{x}_0; \hbar_{\varepsilon} \rangle)_{\varepsilon} \approx x_{0i}, \quad (0.2.3) $$

$$ i = 1, \dots, n. $$

The canonical physical interpretation of these asymptotics shows that the answer is "yes."
without any reference to the corresponding exact quasiclassical asymptotic of the Schrödinger generalized wave function $(\Psi_{\varepsilon}(\mathbf{x},t,\mathbf{x}_0;\hbar))_{\varepsilon}$ given from Colombeau-Schrödinger Eq.(0.1.8)-(0.1.9)

(II) To calculate exact quasiclassical asymptotic of the real $\tilde{\mathbb{R}}$-valued generalized quantum averages

$$ \langle\langle s, t, \mathbf{x}_0; \hbar_{\varepsilon} \rangle\rangle_{\varepsilon} = \frac{\left( \int_{\mathbb{R}^n} x_s |\Psi_{\varepsilon}(\mathbf{x}, t, \mathbf{x}_0; \hbar_{\varepsilon})|^2 d^n x \right)_{\varepsilon}}{\left( \int_{\mathbb{R}^n} |\Psi_{\varepsilon}(\mathbf{x}, t, \mathbf{x}_0; \hbar_{\varepsilon})|^2 d^n x \right)_{\varepsilon}}, \quad (0.2.2) $$

$$ s = 1, \dots, n $$

without any reference to the corresponding exact quasiclassical asymptotic of the Schrödinger wave function $(\Psi_{\varepsilon}(x,t,x_0;\hbar))_{\varepsilon}$ given from Colombeau-Schrödinger Eq.(0.1.8)-(0.1.9).

**Remark.** Note that such quasiclassical asymptotic very important from point of view of the "quantum jumps" problem, well known in modern quantum mechanics. The existence of such jumps was required by Bohr in his theory of the atom. He assumed that an atom remained in an atomic eigenstate until it made an instantaneous jump to another state with the emission or absorption of a photon. Since these jumps do not appear to occur in solutions of the Schrödinger equation, something similar to Bohr's idea has been added as an extra postulate in modern quantum mechanics. The question arises whether an explanation of these jumps can be found to result from a solution $\Psi(\mathbf{x}, t, \mathbf{x}_0; \hbar)$ of the Schrödinger equation (0.1.1)-(0.1.2) alone without additional postulates.

We suggest a new asymptotic representation for the quantum averages (0.2.2) with position variable with well localized initial data, i.e.

$$ \langle\langle i, 0, \mathbf{x}_0; \hbar_{\varepsilon} \rangle\rangle_{\varepsilon} \approx x_{0i}, \quad (0.2.3) $$

$$ i = 1, \dots, n. $$

The canonical physical interpretation of these asymptotics shows that the answer is "yes."
As is well-known, when the potential $V$ is sufficiently regular any solution of the Schrödinger equation (0.1.1) can be represented by Feynman path integral of the form:

$$ \Psi(x, t, x_0; \hbar) = \int_{\substack{q(0)=y \\ q(t)=x}} \Psi_0(q(0)) \exp\left\{\frac{i}{\hbar} S[\dot{q}(\tau), q(\tau), t]\right\} [Dq(\tau)] dy, $$

$$ S[\dot{q}(\tau), q(\tau), t] = \int_{0}^{t} \mathcal{L}(\dot{q}(\tau), q(\tau), \tau) d\tau, \qquad (0.2.4) $$

$$ \mathcal{L}(\dot{q}(\tau), q(\tau), \tau) = m \frac{\dot{q}^2(\tau)}{2} - V(q(\tau), t). $$

Maslov quasiclassical asymptotic of the wave function $\Psi(x, t; \hbar)$ is [22],[25]:

$$ \Psi(x,t;\hbar) = (2\pi i\hbar)^{-\frac{n}{2}} \sum_{k=1}^{m} \int dy \Psi_0(y) |D(q^k(\tau;y,x,t))|^{\frac{1}{2}} \times \\ \times \exp\left\{\frac{i}{\hbar}\left[S[q^k(\tau;y,x,t)] - \frac{i\pi\gamma_k}{2}\right]\right\} + O(\hbar). \qquad (0.2.5) $$

Here
As is well-known, when the potential $V$ is sufficiently regular any solution of the Schrödinger equation (0.1.1) can be represented by Feynman path integral of the form:

$$ 
\begin{align} 
\Psi(x,t,x_0;\hbar) &= \int_{\substack{q(0)=y \\ q(t)=x}} \Psi_0(q(0)) \exp\left\{\frac{i}{\hbar} S[\dot{q}(\tau), q(\tau), t]\right\} [Dq(\tau)] dy, \nonumber \\ 
S[\dot{q}(\tau), q(\tau), t] &= \int_0^t \mathcal{L}(\dot{q}(\tau), q(\tau), \tau) d\tau, \tag{0.2.4} \\ 
\mathcal{L}(\dot{q}(\tau), q(\tau), \tau) &= m \frac{\dot{q}^2(\tau)}{2} - V(q(\tau), t). \nonumber 
\end{align} 
$$

Maslov quasiclassical asymptotic of the wave function $\Psi(x,t;\hbar)$ is [22],[25]:

$$ 
\begin{align} 
\Psi(x,t;\hbar) ={}& (2\pi i\hbar)^{-\frac{n}{2}} \sum_{k=1}^{m} \int dy \Psi_0(y) |D(q^k(\tau;y,x,t))|^{\frac{1}{2}} \times \tag{0.2.5} \\ 
& \times \exp\left\{\frac{i}{\hbar} \left[S[q^k(\tau;y,x,t)] - \frac{i\pi\gamma_k}{2}\right]\right\} + O(\hbar). \nonumber 
\end{align} 
$$

Here
$$
\begin{align}
S[q^k(\tau; y, x, t), t] &= \int_0^t \mathcal{L}(\dot{q}^k(\tau; y, x, t), q^k(\tau; y, x, t), \tau) d\tau, \nonumber \\
\mathcal{L}(\dot{q}(\tau), q(\tau), \tau) &= m \frac{\dot{q}^2}{2} - V(q(\tau), \tau), \tag{0.2.6}
\end{align}
$$

$$
\delta S[q^k(\tau; y, x, t), t] = 0, q^k(0; y, x, t) = y, q^k(t; y, x, t) = x,
$$

$$
D(q^k(\tau; y, x, t)) = \det \left\| \frac{\partial S[q^k(\tau; y, x, t), t]}{\partial x \partial y} \right\| \neq 0, x \neq y.
$$

and the number $\gamma_k = \gamma_k[q^k(\tau; y, x, t)]$ is the Morse index of the trajectory $q^k(\tau; y, x, t)$.
Assume that $\Psi_0(x) = \frac{1}{\epsilon} \exp\left(-\frac{(x-x_0)^2}{\epsilon^2}\right)$, $\epsilon \ll \hbar \ll 1$, i.e. $\Psi_0(x) \simeq \delta(x-x_0)$. From
Eq.(0.2.5) we obtain

$$
\Psi(x,t,x_0;\hbar) = (2\pi i\hbar)^{-\frac{n}{2}} \sum_{k=1}^{m} \int dy \Psi_0(y-x_0) |D(q^k(\tau;y,x,t))|^{\frac{1}{2}} \times \\
\times \exp\left\{\frac{i}{\hbar} \left[S[q^k(\tau;y,x,t)] - \frac{i\pi\gamma_k}{2}\right]\right\} + O(\hbar) = \tag{0.2.7}
$$

$$
\sum_{k=1}^{m} |D_1(q^k(\tau; x_0, x, t))|^{\frac{1}{2}} \exp\left\{\frac{i}{\hbar} \left[S[q^k(\tau; x_0, x, t), t] - \frac{i\pi\gamma_k}{2}\right]\right\} + O(\hbar).
$$

Let us consider quantum harmonic oscillator. Corresponding Schrödinger equation
is:
$$
\begin{align}
S[q^k(\tau; y, x, t), t] &= \int_0^t \mathcal{L}(\dot{q}^k(\tau; y, x, t), q^k(\tau; y, x, t), \tau) d\tau, \nonumber \\
\mathcal{L}(\dot{q}(\tau), q(\tau), \tau) &= m \frac{\dot{q}^2}{2} - V(q(\tau), \tau), \tag{0.2.6}
\end{align}
$$

$$
\delta S[q^k(\tau; y, x, t), t] = 0, q^k(0; y, x, t) = y, q^k(t; y, x, t) = x,
$$

$$
D(q^k(\tau; y, x, t)) = \det \left\| \frac{\partial S[q^k(\tau; y, x, t), t]}{\partial x \partial y} \right\| \neq 0, x \neq y.
$$

and the number $\gamma_k = \gamma_k[q^k(\tau; y, x, t)]$ is the Morse index of the trajectory $q^k(\tau; y, x, t)$.
Assume that $\Psi_0(x) = \frac{1}{\epsilon} \exp\left(-\frac{(x-x_0)^2}{\epsilon^2}\right)$, $\epsilon \ll \hbar \ll 1$, i.e. $\Psi_0(x) \simeq \delta(x-x_0)$. From
Eq.(0.2.5) we obtain

$$
\Psi(x,t,x_0;\hbar) = (2\pi i\hbar)^{-\frac{n}{2}} \sum_{k=1}^{m} \int dy \Psi_0(y-x_0) |D(q^k(\tau;y,x,t))|^{\frac{1}{2}} \times \\
\times \exp\left\{\frac{i}{\hbar} \left[S[q^k(\tau;y,x,t)] - \frac{i\pi\gamma_k}{2}\right]\right\} + O(\hbar) = \tag{0.2.7}
$$

$$
\sum_{k=1}^{m} |D_1(q^k(\tau; x_0, x, t))|^{\frac{1}{2}} \exp\left\{\frac{i}{\hbar} \left[S[q^k(\tau; x_0, x, t), t] - \frac{i\pi\gamma_k}{2}\right]\right\} + O(\hbar).
$$

Let us consider quantum harmonic oscillator. Corresponding Schrödinger equation
is:
$$ -i\hbar \frac{\partial \Psi(x,t)}{\partial t} = \frac{\hbar^2}{2} \Delta \Psi(x,t) - \frac{\omega^2}{2} x^2 \Psi(x,t), \qquad (0.2.8) $$

$$ \Psi(0) \approx \delta(x - x_0). $$

By canonical calculation one obtain

$$ S[q^k(\tau; y, x, t), t] = \frac{\omega}{2 \sin \omega t} [(\cos \omega t)(x^2 + y^2) - 2xy], \qquad (0.2.9) $$

$$ D(q^k(\tau; y, x, t)) = \det \left\| \frac{\partial^2 S}{\partial x \partial y} \right\| = \frac{\omega^n}{(\sin \omega t)^n}. $$

Substitution Eqs.(0.2.9) into Eq.(0.2.5) gives [22]:

$$ \Psi(x,t,x_0;\hbar) = (2\pi i\hbar)^{\frac{n}{2}} \int dy \delta(x-x_0) |D(q^k(\tau;y,x,t))|^{\frac{1}{2}} \times \\ \times \exp\left\{\frac{i}{\hbar}\left[S[q^k(\tau;y,x,t)] - \frac{i\pi\gamma_k}{2}\right]\right\} + O(\hbar) = \qquad (0.2.10) $$

$$ \omega^{n/2} (2\pi i\hbar \sin \omega t)^{-\frac{n}{2}} \times $$

$$ \exp\left\{\frac{i}{\hbar}\left[\frac{\omega}{2 \sin \omega t}[(\cos \omega t)(x^2 + x_0^2) - 2xx_0] - \frac{i\pi\gamma_k}{2}\right]\right\} + O(\hbar). $$

From Eq.(0.2.10) one obtain
$$ -i\hbar \frac{\partial \Psi(x,t)}{\partial t} = \frac{\hbar^2}{2} \Delta \Psi(x,t) - \frac{\omega^2}{2} x^2 \Psi(x,t), \qquad (0.2.8) $$

$$ \Psi(0) \approx \delta(x - x_0). $$

By canonical calculation one obtain

$$ S[q^k(\tau; y, x, t), t] = \frac{\omega}{2 \sin \omega t} [(\cos \omega t)(x^2 + y^2) - 2xy], \qquad (0.2.9) $$

$$ D(q^k(\tau; y, x, t)) = \det \left\| \frac{\partial^2 S}{\partial x \partial y} \right\| = \frac{\omega^n}{(\sin \omega t)^n}. $$

Substitution Eqs.(0.2.9) into Eq.(0.2.5) gives [22]:

$$ \Psi(x,t,x_0;\hbar) = (2\pi i\hbar)^{\frac{n}{2}} \int dy \delta(x-x_0) |D(q^k(\tau;y,x,t))|^{\frac{1}{2}} \times \\ \times \exp\left\{\frac{i}{\hbar}\left[S[q^k(\tau;y,x,t)] - \frac{i\pi\gamma_k}{2}\right]\right\} + O(\hbar) = \qquad (0.2.10) $$

$$ \omega^{n/2} (2\pi i\hbar \sin \omega t)^{-\frac{n}{2}} \times $$

$$ \exp\left\{\frac{i}{\hbar}\left[\frac{\omega}{2 \sin \omega t}[(\cos \omega t)(x^2 + x_0^2) - 2xx_0] - \frac{i\pi\gamma_k}{2}\right]\right\} + O(\hbar). $$

From Eq.(0.2.10) one obtain
$$ \Psi(x, t, x_0; \hbar) = $$

$$ \omega^{n/2} (2\pi i\hbar \sin \omega t)^{-n/2} \times \qquad (0.2.11) $$

$$ \exp\left\{\frac{i}{\hbar}\left[\frac{\omega}{2\sin\omega t}\left[(\cos\omega t)(x^2 + x_0^2) - 2xx_0\right] - \frac{i\pi\gamma_k}{2}\right]\right\} $$

and therefore

$$ \int x_s \Psi(x, t, x_0; \hbar) d^n x = $$

$$ \omega^{n/2} (2\pi i\hbar \sin \omega t)^{-n/2} \times \qquad (0.2.12) $$

$$ \int d^n x x_s \exp\left\{\frac{i}{\hbar}\left[\frac{\omega}{2\sin\omega t}\left[(\cos\omega t)(\mathbf{x}^2 + \mathbf{x}_0^2) - 2\mathbf{x}\mathbf{x}_0\right] - \frac{i\pi\gamma_k}{2}\right]\right\} = $$

$$ \frac{x_{0,s}}{\cos \omega t} \exp \left\{ \frac{i}{\hbar} \left[ \omega x_0^2 (\tan \omega t) - \frac{i\pi\gamma_k}{2} \right] \right\}. $$

Substitution Eq.(0.2.11) and Eq.(0.2.12) into Eq.(0.2.1) gives

$$ \langle x_i, t, x_0; \hbar \rangle = \frac{x_{s0}}{\cos \omega t}, \qquad (0.2.13) $$

$$ s = 1, \dots, n. $$

Formula (0.2.5) can be written by using Maslov canonical operator $K_{\Lambda^n(t)}^{1/\hbar}[\cdot]$.

Let $\Lambda^n(t)$ be an $n$-dimensional Lagrangian manifold of class $C^\infty$ in the phase space $\mathbb{R}_{x,p}^{2n}$

where $x \in \mathbb{R}^n$ and let $d\sigma$ be the volume element on $\Lambda$. A canonical atlas is a locally finite countable covering of $\Lambda^n(t)$ by bounded simply-connected domains $\Omega_j$ (the charts) in each of which one can take as coordinates either the variables $x$ or $p$ a mixed collection
$$ \Psi(x, t, x_0; \hbar) = $$

$$ \omega^{n/2} (2\pi i\hbar \sin \omega t)^{-n/2} \times \qquad (0.2.11) $$

$$ \exp\left\{\frac{i}{\hbar}\left[\frac{\omega}{2\sin\omega t}\left[(\cos\omega t)(x^2 + x_0^2) - 2xx_0\right] - \frac{i\pi\gamma_k}{2}\right]\right\} $$

and therefore

$$ \int x_s \Psi(x, t, x_0; \hbar) d^n x = $$

$$ \omega^{n/2} (2\pi i\hbar \sin \omega t)^{-n/2} \times \qquad (0.2.12) $$

$$ \int d^n x x_s \exp\left\{\frac{i}{\hbar}\left[\frac{\omega}{2\sin\omega t}\left[(\cos\omega t)(\mathbf{x}^2 + \mathbf{x}_0^2) - 2\mathbf{x}\mathbf{x}_0\right] - \frac{i\pi\gamma_k}{2}\right]\right\} = $$

$$ \frac{x_{0,s}}{\cos \omega t} \exp \left\{ \frac{i}{\hbar} \left[ \omega x_0^2 (\tan \omega t) - \frac{i\pi\gamma_k}{2} \right] \right\}. $$

Substitution Eq.(0.2.11) and Eq.(0.2.12) into Eq.(0.2.1) gives

$$ \langle x_i, t, x_0; \hbar \rangle = \frac{x_{s0}}{\cos \omega t}, \qquad (0.2.13) $$

$$ s = 1, \dots, n. $$

Formula (0.2.5) can be written by using Maslov canonical operator $K_{\Lambda^n(t)}^{1/\hbar}[\cdot]$.

Let $\Lambda^n(t)$ be an $n$-dimensional Lagrangian manifold of class $C^\infty$ in the phase space $\mathbb{R}_{x,p}^{2n}$

where $x \in \mathbb{R}^n$ and let $d\sigma$ be the volume element on $\Lambda$. A canonical atlas is a locally finite countable covering of $\Lambda^n(t)$ by bounded simply-connected domains $\Omega_j$ (the charts) in each of which one can take as coordinates either the variables $x$ or $p$ a mixed collection
$(p_\alpha, x_\beta),$

$$
\alpha = \{\alpha_1, \dots, \alpha_s\}, \tag{0.2.14}
$$

$\beta = \{\beta_1, \dots, \beta_{n-s}\},$

not containing dual pairs $(p_j, x_j)$. The Maslov canonical operator $K_{\Lambda^n(t)}^{1/\hbar}$ maps
$C_0^\infty(\Lambda^n(t))$ into $C(\mathbb{R}_x^n)$. The canonical operators are introduced as follows:

(1) Let the chart $\Omega_j$ be non-degenerate, that is, $\Omega_j$ is given by an equation $p = p(x,t)$ and

$$
(K_{\Omega_j}^{1/\hbar}[\varphi])(x) = \sqrt{|\frac{d\sigma}{dx}|} \exp\left[\frac{i}{\hbar} \int_{r^0}^{r} (p, dx)\right] \varphi(r), \qquad (0.2.15)
$$

$r = (x, p(x, t)),$

$(p, dx) = \sum_{j=1}^{n} p_j dx_j.$

Here $\hbar \ll 1$ is a parameter, $r^0 \in \Omega_j$ is a fixed point and $\varphi \in C_0^\infty(\Omega)$.

(2) Let the local coordinates in the chart $\Omega_j$ be $p$, that is $\Omega_j$, is given by an equation $x = x(p,t)$, and let

$$
(K_{\Omega_j}^{1/\hbar}(\Omega_j[\varphi]))(x) =
$$

$$
\mathcal{F}_{1/\hbar, p \to x}^{-1} \left\{ \sqrt{|\frac{d\sigma}{dx}|} \exp \left[ \frac{i}{\hbar} \left( \int_{r^0}^{r} (p, dx) - (x(p,t), p) \right) \right] \varphi(r) \right\}, \quad (0.2.16)
$$

$r = (x(p, t), p).$

Here
$(p_{\alpha}, x_{\beta}),$

$$ \alpha = \{\alpha_1, \dots, \alpha_s\}, \qquad (0.2.14) $$

$$ \beta = \{\beta_1, \dots, \beta_{n-s}\}, $$

not containing dual pairs $(p_j, x_j)$. The Maslov canonical operator $K_{\Lambda^n(t)}^{1/\hbar}$ maps $C_0^\infty(\Lambda^n(t))$ into $C(\mathbb{R}_x^n)$. The canonical operators are introduced as follows:

(1) Let the chart $\Omega_j$ be non-degenerate, that is, $\Omega_j$ is given by an equation $p = p(x, t)$ and

$$ (K_{\Omega_j}^{1/\hbar}[\varphi])(x) = \sqrt{|\frac{d\sigma}{dx}|} \exp\left[\frac{i}{\hbar} \int_{r^0}^{r} (p, dx)\right] \varphi(r), \qquad (0.2.15) $$

$$ r = (x, p(x, t)), $$

$$ (p, dx) = \sum_{j=1}^{n} p_j dx_j. $$

Here $\hbar \ll 1$ is a parameter, $r^0 \in \Omega_j$ is a fixed point and $\varphi \in C_0^\infty(\Omega)$.

(2) Let the local coordinates in the chart $\Omega_j$ be $p$, that is $\Omega_j$, is given by an equation $x = x(p, t)$, and let

$$ (K_{\Omega_j}^{1/\hbar}(\Omega_j[\varphi]))(x) = $$

$$ \mathcal{F}_{1/\hbar, p \to x}^{-1} \left\{ \sqrt{|\frac{d\sigma}{dx}|} \exp \left[ \frac{i}{\hbar} \left( \int_{r^0}^{r} (p, dx) - (x(p, t), p) \right) \right] \varphi(r) \right\}, \qquad (0.2.16) $$

$$ r = (x(p, t), p). $$

Here
$$ \mathcal{F}_{1/\hbar, p \to x}^{-1}[\phi](x) = \frac{1}{(-2\pi i\hbar)^{n/2}} \int_{\mathbb{R}^n} d^n p \phi(p) \exp\left[\frac{i}{\hbar}(x, p)\right]. \qquad (0.2.17) $$

$K_{\Omega_j}^{1/\hbar}$ is defined analogously in the case when the coordinates in are some collection $(p_\alpha, x_\beta)$. Let $\oint_l (p, dx) = 0$ and let the Maslov index $\text{ind}(l) = 0$ for any closed path lying on $\Lambda^n(t)$. One introduces a partition of unity of class $C^\infty$ on $\Lambda^n(t)$

$$ \sum_{j=1}^{\infty} e_j(x) = 1, \mathbf{supp}(e_j) \subset \Omega_j \qquad (0.2.18) $$

and one fixes a point $r^0 = \Omega_{j0}$. The Maslov canonical operator is defined by

$$ (K_{\Lambda^n(t)}^{1/\hbar}[\varphi(r)])(x) = \sum_j c_j (K_{\Omega_j}^{1/\hbar}[e_j\varphi])(x), \qquad (0.2.19) $$

$$ c_j = \exp\left(\frac{i\pi}{2} \gamma_j^M\right). $$

Here $\gamma_j^M$ is the Maslov index of a chain of charts joining the charts $\Omega_{j0}$ and $\Omega_j$. A point $r \in \Lambda^n(t)$ is called non-singular if it has a neighbourhood in given by an equation $p = p(x)$. Let the intersection of the charts $\Omega_i$ and $\Omega_j$ be non-empty and connected, let $r \in \Omega_i \cap \Omega_j$ be a non-singular point and let $(p_\alpha, x_\beta), (p_{\tilde{\alpha}}, x_{\tilde{\beta}})$ be the coordinates in these charts. The number

$$ \gamma_{ij}^{M} = \text{inerdex}\left(\frac{\partial x_{\alpha}(r)}{\partial p_{\alpha}}\right) - \text{inerdex}\left(\frac{\partial x_{\tilde{\alpha}}(r)}{\partial p_{\tilde{\alpha}}}\right) \qquad (0.2.20) $$

is the Maslov index of the pair of charts $\Omega_i$ and $\Omega_j$. The Maslov index of a chain of charts is defined by additivity. The Maslov index of a path is defined analogously. The Maslov index of a path (mod 4) on a Lagrangian manifold is an integer homotopy invariant. The Maslov canonical operator is invariant under the choice of the canonical atlas, of local coordinates in the charts and the partition of
$$ \mathcal{F}_{1/\hbar, p \to x}^{-1}[\phi](x) = \frac{1}{(-2\pi i\hbar)^{n/2}} \int_{\mathbb{R}^n} d^n p \phi(p) \exp\left[\frac{i}{\hbar}(x, p)\right]. \qquad (0.2.17) $$

$K_{\Omega_j}^{1/\hbar}$ is defined analogously in the case when the coordinates in are some collection $(p_\alpha, x_\beta)$. Let $\oint_l (p, dx) = 0$ and let the Maslov index $\mathrm{ind}(l) = 0$ for any closed path lying on $\Lambda^n(t)$. One introduces a partition of unity of class $C^\infty$ on $\Lambda^n(t)$

$$ \sum_{j=1}^{\infty} e_j(x) = 1, \mathbf{supp}(e_j) \subset \Omega_j \qquad (0.2.18) $$

and one fixes a point $r^0 = \Omega_{j0}$. The Maslov canonical operator is defined by

$$ (K_{\Lambda^n(t)}^{1/\hbar}[\varphi(r)])(x) = \sum_j c_j (K_{\Omega_j}^{1/\hbar}[e_j\varphi])(x), \qquad (0.2.19) $$

$$ c_j = \exp\left(\frac{i\pi}{2} \gamma_j^M\right). $$

Here $\gamma_j^M$ is the Maslov index of a chain of charts joining the charts $\Omega_{j0}$ and $\Omega_j$. A point $r \in \Lambda^n(t)$ is called non-singular if it has a neighbourhood in given by an equation $p = p(x)$. Let the intersection of the charts $\Omega_i$ and $\Omega_j$ be non-empty and connected, let $r \in \Omega_i \cap \Omega_j$ be a non-singular point and let $(p_\alpha, x_\beta), (p_{\tilde{\alpha}}, x_{\tilde{\beta}})$ be the coordinates in these charts. The number

$$ \gamma_{ij}^{M} = \mathbf{inerdex} \left( \frac{\partial x_{\alpha}(r)}{\partial p_{\alpha}} \right) - \mathbf{inerdex} \left( \frac{\partial x_{\tilde{\alpha}}(r)}{\partial p_{\tilde{\alpha}}} \right) \qquad (0.2.20) $$

is the Maslov index of the pair of charts $\Omega_i$ and $\Omega_j$. The Maslov index of a chain of charts is defined by additivity. The Maslov index of a path is defined analogously. The Maslov index of a path (mod 4) on a Lagrangian manifold is an integer homotopy invariant. The Maslov canonical operator is invariant under the choice of the canonical atlas, of local coordinates in the charts and the partition of
unity in the following sense: If $K_{\Lambda^n(t)}^{1/\hbar}$, $\tilde{K}_{\Lambda^n(t)}^{1/\hbar}$ are two Maslov canonical operators, then in $L_2(\mathbb{R}^n)$,

$$ (K_{\Lambda^n(t)}^{1/\hbar}\varphi - \tilde{K}_{\Lambda^n(t)}^{1/\hbar}\varphi - ) (x) = O(\hbar) \qquad (0.2.21) $$

for any function $\varphi \in C_0^\infty(\Lambda^n(t))$. Let $L(x, \hbar D)$ be a differential operator with real symbol $L(x, p)$ of class $C^\infty$ and suppose that $L(x, p) = 0$ on $\Lambda^n(t)$. Suppose that $\Lambda^n(t)$ and the volume element $d\sigma$ are invariant under the Hamiltonian system

$$ \frac{dx}{d\tau} = \frac{\partial L}{\partial p}, \frac{dp}{d\tau} = -\frac{\partial L}{\partial x}. \qquad (0.2.22) $$

Then the following commutation formula is true

$$ L(x, \hbar D) (K_{\Lambda^n(t)}^{1/\hbar} \varphi)(x) = \frac{\hbar}{i} K_{\Lambda^n(t)}^{1/\hbar} [\mathfrak{R}\varphi + O(\hbar)], \qquad (0.2.23) \\ \mathfrak{R}\varphi = \left[ \frac{d}{d\tau} - \frac{1}{2} \sum_{j=1}^{n} \frac{\partial^2 L(x, p)}{\partial x_j \partial p_j} \right] \varphi $$

where $\varphi \in C_0^\infty(\Lambda^n(t))$ and $d/d\tau$ is the derivative along the integral curves of the flow of the Hamiltonian system. The equation $\mathfrak{R}\varphi = 0$ is called the transport equation. The commutation formula implies that if $\mathfrak{R}\varphi = 0$, then the function is a formal asymptotic solution of the equation $L(x, \hbar D)u = 0$.

In general case, when the potential $(V_\epsilon)_\epsilon$ in $G(\mathbb{R}^n)$ any solution $(\Psi_\epsilon(x, t, x_0; \hbar_\epsilon))_\epsilon$ of the Colombeau-Schrödinger equation (0.1.8) can be represented by Colombeau-Feynman path integral of the form:
unity in the following sense: If $K_{\Lambda^n(t)}^{1/\hbar}$, $\tilde{K}_{\Lambda^n(t)}^{1/\hbar}$ are two Maslov canonical operators, then in $L_2(\mathbb{R}^n)$,

$$ (K_{\Lambda^n(t)}^{1/\hbar}\varphi - \tilde{K}_{\Lambda^n(t)}^{1/\hbar}\varphi - ) (x) = O(\hbar) \qquad (0.2.21) $$

for any function $\varphi \in C_0^\infty(\Lambda^n(t))$. Let $L(x, \hbar D)$ be a differential operator with real symbol $L(x, p)$ of class $C^\infty$ and suppose that $L(x, p) = 0$ on $\Lambda^n(t)$. Suppose that $\Lambda^n(t)$ and the volume element $d\sigma$ are invariant under the Hamiltonian system

$$ \frac{dx}{d\tau} = \frac{\partial L}{\partial p}, \frac{dp}{d\tau} = -\frac{\partial L}{\partial x}. \qquad (0.2.22) $$

Then the following commutation formula is true

$$ L(x, \hbar D) (K_{\Lambda^n(t)}^{1/\hbar} \varphi)(x) = \frac{\hbar}{i} K_{\Lambda^n(t)}^{1/\hbar} [\mathfrak{R}\varphi + O(\hbar)], \qquad (0.2.23) \\ \mathfrak{R}\varphi = \left[ \frac{d}{d\tau} - \frac{1}{2} \sum_{j=1}^{n} \frac{\partial^2 L(x, p)}{\partial x_j \partial p_j} \right] \varphi $$

where $\varphi \in C_0^\infty(\Lambda^n(t))$ and $d/d\tau$ is the derivative along the integral curves of the flow of the Hamiltonian system. The equation $\mathfrak{R}\varphi = 0$ is called the transport equation. The commutation formula implies that if $\mathfrak{R}\varphi = 0$, then the function is a formal asymptotic solution of the equation $L(x, \hbar D)u = 0$.

In general case, when the potential $(V_\varepsilon)_\varepsilon$ in $G(\mathbb{R}^n)$ any solution $(\Psi_\varepsilon(x, t, x_0; \hbar_\varepsilon))_\varepsilon$ of the Colombeau-Schrödinger equation (0.1.8) can be represented by Colombeau-Feynman path integral of the form:
$$ (\Psi_{\varepsilon}(x, t, x_0; \hbar_{\varepsilon}))_{\varepsilon} = $$

$$ \left( \int_{\substack{q(0)=y \\ q(t)=x}} (\Psi_{0,\varepsilon}(q(0)))_{\varepsilon} \exp\left\{\frac{i}{\hbar_{\varepsilon}} \mathbf{S}_{\varepsilon}[\dot{q}(\tau), q(\tau), t]\right\} [Dq(\tau), \varepsilon] dy \right)_{\varepsilon}, \qquad (0.2.24) $$

$$ (\mathbf{S}_{\varepsilon}[\dot{q}(\tau), q(\tau), t])_{\varepsilon} = \left( \int_{0}^{t} \mathcal{L}_{\varepsilon}(\dot{q}(\tau), q(\tau), \tau) d\tau \right)_{\varepsilon}, $$

$$ (\mathcal{L}_{\varepsilon}(\dot{q}(\tau), q(\tau), \tau))_{\varepsilon} = m \frac{\dot{q}^2(\tau)}{2} - (V_{\varepsilon}(q(\tau), t))_{\varepsilon}. $$

Maslov quasiclassical asymptotic of the generalized wave function $(\Psi_{\varepsilon}(x, t; \hbar))_{\varepsilon}$ is:

$$ (\Psi_{\varepsilon}(x, t; \hbar_{\varepsilon}))_{\varepsilon} = (2\pi i(\hbar_{\varepsilon}))_{\varepsilon}^{-\frac{n}{2}} \left( \sum_{k=1}^{m(\varepsilon)} \int dy \Psi_{0,\varepsilon}(y) |D(q_{\varepsilon}^{k}(\tau; t, y, x))|^{\frac{1}{2}} \times \right. \qquad (0.2.25) $$

$$ \exp\left\{\frac{i}{\hbar_{\varepsilon}}\left[S_{\varepsilon}[q_{\varepsilon}^{k}(\tau; t, y, x)] - \frac{i\pi\gamma_{k,\varepsilon}}{2}\right]\right\}_{\varepsilon} + (O(\hbar_{\varepsilon}))_{\varepsilon}. $$

Here
$$ (\Psi_{\varepsilon}(x, t, x_0; \hbar_{\varepsilon}))_{\varepsilon} = $$

$$ \left( \int_{\substack{q(0)=y \\ q(t)=x}} (\Psi_{0,\varepsilon}(q(0)))_{\varepsilon} \exp\left\{\frac{i}{\hbar_{\varepsilon}} \mathbf{S}_{\varepsilon}[\dot{q}(\tau), q(\tau), t]\right\} [Dq(\tau), \varepsilon] dy \right)_{\varepsilon}, \qquad (0.2.24) $$

$$ (S_{\varepsilon}[\dot{q}(\tau), q(\tau), t])_{\varepsilon} = \left( \int_{0}^{t} \mathcal{L}_{\varepsilon}(\dot{q}(\tau), q(\tau), \tau) d\tau \right)_{\varepsilon}, $$

$$ (\mathcal{L}_{\varepsilon}(\dot{q}(\tau), q(\tau), \tau))_{\varepsilon} = m \frac{\dot{q}^2(\tau)}{2} - (V_{\varepsilon}(q(\tau), t))_{\varepsilon}. $$

Maslov quasiclassical asymptotic of the generalized wave function $(\Psi_{\varepsilon}(x, t; \hbar))_{\varepsilon}$ is:

$$ (\Psi_{\varepsilon}(x, t; \hbar_{\varepsilon}))_{\varepsilon} = (2\pi i(\hbar_{\varepsilon}))_{\varepsilon}^{-\frac{n}{2}} \left( \sum_{k=1}^{m(\varepsilon)} \int dy \Psi_{0,\varepsilon}(y) |D(q_{\varepsilon}^{k}(\tau; t, y, x))|^{\frac{1}{2}} \times \right. \qquad (0.2.25) $$

$$ \exp\left\{\frac{i}{\hbar_{\varepsilon}}\left[S_{\varepsilon}[q_{\varepsilon}^{k}(\tau; t, y, x)] - \frac{i\pi\gamma_{k,\varepsilon}}{2}\right]\right\}_{\varepsilon} + (O(\hbar_{\varepsilon}))_{\varepsilon}. $$

Here
$$ (\mathbf{S}_{\varepsilon}[q_{\varepsilon}^{k}(\tau; t, y, x), t])_{\varepsilon} = \left( \int_{0}^{t} \mathcal{L}_{\varepsilon}(\dot{q}_{\varepsilon}^{k}(\tau; t, y, x), q_{\varepsilon}^{k}(\tau; t, y, x), \tau) d\tau \right)_{\varepsilon}, \\ (\mathcal{L}_{\varepsilon}(\dot{q}_{\varepsilon}(\tau; t, y, x), q_{\varepsilon}(\tau; t, y, x), \tau))_{\varepsilon} = \left( m \frac{\dot{q}_{\varepsilon}^{2}(\tau; t, y, x)}{2} - V_{\varepsilon}(q_{\varepsilon}(\tau; t, y, x), \tau) \right)_{\varepsilon}, \quad (0.2.26) $$

$$ (\delta \mathbf{S}_{\varepsilon}[q_{\varepsilon}^{k}(\tau; t, y, x), t])_{\varepsilon} = 0, (q_{\varepsilon}^{k}(0; y, x, t))_{\varepsilon} = y, (q_{\varepsilon}^{k}(t; y, x))_{\varepsilon} = x, $$

$$ (D(q_{\varepsilon}^{k}(\tau; t, y, x)))_{\varepsilon} = \left( \det \left\| \frac{\partial \mathbf{S}[q_{\varepsilon}^{k}(\tau; t, y, x), t]}{\partial x \partial y} \right\| \right)_{\varepsilon} \neq 0, x \neq y. $$

Formula (0.2.25) can be written by using Colombeau-Maslov canonical operator $(K_{\Lambda_{\varepsilon}^{n}(t)}^{1/h}[\cdot])_{\varepsilon}$. Let $\Lambda_{\varepsilon}^{n}(t), \forall \varepsilon \in (0,1]$ be an $n$-dimensional Colombeau-Lagrangian manifold of class $G(\mathbb{R}^{2n})$ in the phase space $\mathbb{R}_{x,p}^{2n}$ where $x \in \mathbb{R}^n$ and let $d\sigma_{\varepsilon}$ be the volume element on $\Lambda_{\varepsilon}^{n}(t)$. A canonical atlas is a locally finite countable covering of $\Lambda_{\varepsilon}^{n}(t)$ by bounded simply-connected domains $\Omega_{j,\varepsilon}$ (the $\varepsilon$-charts) in each of which one can take as coordinates either the variables $x$ or $p$ a mixed collection

$(p_{\alpha}, x_{\beta}),$

$$ \alpha = \{\alpha_1, \dots, \alpha_s\}, \quad (0.2.27) $$

$$ \beta = \{\beta_1, \dots, \beta_{n-s}\}, $$

not containing dual pairs $(p_j, x_j)$. The Colombeau-Maslov canonical operator $(K_{\Lambda_{\varepsilon}^{n}(t)}^{1/h})_{\varepsilon}$ maps $(C_0^{\infty}(\Lambda^n(t)))^I$ into $(C(\mathbb{R}_x^n))^I, I = (0,1]$. The generalized canonical operators are introduced as follows:

(1) Let the $\varepsilon$-chart $\Omega_{j,\varepsilon}$ be non-degenerate, that is, $\Omega_{j,\varepsilon}$ is given by an equation $p_{\varepsilon} = p_{\varepsilon}(x,t)$ and
$$ (\mathbf{S}_{\varepsilon}[q_{\varepsilon}^{k}(\tau; t, y, x), t])_{\varepsilon} = \left( \int_{0}^{t} \mathcal{L}_{\varepsilon}(\dot{q}_{\varepsilon}^{k}(\tau; t, y, x), q_{\varepsilon}^{k}(\tau; t, y, x), \tau) d\tau \right)_{\varepsilon}, $$ $$ (\mathcal{L}_{\varepsilon}(\dot{q}_{\varepsilon}(\tau; t, y, x), q_{\varepsilon}(\tau; t, y, x), \tau))_{\varepsilon} = \left( m \frac{\dot{q}_{\varepsilon}^{2}(\tau; t, y, x)}{2} - V_{\varepsilon}(q_{\varepsilon}(\tau; t, y, x), \tau) \right)_{\varepsilon}, \quad (0.2.26) $$ $$ (\delta \mathbf{S}_{\varepsilon}[q_{\varepsilon}^{k}(\tau; t, y, x), t])_{\varepsilon} = 0, (q_{\varepsilon}^{k}(0; y, x, t))_{\varepsilon} = y, (q_{\varepsilon}^{k}(t; y, x))_{\varepsilon} = x, $$ $$ (D(q_{\varepsilon}^{k}(\tau; t, y, x)))_{\varepsilon} = \left( \det \left\| \frac{\partial \mathbf{S}[q_{\varepsilon}^{k}(\tau; t, y, x), t]}{\partial x \partial y} \right\| \right)_{\varepsilon} \neq 0, x \neq y. $$

Formula (0.2.25) can be written by using Colombeau-Maslov canonical operator $(K_{\Lambda_{\varepsilon}^{n}(t)}^{1/h}[\cdot])_{\varepsilon}$. Let $\Lambda_{\varepsilon}^{n}(t), \forall \varepsilon \in (0,1]$ be an $n$-dimensional Colombeau-Lagrangian manifold of class $G(\mathbb{R}^{2n})$ in the phase space $\mathbb{R}_{x,p}^{2n}$ where $x \in \mathbb{R}^{n}$ and let $d\sigma_{\varepsilon}$ be the volume element on $\Lambda_{\varepsilon}^{n}(t)$. A canonical atlas is a locally finite countable covering of $\Lambda_{\varepsilon}^{n}(t)$ by bounded simply-connected domains $\Omega_{j,\varepsilon}$ (the $\varepsilon$-charts) in each of which one can take as coordinates either the variables $x$ or $p$ a mixed collection

$$ (p_{\alpha}, x_{\beta}), $$ $$ \alpha = \{\alpha_1, \dots, \alpha_s\}, \quad (0.2.27) $$ $$ \beta = \{\beta_1, \dots, \beta_{n-s}\}, $$

not containing dual pairs $(p_j, x_j)$. The Colombeau-Maslov canonical operator $(K_{\Lambda_{\varepsilon}^{n}(t)}^{1/h})_{\varepsilon}$ maps $(C_0^{\infty}(\Lambda^n(t)))^I$ into $(C(\mathbb{R}_x^n))^I, I = (0,1]$. The generalized canonical operators are introduced as follows:

(1) Let the $\varepsilon$-chart $\Omega_{j,\varepsilon}$ be non-degenerate, that is, $\Omega_{j,\varepsilon}$ is given by an equation $p_{\varepsilon} = p_{\varepsilon}(x,t)$ and
$$ ((K_{\Omega_{j,\varepsilon}}^{1/\hbar_{\varepsilon}}[\varphi_{\varepsilon}))(x))_{\varepsilon} = \sqrt{|\left(\frac{d\sigma_{\varepsilon}}{dx}\right)_{\varepsilon}|} \left( \exp\left[\frac{i}{\hbar_{\varepsilon}}\int_{r_{\varepsilon}}^{r_{\varepsilon}} (p_{\varepsilon}, dx)\right] \varphi_{\varepsilon}(r) \right)_{\varepsilon}, \quad (0.2.28) $$

$$ r_{\varepsilon} = (x, p_{\varepsilon}(x, t)), $$

$$ (p_{\varepsilon}, dx) = \sum_{j=1}^{n} p_{j,\varepsilon} dx_j. $$

Here $(\hbar_{\varepsilon})_{\varepsilon} \in \tilde{\mathbb{R}}$ is a parameter, $r_{\varepsilon}^0 \in \Omega_{j,\varepsilon}$ is a fixed point and $\varphi_{\varepsilon} \in G(\Omega)$.

(2) Let the local coordinates in the $\varepsilon$-chart $\Omega_{j,\varepsilon}$ be $p$, that is $\Omega_{j,\varepsilon}$, is given by an equation $x_{\varepsilon} = x_{\varepsilon}(p,t)$, and let

$$ ((K_{\Omega_{j,\varepsilon}}^{1/\hbar_{\varepsilon}}(\Omega_{j,\varepsilon}[\varphi_{\varepsilon}]))(x))_{\varepsilon} = $$

$$ \left( \mathcal{F}_{1/\hbar_{\varepsilon}, p \to x}^{-1} \left\{ \sqrt{|\frac{d\sigma_{\varepsilon}}{dx}|} \exp\left[ \frac{i}{\hbar} \left( \int_{r_{\varepsilon}}^{r_{\varepsilon}} (p, dx) - (x_{\varepsilon}(p, t), p) \right) \right] \varphi_{\varepsilon}(r) \right\} \right)_{\varepsilon}, \quad (0.2.29) $$

$$ r_{\varepsilon} = (x_{\varepsilon}(p, t), p). $$

Here

$$ ((\mathcal{F}_{1/\hbar_{\varepsilon}, p \to x}^{-1}[\varphi_{\varepsilon}]))_{\varepsilon}(x) = \left( \frac{1}{(-2\pi i \hbar_{\varepsilon})^{n/2}} \int_{\mathbb{R}^n} d^n p \varphi_{\varepsilon}(p) \exp\left[\frac{i}{\hbar}(x, p)\right] \right)_{\varepsilon}. \quad (0.2.30) $$

$(K_{\Omega_{j,\varepsilon}}^{1/\hbar_{\varepsilon}})_{\varepsilon}$ is defined analogously in the case when the coordinates are some collection $(p_a, x_{\beta})$. Let $\left(\oint_{l} (p_{\varepsilon}, dx)\right)_{\varepsilon} = 0$ and let the Maslov index $(\mathrm{ind}(l_{\varepsilon}))_{\varepsilon} = 0$ for any closed path $(l_{\varepsilon})_{\varepsilon}$ lying on $\Lambda_{\varepsilon}^n(t)$. One introduces a partition of unity of class $C^{\infty}$ on $\Lambda_{\varepsilon}^n(t)$
$$ ((K_{\Omega_{j,\varepsilon}}^{1/\hbar_{\varepsilon}}[\varphi_{\varepsilon}]) (x))_{\varepsilon} = \sqrt{|\left(\frac{d\sigma_{\varepsilon}}{dx}\right)_{\varepsilon}|} \left( \exp\left[\frac{i}{\hbar_{\varepsilon}}\int_{r_{\varepsilon}}^{r_{\varepsilon}} (p_{\varepsilon}, dx)\right] \varphi_{\varepsilon}(r) \right)_{\varepsilon}, \quad (0.2.28) $$

$$ r_{\varepsilon} = (x, p_{\varepsilon}(x, t)), $$

$$ (p_{\varepsilon}, dx) = \sum_{j=1}^{n} p_{j,\varepsilon} dx_j. $$

Here $(\hbar_{\varepsilon})_{\varepsilon} \in \tilde{\mathbb{R}}$ is a parameter, $r_{\varepsilon}^0 \in \Omega_{j,\varepsilon}$ is a fixed point and $\varphi_{\varepsilon} \in G(\Omega)$.

(2) Let the local coordinates in the $\varepsilon$-chart $\Omega_{j,\varepsilon}$ be $p$, that is $\Omega_{j,\varepsilon}$, is given by an equation $x_{\varepsilon} = x_{\varepsilon}(p,t)$, and let

$$ ((K_{\Omega_{j,\varepsilon}}^{1/\hbar_{\varepsilon}}(\Omega_{j,\varepsilon}[\varphi_{\varepsilon]}))(x))_{\varepsilon} = $$

$$ \left( \mathcal{F}_{1/\hbar, \varepsilon, p \to x}^{-1} \left\{ \sqrt{|\frac{d\sigma_{\varepsilon}}{dx}|} \exp\left[ \frac{i}{\hbar} \left( \int_{r_{\varepsilon}}^{r_{\varepsilon}} (p, dx) - (x_{\varepsilon}(p, t), p) \right) \right] \varphi_{\varepsilon}(r) \right\} \right)_{\varepsilon}, \quad (0.2.29) $$

$$ r_{\varepsilon} = (x_{\varepsilon}(p, t), p). $$

Here

$$ ((\mathcal{F}_{1/\hbar, p \to x}^{-1}[\varphi_{\varepsilon}])_{\varepsilon})(x) = \left( \frac{1}{(-2\pi i \hbar_{\varepsilon})^{n/2}} \int_{\mathbb{R}^n} d^n p \varphi_{\varepsilon}(p) \exp\left[\frac{i}{\hbar}(x, p)\right] \right)_{\varepsilon}. \quad (0.2.30) $$

$(K_{\Omega_{j,\varepsilon}}^{1/\hbar_{\varepsilon}})_{\varepsilon}$ is defined analogously in the case when the coordinates are some collection $(p_a, x_{\beta})$. Let $\left(\oint_{l}(p_{\varepsilon}, dx)\right)_{\varepsilon} = 0$ and let the Maslov index $(\mathrm{ind}(l_{\varepsilon}))_{\varepsilon} = 0$ for any closed path $(l_{\varepsilon})_{\varepsilon}$ lying on $\Lambda_{\varepsilon}^n(t)$. One introduces a partition of unity of class $C^{\infty}$ on $\Lambda_{\varepsilon}^n(t)$
$$ \sum_{j=1}^{\infty} e_{j,\varepsilon}(x) = 1, \mathbf{supp}(e_{j,\varepsilon}) \subset \Omega_{j,\varepsilon} \qquad (0.2.31) $$

and one fixes a point $r_{\varepsilon}^0 = \Omega_{j_0,\varepsilon}$. The Colombeau-Maslov canonical operator is defined by

$$ \left( \left( K_{\Lambda_{\varepsilon}^n(t)}^{1/\hbar_{\varepsilon}}[\varphi_{\varepsilon}(r)] \right)_{\varepsilon} \right)(x) = \left( \left( \sum_j c_{j,\varepsilon} K_{\Omega_{j,\varepsilon}}^{1/\hbar_{\varepsilon}}[e_{j,\varepsilon}\varphi_{\varepsilon}] \right)_{\varepsilon} \right)(x), \qquad (0.2.32) $$

$$ c_{j,\varepsilon} = \exp\left(\frac{i\pi}{2} \gamma_{j,\varepsilon}^{M}\right). $$

Here $\gamma_{j,\varepsilon}^M$ is the Maslov index of a chain of charts joining the $\varepsilon$-charts $\Omega_{j_0,\varepsilon}$ and $\Omega_{j,\varepsilon}$.

A point $r_{\varepsilon} \in \Lambda_{\varepsilon}^n(t)$ is called non-singular if it has a neighbourhood in given by an equation $p = p_{\varepsilon}(x)$. Let the intersection of the $\varepsilon$-charts $\Omega_{i,\varepsilon}$ and $\Omega_{j,\varepsilon}$ be non-empty and connected, let $\forall \varepsilon \in (0,1] : r_{\varepsilon} \in \Omega_{i,\varepsilon} \cap \Omega_{j,\varepsilon}$ be a non-singular point and let $(p_a, x_{\beta}), (p_{\tilde{a}}, x_{\tilde{\beta}})$ be the coordinates in these charts. The number

$$ (\gamma_{ij,\varepsilon}^M)_{\varepsilon} = \left( \mathbf{inerdex} \left( \frac{\partial x_a(r_{\varepsilon})}{\partial p_a} \right) \right)_{\varepsilon} - \left( \mathbf{inerdex} \left( \frac{\partial x_{\tilde{a}}(r_{\varepsilon})}{\partial p_{\tilde{a}}} \right) \right)_{\varepsilon} \qquad (0.2.33) $$

is the generalized Maslov index of the pair of $\varepsilon$-nets $\Omega_{i,\varepsilon}$ and $\Omega_{j,\varepsilon}$, $\varepsilon \in (0,1]$. The Maslov index of a chain of nets is defined by additivity. The generalized Maslov index of a path is defined analogously. The Maslov index of a path (mod 4) on a Colombeau-Lagrangian manifold is an integer homotopy invariant. The Maslov canonical operator is invariant under the choice of the canonical atlas, of local coordinates in the charts and the partition of unity in the following sense: if

$$ (K_{\Lambda_{\varepsilon}^n(t)}^{1/\hbar_{\varepsilon}})_{\varepsilon}, (\tilde{K}_{\Lambda_{\varepsilon}^n(t)}^{1/\hbar_{\varepsilon}})_{\varepsilon} \text{ are two Colombeau-Maslov canonical operators, then in } G_{\mathcal{L}_2(\mathbb{R}^n)} $$

$$ \left( \left( K_{\Lambda_{\varepsilon}^n(t)}^{1/\hbar_{\varepsilon}} \varphi_{\varepsilon} - \tilde{K}_{\Lambda_{\varepsilon}^n(t)}^{1/\hbar_{\varepsilon}} \varphi_{\varepsilon} \right) \right)_{\varepsilon} (x) = (O(\hbar_{\varepsilon}))_{\varepsilon} \qquad (0.2.34) $$

for any generalized function $(\varphi_{\varepsilon})_{\varepsilon} \in G_{\mathcal{L}_2}(\Lambda_{\varepsilon}^n(t))$. Let $(L_{\varepsilon}(x, \hbar_{\varepsilon}D))_{\varepsilon}$ be a differential operator with real symbol $(L_{\varepsilon}(x, p))_{\varepsilon}$ of class $G(\mathbb{R}^{2n}_{x,p})$ and suppose that $\forall \varepsilon \in (0,1] : L_{\varepsilon}(x, p) = 0$ on $\Lambda_{\varepsilon}^n(t)$. Suppose that $(\Lambda_{\varepsilon}^n(t))_{\varepsilon}$ and the volume element $(d\sigma_{\varepsilon})_{\varepsilon}$ are invariant under the Hamiltonian system
$$ \sum_{j=1}^{\infty} e_{j,\varepsilon}(x) = 1, \mathbf{supp}(e_{j,\varepsilon}) \subset \Omega_{j,\varepsilon} \qquad (0.2.31) $$

and one fixes a point $r_{\varepsilon}^0 = \Omega_{j_0,\varepsilon}$. The Colombeau-Maslov canonical operator is defined by

$$ \left( \left( K_{\Lambda_{\varepsilon}^n(t)}^{1/\hbar_{\varepsilon}}[\varphi_{\varepsilon}(r)] \right)_{\varepsilon} \right)(x) = \left( \left( \sum_j c_{j,\varepsilon} \left( K_{\Omega_{j,\varepsilon}}^{1/\hbar_{\varepsilon}}[e_{j,\varepsilon}\varphi_{\varepsilon}] \right) \right)_{\varepsilon} \right)(x), \qquad (0.2.32) $$

$$ c_{j,\varepsilon} = \exp\left(\frac{i\pi}{2} \gamma_{j,\varepsilon}^{M}\right). $$

Here $\gamma_{j,\varepsilon}^M$ is the Maslov index of a chain of charts joining the $\varepsilon$-charts $\Omega_{j_0,\varepsilon}$ and $\Omega_{j,\varepsilon}$.

A point $r_{\varepsilon} \in \Lambda_{\varepsilon}^n(t)$ is called non-singular if it has a neighbourhood in given by an equation $p = p_{\varepsilon}(x)$. Let the intersection of the $\varepsilon$-charts $\Omega_{i,\varepsilon}$ and $\Omega_{j,\varepsilon}$ be non-empty and connected, let $\forall \varepsilon \in (0,1] : r_{\varepsilon} \in \Omega_{i,\varepsilon} \cap \Omega_{j,\varepsilon}$ be a non-singular point and let $(p_a, x_{\beta}), (p_{\tilde{a}}, x_{\tilde{\beta}})$ be the coordinates in these charts. The number

$$ (\gamma_{ij,\varepsilon}^M)_{\varepsilon} = \left( \mathbf{inerdex} \left( \frac{\partial x_a(r_{\varepsilon})}{\partial p_a} \right) \right)_{\varepsilon} - \left( \mathbf{inerdex} \left( \frac{\partial x_{\tilde{a}}(r_{\varepsilon})}{\partial p_{\tilde{a}}} \right) \right)_{\varepsilon} \qquad (0.2.33) $$

is the generalized Maslov index of the pair of $\varepsilon$-nets $\Omega_{i,\varepsilon}$ and $\Omega_{j,\varepsilon}$, $\varepsilon \in (0,1]$. The Maslov index of a chain of nets is defined by additivity. The generalized Maslov index of a path is defined analogously. The Maslov index of a path (mod 4) on a Colombeau-Lagrangian manifold is an integer homotopy invariant. The Maslov canonical operator is invariant under the choice of the canonical atlas, of local coordinates in the charts and the partition of unity in the following sense: if

$$ (K_{\Lambda_{\varepsilon}^n(t)}^{1/\hbar_{\varepsilon}})_{\varepsilon}, (\tilde{K}_{\Lambda_{\varepsilon}^n(t)}^{1/\hbar_{\varepsilon}})_{\varepsilon} \text{ are two Colombeau-Maslov canonical operators, then in } G_{\mathcal{L}_2(\mathbb{R}^n)} $$

$$ \left( \left( K_{\Lambda_{\varepsilon}^n(t)}^{1/\hbar_{\varepsilon}} \varphi_{\varepsilon} - \tilde{K}_{\Lambda_{\varepsilon}^n(t)}^{1/\hbar_{\varepsilon}} \varphi_{\varepsilon} \right) \right)_{\varepsilon} (x) = (O(\hbar_{\varepsilon}))_{\varepsilon} \qquad (0.2.34) $$

for any generalized function $(\varphi_{\varepsilon})_{\varepsilon} \in G_{\mathcal{L}_2}(\Lambda_{\varepsilon}^n(t))$. Let $(L_{\varepsilon}(x, \hbar_{\varepsilon}D))_{\varepsilon}$ be a differential operator with real symbol $(L_{\varepsilon}(x, p))_{\varepsilon}$ of class $G(\mathbb{R}^{2n}_{x,p})$ and suppose that $\forall \varepsilon \in (0,1] : L_{\varepsilon}(x, p) = 0$ on $\Lambda_{\varepsilon}^n(t)$. Suppose that $(\Lambda_{\varepsilon}^n(t))_{\varepsilon}$ and the volume element $(d\sigma_{\varepsilon})_{\varepsilon}$ are invariant under the Hamiltonian system
$$ \left( \frac{dx_{\varepsilon}(\tau)}{d\tau} \right)_{\varepsilon} = \left( \left[ \frac{\partial L_{\varepsilon}(x_{\varepsilon}(\tau), p)}{\partial p} \right]_{p=p_{\varepsilon}(\tau)} \right)_{\varepsilon}, \qquad (0.2.35) $$

$$ \left( \frac{dp_{\varepsilon}(\tau)}{d\tau} \right)_{\varepsilon} = - \left( \left[ \frac{\partial L_{\varepsilon}(x, p_{\varepsilon}(\tau))}{\partial x} \right]_{x=x_{\varepsilon}(\tau)} \right)_{\varepsilon}. $$

Then the following commutation formula is true

$$ \left( \left( L_{\varepsilon}(x, \hbar_{\varepsilon}D) (K_{\Lambda_{\varepsilon}^{n}(t)}^{1/\hbar_{\varepsilon}} \varphi_{\varepsilon}) \right)_{\varepsilon} \right) (x) = \frac{(\hbar_{\varepsilon})_{\varepsilon}}{i} \left( (K_{\Lambda_{\varepsilon}^{n}(t)}^{1/\hbar_{\varepsilon}})_{\varepsilon} \right) \left[ (\Re_{\varepsilon}\varphi_{\varepsilon})_{\varepsilon} + (O(\hbar_{\varepsilon}))_{\varepsilon} \right], \qquad (0.2.36) $$

$$ (\Re_{\varepsilon}\varphi_{\varepsilon})_{\varepsilon} = \left( \left[ \frac{d_{\varepsilon}}{d_{\varepsilon}\tau} - \frac{1}{2} \sum_{j=1}^{n} \frac{\partial^2 L_{\varepsilon}(x,p)}{\partial x_j \partial p_j} \right] \varphi_{\varepsilon} \right)_{\varepsilon} $$

where $\forall \varepsilon \in (0,1] : \varphi_{\varepsilon} \in C_{0}^{\infty}(\Lambda_{\varepsilon}^{n}(t))$ and $d_{\varepsilon}/d_{\varepsilon}\tau$ is the derivative along the integral curves of the flow of the Hamiltonian system (0.2.35). The equation $(\Re_{\varepsilon}\varphi_{\varepsilon})_{\varepsilon} = 0$ is called the generalized (Colombeau) transport equation. The commutation formula implies that if $(\Re_{\varepsilon}\varphi_{\varepsilon})_{\varepsilon} = 0$, then the function $(u_{\varepsilon})_{\varepsilon} = (K_{\Lambda_{\varepsilon}^{n}(t)}^{1/\hbar_{\varepsilon}} \varphi_{\varepsilon})$ is a formal Colombeau asymptotic solution of the equation $(L_{\varepsilon}(x, \hbar_{\varepsilon}D)u_{\varepsilon})_{\varepsilon} = 0$.

Let $\mathbf{H}$ be a complex infinite dimensional separable Hilbert space, with inner product $\langle \cdot, \cdot \rangle$ and norm $\| \cdot \|$. Let us consider now Colombeau-Couchy problem:

$$ -i((\hbar_{\varepsilon})_{\varepsilon}) \left( \frac{\partial \Psi_{\varepsilon}(t)}{\partial t} \right)_{\varepsilon} + (\hat{H}_{\varepsilon} \Psi_{\varepsilon}(t))_{\varepsilon} + ((\hbar_{\varepsilon})_{\varepsilon}) (f_{\varepsilon}(t))_{\varepsilon} = 0, \qquad (0.2.37) $$

$$ (\Psi_{\varepsilon}(0))_{\varepsilon} = (\Psi_{0,\varepsilon})_{\varepsilon}, 0 \le t \le T. $$
$$ \left( \frac{dx_{\varepsilon}(\tau)}{d\tau} \right)_{\varepsilon} = \left( \left[ \frac{\partial L_{\varepsilon}(x_{\varepsilon}(\tau), p)}{\partial p} \right]_{p=p_{\varepsilon}(\tau)} \right)_{\varepsilon}, \qquad (0.2.35) $$

$$ \left( \frac{dp_{\varepsilon}(\tau)}{d\tau} \right)_{\varepsilon} = - \left( \left[ \frac{\partial L_{\varepsilon}(x, p_{\varepsilon}(\tau))}{\partial x} \right]_{x=x_{\varepsilon}(\tau)} \right)_{\varepsilon}. $$

Then the following commutation formula is true

$$ \left( \left( L_{\varepsilon}(x, \hbar_{\varepsilon}D) (K_{\Lambda_{\varepsilon}^{n}(t)}^{1/\hbar_{\varepsilon}} \varphi_{\varepsilon}) \right)_{\varepsilon} \right) (x) = \frac{(\hbar_{\varepsilon})_{\varepsilon}}{i} \left( (K_{\Lambda_{\varepsilon}^{n}(t)}^{1/\hbar_{\varepsilon}})_{\varepsilon} \right) \left[ (\Re_{\varepsilon}\varphi_{\varepsilon})_{\varepsilon} + (O(\hbar_{\varepsilon}))_{\varepsilon} \right], \qquad (0.2.36) $$

$$ (\Re_{\varepsilon}\varphi_{\varepsilon})_{\varepsilon} = \left( \left[ \frac{d_{\varepsilon}}{d_{\varepsilon}\tau} - \frac{1}{2} \sum_{j=1}^{n} \frac{\partial^2 L_{\varepsilon}(x,p)}{\partial x_j \partial p_j} \right] \varphi_{\varepsilon} \right)_{\varepsilon} $$

where $\forall \varepsilon \in (0,1] : \varphi_{\varepsilon} \in C_{0}^{\infty}(\Lambda_{\varepsilon}^{n}(t))$ and $d_{\varepsilon}/d_{\varepsilon}\tau$ is the derivative along the integral curves of the flow of the Hamiltonian system (0.2.35). The equation $(\Re_{\varepsilon}\varphi_{\varepsilon})_{\varepsilon} = 0$ is called the generalized (Colombeau) transport equation. The commutation formula implies that if $(\Re_{\varepsilon}\varphi_{\varepsilon})_{\varepsilon} = 0$, then the function $(u_{\varepsilon})_{\varepsilon} = (K_{\Lambda_{\varepsilon}^{n}(t)}^{1/\hbar_{\varepsilon}} \varphi_{\varepsilon})$ is a formal Colombeau asymptotic solution of the equation $(L_{\varepsilon}(x, \hbar_{\varepsilon}D)u_{\varepsilon})_{\varepsilon} = 0$.

Let $\mathbf{H}$ be a complex infinite dimensional separable Hilbert space, with inner product $\langle \cdot, \cdot \rangle$ and norm $\| \cdot \|$. Let us consider now Colombeau-Couchy problem:

$$ -i((\hbar_{\varepsilon})_{\varepsilon}) \left( \frac{\partial \Psi_{\varepsilon}(t)}{\partial t} \right)_{\varepsilon} + (\hat{H}_{\varepsilon} \Psi_{\varepsilon}(t))_{\varepsilon} + ((\hbar_{\varepsilon})_{\varepsilon}) (f_{\varepsilon}(t))_{\varepsilon} = 0, \qquad (0.2.37) $$

$$ (\Psi_{\varepsilon}(0))_{\varepsilon} = (\Psi_{0,\varepsilon})_{\varepsilon}, 0 \le t \le T. $$
**Lemma 0.2.1.** Assume that:

(i) $\forall \varepsilon \in (0,1]$ : operator $H_\varepsilon$ is essentially self-adjoint,

(ii) $\forall \varepsilon \in (0,1]$ : operator $\hat{H}_\varepsilon$ is the closure of $H_\varepsilon, \Psi_\varepsilon(t) : \mathbb{R} \to \mathbf{H}$ and

(iii) $\forall \varepsilon \in (0,1]$ and $\forall t(0 \le t \le T)$ : function $f_\varepsilon(t) : \mathbb{R}_+ \to \mathbf{H}$ is continuous.

(iv) $(\Psi_{0,\varepsilon})_\varepsilon \in D((\hat{H}_\varepsilon)_\varepsilon)$.

Then there exists unique Colombeau solution $(\Psi_\varepsilon(t))_\varepsilon, 0 \le t \le T$ of the Colombeau-Couchy problem (0.2.37) and

$$ (\|\Psi_{\varepsilon}(t)\|)_{\varepsilon} \le (\|\Psi_{\varepsilon}(0)\|)_{\varepsilon} + \left( \int_{0}^{T} \|f_{\varepsilon}(t)\| dt \right)_{\varepsilon} \quad (0.2.38) $$

**Proof.** From Eq.(0.2.37) one obtain

$$ i\left(\frac{\partial \Psi_{\varepsilon}(t)}{\partial t}\right)_{\varepsilon} = \frac{1}{(\hbar_{\varepsilon})_{\varepsilon}}\left(\hat{H}_{\varepsilon}\Psi_{\varepsilon}(t)\right)_{\varepsilon} + (f_{\varepsilon}(t))_{\varepsilon}, \quad (0.2.39) $$

$$ (\Psi_{\varepsilon}(0))_{\varepsilon} = (\Psi_{0,\varepsilon})_{\varepsilon}, 0 \le t \le T. $$

Under assumptions (i)-(iv) from Eq.(0.2.39) one obtain $\forall t \in [0, T]$ :

$$ (\Psi_{\varepsilon}(t))_{\varepsilon} = \exp\left(-it\left(\frac{\hat{H}_{\varepsilon}}{\hbar_{\varepsilon}}\right)_{\varepsilon}\right) + \left(\int_{0}^{t} \exp\left(\frac{i(\tau - t)\hat{H}_{\varepsilon}}{\hbar_{\varepsilon}}\right) f_{\varepsilon}(\tau) d\tau\right)_{\varepsilon} \quad (0.2.40) $$

Thus estimate (0.2.38) follows immediately from Eq.(0.2.40) and unitarity $\forall \alpha(\alpha \in \mathbb{R})$ of the operator $\exp(i\alpha(\hat{H}_\varepsilon))$.

The next simple lemma for the classical case, given in [22] (see [22] chapter II, section 10.3).

**Lemma 0.2.2.** Let $(\Psi_\varepsilon(t,x;\hbar_\varepsilon))_\varepsilon, 0 \le t \le T$ be the Colombeau solution of the Couchy problem for the Colombeau-Schrödinger equation (0.2.42) and let $(\Psi_\varepsilon^{\text{FAS}}(t,x;\hbar_\varepsilon))_\varepsilon$ be formal asymptotic Colombeau solution of the Colombeau-Schrödinger equation
**Lemma 0.2.1.** Assume that:

(i) $\forall \varepsilon \in (0,1]$ : operator $H_\varepsilon$ is essentially self-adjoint,

(ii) $\forall \varepsilon \in (0,1]$ : operator $\hat{H}_\varepsilon$ is the closure of $H_\varepsilon, \Psi_\varepsilon(t) : \mathbb{R} \to \mathbf{H}$ and

(iii) $\forall \varepsilon \in (0,1]$ and $\forall t(0 \le t \le T)$ : function $f_\varepsilon(t) : \mathbb{R}_+ \to \mathbf{H}$ is continuous.

(iv) $(\Psi_{0,\varepsilon})_\varepsilon \in D((\hat{H}_\varepsilon)_\varepsilon)$.

Then there exists unique Colombeau solution $(\Psi_\varepsilon(t))_\varepsilon, 0 \le t \le T$ of the Colombeau-Couchy problem (0.2.37) and

$$ (\|\Psi_{\varepsilon}(t)\|)_{\varepsilon} \le (\|\Psi_{\varepsilon}(0)\|)_{\varepsilon} + \left( \int_{0}^{T} \|f_{\varepsilon}(t)\| dt \right)_{\varepsilon} \quad (0.2.38) $$

**Proof.** From Eq.(0.2.37) one obtain

$$ i \left( \frac{\partial \Psi_{\varepsilon}(t)}{\partial t} \right)_{\varepsilon} = \frac{1}{(\hbar_{\varepsilon})_{\varepsilon}} \left( \hat{H}_{\varepsilon} \Psi_{\varepsilon}(t) \right)_{\varepsilon} + (f_{\varepsilon}(t))_{\varepsilon}, \quad (0.2.39) $$

$$ (\Psi_{\varepsilon}(0))_{\varepsilon} = (\Psi_{0,\varepsilon})_{\varepsilon}, 0 \le t \le T. $$

Under assumptions (i)-(iv) from Eq.(0.2.39) one obtain $\forall t \in [0, T]$ :

$$ (\Psi_{\varepsilon}(t))_{\varepsilon} = \exp\left(-it\left(\frac{\hat{H}_{\varepsilon}}{\hbar_{\varepsilon}}\right)_{\varepsilon}\right) + \left(\int_{0}^{t} \exp\left(\frac{i(\tau - t)\hat{H}_{\varepsilon}}{\hbar_{\varepsilon}}\right) f_{\varepsilon}(\tau) d\tau\right)_{\varepsilon} \quad (0.2.40) $$

Thus estimate (0.2.38) follows immediately from Eq.(0.2.40) and unitarity $\forall \alpha(\alpha \in \mathbb{R})$ of the operator $\exp(i\alpha(\hat{H}_\varepsilon))$.

The next simple lemma for the classical case, given in [22] (see [22] chapter II, section 10.3).

**Lemma 0.2.2.** Let $(\Psi_\varepsilon(t,x;\hbar_\varepsilon))_\varepsilon, 0 \le t \le T$ be the Colombeau solution of the Couchy problem for the Colombeau-Schrödinger equation (0.2.42) and let $(\Psi_\varepsilon^{FAS}(t,x;\hbar_\varepsilon))_\varepsilon$ be formal asymptotic Colombeau solution of the Colombeau-Schrödinger equation
(0.2.42) given by using Colombeau-Maslov operator, i.e.

$$ (\Psi_{\varepsilon}^{\text{FAS}}(t, x; \hbar_{\varepsilon}))_{\varepsilon} = \left( \left( K_{\Lambda_{\varepsilon}^{n}(t)}^{1/\hbar_{\varepsilon}} \Psi_{0,\varepsilon}(x) \right)_{\varepsilon} \right). \quad (0.2.41) $$

Then an asymptotic expansion [55],[56] of the net $(\Psi_{\varepsilon}(t,x;\hbar_{\varepsilon}))_{\varepsilon}$ coincide with $(\Psi_{\varepsilon}^{\text{FAS}}(t,x;\hbar_{\varepsilon}))_{\varepsilon}$.

**The main result of this paper is:**

**Theorem 0.2.1.** Let $\mathbf{H}$ be a complex infinite dimensional separable Hilbertspace, with inner product $\langle \cdot, \cdot \rangle$ and norm $|\cdot|$. Let us consider Colombeau-Schrödinger

equation:

$$ -i((\hbar_{\varepsilon})_{\varepsilon}) \left( \frac{\partial \Psi_{\varepsilon}(t)}{\partial t} \right)_{\varepsilon} + (\hat{H}_{\varepsilon}(t) \Psi_{\varepsilon}(t))_{\varepsilon} = 0, \quad (0.2.42) $$

$$ (\Psi_{\varepsilon}(0))_{\varepsilon} = (\Psi_{0,\varepsilon}(x))_{\varepsilon}, x \in \mathbb{R}^{n}, $$

where $\forall \varepsilon \in (0,1]$ operator $H_{\varepsilon}(t)$ given via formula

$$ H_{\varepsilon}(t) = -\frac{\hbar_{\varepsilon}^2}{2m} \Delta + V_{\varepsilon}(x, t), \quad (0.2.43) $$

where $\hbar_{\varepsilon} = O(\varepsilon^k), k \in \mathbb{N}$ if $\varepsilon \to 0$ and:

(i) $\text{cl}[(V_{\varepsilon}(x, \cdot))_{\varepsilon}] \in G(\mathbb{R}^n)$,

(ii) $\forall \varepsilon \in (0,1]$ : operator $H_{\varepsilon}$ is essentially self-adjoint,

(iii) $\forall \varepsilon \in (0,1]$ : operator $\hat{H}_{\varepsilon}$ is the closure of $H_{\varepsilon}, \Psi_{\varepsilon}(t) : \mathbb{R} \to \mathbf{H}$ and

(iv) $\forall \varepsilon \in (0,1]$ : $\hat{H}_{\varepsilon}(t) : \mathbb{R}_{+} \times \mathbf{H} \to \mathbf{H}$.

Let us consider Colombeau-Schrödinger equation (0.2.42) with initial condition

$$ (\Psi_{0,\varepsilon}(\mathbf{x}))_{\varepsilon} = (\sqrt{\Psi_{\varepsilon}(\mathbf{x}, \mathbf{x}_0)})_{\varepsilon}, \text{ where} $$
(0.2.42) given by using Colombeau-Maslov operator, i.e.

$$ (\Psi_{\varepsilon}^{\text{FAS}}(t, x; \hbar_{\varepsilon}))_{\varepsilon} = \left( \left( K_{\Lambda_{\varepsilon}^{n}(t)}^{1/\hbar_{\varepsilon}} \Psi_{0,\varepsilon}(x) \right)_{\varepsilon} \right). \quad (0.2.41) $$

Then an asymptotic expansion [55],[56] of the net $(\Psi_{\varepsilon}(t,x;\hbar_{\varepsilon}))_{\varepsilon}$ coincide with $(\Psi_{\varepsilon}^{\text{FAS}}(t,x;\hbar_{\varepsilon}))_{\varepsilon}$.

**The main result of this paper is:**

**Theorem 0.2.1.** Let $\mathbf{H}$ be a complex infinite dimensional separable Hilbertspace, with inner product $\langle \cdot, \cdot \rangle$ and norm $|\cdot|$. Let us consider Colombeau-Schrödinger

equation:

$$ -i((\hbar_{\varepsilon})_{\varepsilon}) \left( \frac{\partial \Psi_{\varepsilon}(t)}{\partial t} \right)_{\varepsilon} + (\hat{H}_{\varepsilon}(t) \Psi_{\varepsilon}(t))_{\varepsilon} = 0, \quad (0.2.42) $$

$$ (\Psi_{\varepsilon}(0))_{\varepsilon} = (\Psi_{0,\varepsilon}(x))_{\varepsilon}, x \in \mathbb{R}^{n}, $$

where $\forall \varepsilon \in (0,1]$ operator $H_{\varepsilon}(t)$ given via formula

$$ H_{\varepsilon}(t) = -\frac{\hbar_{\varepsilon}^2}{2m} \Delta + V_{\varepsilon}(x, t), \quad (0.2.43) $$

where $\hbar_{\varepsilon} = O(\varepsilon^k), k \in \mathbb{N}$ if $\varepsilon \to 0$ and :

(i) $\mathbf{cl}[(V_{\varepsilon}(x, \cdot))_{\varepsilon}] \in G(\mathbb{R}^n)$,

(ii) $\forall \varepsilon \in (0,1]$ : operator $H_{\varepsilon}$ is essentially self-adjoint,

(iii) $\forall \varepsilon \in (0,1]$ : operator $\hat{H}_{\varepsilon}$ is the closure of $H_{\varepsilon}, \Psi_{\varepsilon}(t) : \mathbb{R} \to \mathbf{H}$ and

(iv) $\forall \varepsilon \in (0,1]$ : $\hat{H}_{\varepsilon}(t) : \mathbb{R}_{+} \times \mathbf{H} \to \mathbf{H}$.

Let us consider Colombeau-Schrödinger equation (0.2.42) with initial condition

$$ (\Psi_{0,\varepsilon}(\mathbf{x}))_{\varepsilon} = (\sqrt{\Psi_{\varepsilon}(\mathbf{x}, \mathbf{x}_0)})_{\varepsilon}, \text{ where} $$
$$ \Psi_{\varepsilon}(\mathbf{x}, \mathbf{x}_0) = \frac{\eta_{\varepsilon}^{n/2}}{(2\pi)^{n/2} \hbar_{\varepsilon}^{n/2}} \exp\left(-\frac{\eta_{\varepsilon}(\mathbf{x} - \mathbf{x}_0)^2}{\hbar_{\varepsilon}}\right), \qquad (0.2.44) $$

$$ 0 < \hbar_{\varepsilon} \ll \eta_{\varepsilon} \ll 1 \text{ if } \varepsilon \to 0, $$

i.e. $(\Psi_{0,\varepsilon}^2(x))_\varepsilon = \delta(x - x_0)$. We assume that $V_\varepsilon(\cdot, t)|_{\varepsilon=0} = V(\cdot, t) : \mathbb{R}^n \to \mathbb{R}$ is an polynomial function on variable $\mathbf{x}=(x_1,x_2,\dots,x_n)$, i.e.

$$ V(\mathbf{x}, t) = \sum_{\|\alpha\| \le n} g_{\alpha}(t) x^{\alpha}, \alpha = (i_1, \dots, i_n), \|\alpha\| = \sum_{r} i_r, i = 1, \dots, n, \qquad (0.2.45) $$

$$ x^{\alpha} = x_1^{i_1} x_2^{i_2} \dots x_n^{i_n} $$

Let $\mathbf{u}(\tau, t, \lambda, \mathbf{x}) = (u_1(\tau, t, \lambda, \mathbf{x}), \dots, u_n(\tau, t, \lambda, \mathbf{x}))$ be the solution of the linear differential boundary problem (differential master equation):

$$ \frac{d^2 \mathbf{u}(\tau, t, \lambda, \mathbf{y}, \mathbf{x})}{d\tau^2} = \mathbf{Hess}[V(\lambda, \tau)]\mathbf{u}^T(\tau, t, \lambda, \mathbf{y}, \mathbf{x}) + [\mathbf{V}'(\lambda, \tau)]^T, \qquad (0.2.46) $$

$$ \mathbf{u}(0, t, \lambda, \mathbf{y}, \mathbf{x}) = \mathbf{y}, \mathbf{u}(t, t, \lambda, \mathbf{y}, \mathbf{x}) = \mathbf{x}. $$

Here $\mathbf{y}, \mathbf{x} \in \mathbb{R}^n$, $\lambda = (\lambda_1, \dots, \lambda_n)$, $\mathbf{u}^T(\tau, t, \lambda, \mathbf{y}, \mathbf{x}) = (u_1(\tau, t, \lambda, \mathbf{y}, \mathbf{x}), \dots, u_n(\tau, t, \lambda, \mathbf{y}, \mathbf{x}))^T$,

$$ \mathbf{V}'(\lambda, \tau) = \left( \frac{\partial V(\mathbf{x}, \tau)}{\partial x_1}, \dots, \frac{\partial V(\mathbf{x}, \tau)}{\partial x_n} \right)_{\mathbf{x}=\lambda} \qquad (0.2.47) $$

and here $\mathbf{Hess}[V(\lambda, \tau)]$ is the Hessian of the function $V(\cdot, \tau) : \mathbb{R}^n \to \mathbb{R}$ at point $\mathbf{x} = \lambda$, i.e.
$$ \Psi_{\varepsilon}(\mathbf{x}, \mathbf{x}_0) = \frac{\eta_{\varepsilon}^{n/2}}{(2\pi)^{n/2} \hbar_{\varepsilon}^{n/2}} \exp\left(-\frac{\eta_{\varepsilon}(\mathbf{x} - \mathbf{x}_0)^2}{\hbar_{\varepsilon}}\right), \qquad (0.2.44) $$

$$ 0 < \hbar_{\varepsilon} \ll \eta_{\varepsilon} \ll 1 \text{ if } \varepsilon \to 0, $$

i.e. $(\Psi_{0,\varepsilon}^2(x))_\varepsilon = \delta(x - x_0)$. We assume that $V_\varepsilon(\cdot, t)|_{\varepsilon=0} = V(\cdot, t) : \mathbb{R}^n \to \mathbb{R}$ is an polynomial function on variable $\mathbf{x} = (x_1, x_2, \dots, x_n)$, i.e.

$$ V(\mathbf{x}, t) = \sum_{\|\alpha\| \le n} g_{\alpha}(t) x^{\alpha}, \alpha = (i_1, \dots, i_n), \|\alpha\| = \sum_{r} i_r, i = 1, \dots, n, \qquad (0.2.45) $$

$$ x^{\alpha} = x_1^{i_1} x_2^{i_2} \dots x_n^{i_n} $$

Let $\mathbf{u}(\tau, t, \lambda, \mathbf{x}) = (u_1(\tau, t, \lambda, \mathbf{x}), \dots, u_n(\tau, t, \lambda, \mathbf{x}))$ be the solution of the linear differential boundary problem (differential master equation):

$$ \frac{d^2 \mathbf{u}(\tau, t, \lambda, \mathbf{y}, \mathbf{x})}{d\tau^2} = \mathbf{Hess}[V(\lambda, \tau)] \mathbf{u}^T(\tau, t, \lambda, \mathbf{y}, \mathbf{x}) + [\mathbf{V}'(\lambda, \tau)]^T, \qquad (0.2.46) $$

$$ \mathbf{u}(0, t, \lambda, \mathbf{y}, \mathbf{x}) = \mathbf{y}, \mathbf{u}(t, t, \lambda, \mathbf{y}, \mathbf{x}) = \mathbf{x}. $$

Here $\mathbf{y}, \mathbf{x} \in \mathbb{R}^n$, $\lambda = (\lambda_1, \dots, \lambda_n)$, $\mathbf{u}^T(\tau, t, \lambda, \mathbf{y}, \mathbf{x}) = (u_1(\tau, t, \lambda, \mathbf{y}, \mathbf{x}), \dots, u_n(\tau, t, \lambda, \mathbf{y}, \mathbf{x}))^T$,

$$ \mathbf{V}'(\lambda, \tau) = \left( \frac{\partial V(\mathbf{x}, \tau)}{\partial x_1}, \dots, \frac{\partial V(\mathbf{x}, \tau)}{\partial x_n} \right)_{\mathbf{x}=\lambda} \qquad (0.2.47) $$

and here $\mathbf{Hess}[V(\lambda, \tau)]$ is the Hessian of the function $V(\cdot, \tau) : \mathbb{R}^n \to \mathbb{R}$ at point $\mathbf{x} = \lambda$, i.e.
$$
\begin{align}
\mathbf{Hess}[V(\lambda, \tau)] &= \nonumber \\
&\quad \mathbf{Hess}_{ij}[V(\lambda, \tau)] = \left[ \frac{\partial^2 V(\mathbf{x}, \tau)}{\partial x_i \partial x_j} \right]_{\mathbf{x}=\lambda} \tag{0.2.48}
\end{align}
$$

and here $\mathbf{Hess}_{ij}[V(\lambda, t)]$ is the Hessian of the function $V(\cdot, t) : \mathbb{R}^n \to \mathbb{R}$ at point $\mathbf{x} = \lambda$. Let $S(t, \lambda, \mathbf{y}, \mathbf{x})$ be a function (master action corresponding to Eq.(0.2.46))

$$
S(t, \lambda, \mathbf{y}, \mathbf{x}) = \int_{0}^{t} \tilde{\mathcal{L}}(\dot{\mathbf{u}}(\tau, t, \lambda, \mathbf{y}, \mathbf{x}), \mathbf{u}(\tau, t, \lambda, \mathbf{y}, \mathbf{x}), \tau) d\tau, \quad (0.2.49)
$$

where function $\tilde{\mathcal{L}}(\dot{\mathbf{u}}(\tau), \mathbf{u}(\tau), \tau)$ (master Lagrangian corresponding to Eq.(0.2.46)) is

$$
\begin{align}
\tilde{\mathcal{L}}(\dot{\mathbf{u}}(\tau), \mathbf{u}(\tau), \tau) &= m \frac{\dot{\mathbf{u}}^2(\tau)}{2} - \hat{V}(\mathbf{u}(\tau), \tau), \tag{0.2.50} \\
\hat{V}(\mathbf{u}(\tau), \tau) &= [V'(\lambda, \tau)]\mathbf{u}^T(\tau) + \frac{1}{2}\mathbf{u}(\tau, \lambda)\mathbf{Hess}[V(\lambda, \tau)]\mathbf{u}^T(\tau) \nonumber
\end{align}
$$

Let $\mathbf{y}_{\text{cr}}(t, \lambda, \mathbf{x})$ be solution for a system of equations:

$$
\begin{align}
\frac{\partial S(t, \lambda, \mathbf{x}, \mathbf{y}_{\text{cr}}(t, \lambda, \mathbf{y}))}{\partial y_i} &= 0, \tag{0.2.51} \\
i &= 1, \dots, n. \notag
\end{align}
$$

Let $\hat{\mathbf{u}}(t, \lambda, \mathbf{x}_0) = \mathbf{x}_{\text{cr}}(t, \lambda, \mathbf{x}_0)$ be solution for a system of equations:
$$
\begin{align}
\mathbf{Hess}[V(\lambda, \tau)] &= \nonumber \\
&\quad \mathbf{Hess}_{ij}[V(\lambda, \tau)] = \left[ \frac{\partial^2 V(\mathbf{x}, \tau)}{\partial x_i \partial x_j} \right]_{\mathbf{x}=\lambda} \tag{0.2.48}
\end{align}
$$

and here $\mathbf{Hess}_{ij}[V(\lambda, t)]$ is the Hessian of the function $V(\cdot, t) : \mathbb{R}^n \to \mathbb{R}$ at point $\mathbf{x} = \lambda$. Let $S(t, \lambda, \mathbf{y}, \mathbf{x})$ be a function (master action corresponding to Eq.(0.2.46))

$$
S(t, \lambda, \mathbf{y}, \mathbf{x}) = \int_{0}^{t} \tilde{\mathcal{L}}(\dot{\mathbf{u}}(\tau, t, \lambda, \mathbf{y}, \mathbf{x}), \mathbf{u}(\tau, t, \lambda, \mathbf{y}, \mathbf{x}), \tau) d\tau, \quad (0.2.49)
$$

where function $\tilde{\mathcal{L}}(\dot{\mathbf{u}}(\tau), \mathbf{u}(\tau), \tau)$ (master Lagrangian corresponding to Eq.(0.2.46)) is

$$
\begin{align}
\tilde{\mathcal{L}}(\dot{\mathbf{u}}(\tau), \mathbf{u}(\tau), \tau) &= m \frac{\dot{\mathbf{u}}^2(\tau)}{2} - \hat{V}(\mathbf{u}(\tau), \tau), \tag{0.2.50} \\
\hat{V}(\mathbf{u}(\tau), \tau) &= [V'(\lambda, \tau)]\mathbf{u}^T(\tau) + \frac{1}{2}\mathbf{u}(\tau, \lambda)\mathbf{Hess}[V(\lambda, \tau)]\mathbf{u}^T(\tau) \nonumber
\end{align}
$$

Let $\mathbf{y}_{\text{cr}}(t, \lambda, \mathbf{x})$ be solution for a system of equations:

$$
\begin{align}
\frac{\partial S(t, \lambda, \mathbf{x}, \mathbf{y}_{\text{cr}}(t, \lambda, \mathbf{y}))}{\partial y_i} &= 0, \tag{0.2.51} \\
i &= 1, \dots, n. \notag
\end{align}
$$

Let $\hat{\mathbf{u}}(t, \lambda, \mathbf{x}_0) = \mathbf{x}_{\text{cr}}(t, \lambda, \mathbf{x}_0)$ be solution for a system of equations:
$$y_{cr}(t, \lambda, \mathbf{x}_{cr}) + \lambda - \mathbf{x}_{0} = 0. \qquad (0.2.52)$$

We assume now that: for a fixed $t, \mathbf{x}_0$ and $\lambda$ point $x_{cr}(t, \lambda, \mathbf{x}_0)$ does not focal point on the trajectory, given via solution of the linear boundary problem (0.2.46).

Then for any Colombeau solution of the Colombeau-Schrödinger equation Eq.(0.2.42) with initial condition (0.2.43) the inequalities is satisfied:

$$\limsup_{\epsilon \to 0} |\langle i, t, \mathbf{x}_0; \hbar_\epsilon, \epsilon \rangle - \lambda_i| \le |\hat{u}_i(t, \lambda, \mathbf{x}_0)|, \qquad (0.2.53)$$

$$i = 1, \dots, n.$$

Here the $\langle i, t, \mathbf{x}_0; \hbar_\epsilon, \epsilon \rangle$ quantum averages given by Eq.(0.2.2),i.e.

$$\langle i, t, \mathbf{x}_0; \hbar_\epsilon, \epsilon \rangle = \frac{\int_{\mathbb{R}^n} x_i |\Psi_\epsilon(\mathbf{x}, t, \mathbf{x}_0; \hbar)|^2 d^n x}{\int_{\mathbb{R}^n} |\Psi_\epsilon(\mathbf{x}, t, \mathbf{x}_0; \hbar)|^2 d^n x}, \qquad (0.2.54)$$

$$i = 1, \dots, n.$$

**Example.** Let's calculate master equation for the one dimensional quantum anharmonic oscillator with a cubic potential,supplemented by an additive sinusoidal driving i.e.

$$V(x, \tau) = \frac{m\omega^2}{2}x^2 + ax^3 - bx - [A \sin(\Omega\tau)]x. \qquad (0.2.55)$$

The corresponding classical Lagrangian is
$$y_{cr}(t, \lambda, \mathbf{x}_{cr}) + \lambda - \mathbf{x}_{0} = 0. \qquad (0.2.52)$$

We assume now that: for a fixed $t, \mathbf{x}_0$ and $\lambda$ point $x_{cr}(t, \lambda, \mathbf{x}_0)$ does not focal point on the trajectory, given via solution of the linear boundary problem (0.2.46).

Then for any Colombeau solution of the Colombeau-Schrödinger equation Eq.(0.2.42) with initial condition (0.2.43) the inequalities is satisfied:

$$\limsup_{\epsilon \to 0} |\langle i, t, \mathbf{x}_0; \hbar_\epsilon, \epsilon \rangle - \lambda_i| \le |\hat{u}_i(t, \lambda, \mathbf{x}_0)|, \qquad (0.2.53)$$

$$i = 1, \dots, n.$$

Here the $\langle i, t, \mathbf{x}_0; \hbar_\epsilon, \epsilon \rangle$ quantum averages given by Eq.(0.2.2),i.e.

$$\langle i, t, \mathbf{x}_0; \hbar_\epsilon, \epsilon \rangle = \frac{\int_{\mathbb{R}^n} x_i |\Psi_\epsilon(\mathbf{x}, t, \mathbf{x}_0; \hbar)|^2 d^n x}{\int_{\mathbb{R}^n} |\Psi_\epsilon(\mathbf{x}, t, \mathbf{x}_0; \hbar)|^2 d^n x}, \qquad (0.2.54)$$

$$i = 1, \dots, n.$$

**Example.** Let's calculate master equation for the one dimensional quantum anharmonic oscillator with a cubic potential,supplemented by an additive sinusoidal driving i.e.

$$V(x, \tau) = \frac{m\omega^2}{2}x^2 + ax^3 - bx - [A \sin(\Omega\tau)]x. \qquad (0.2.55)$$

The corresponding classical Lagrangian is
$$ \mathcal{L}(\dot{x}(\tau), x(\tau), \tau) = \frac{m}{2} \dot{x}^2(\tau) - \frac{m\omega^2}{2} x^2(\tau) - ax^3(\tau) + bx(\tau) + [A \sin(\Omega\tau)]x(\tau). \quad (0.2.56) $$

From Eq.(0.2.48) and Eq.(0.2.50) using Eq.(0.2.55) we obtain

$$ \mathbf{V}'(\boldsymbol{\lambda}, \tau) = \left[ \frac{\partial V(x, \tau)}{\partial x} \right]_{x=\boldsymbol{\lambda}} = [m\omega^2 x + 3ax^2 - b - A \sin(\Omega\tau)]_{x=\boldsymbol{\lambda}} = \\ m\omega^2 \boldsymbol{\lambda} + 3a\boldsymbol{\lambda}^2 - b - A \sin(\Omega\tau), $$

$$ \mathbf{Hess}[V(\boldsymbol{\lambda}, \tau)] = \left[ \frac{\partial^2 V(x, \tau)}{\partial x^2} \right]_{x=\boldsymbol{\lambda}} = [m\omega^2 + 6ax]_{x=\boldsymbol{\lambda}} = m\omega^2 + 6a\boldsymbol{\lambda}, \quad (0.2.57) $$

$$ \hat{V}(u, \tau) = [V'(\boldsymbol{\lambda}, \tau)]u + \frac{1}{2} \mathbf{Hess}[V(\boldsymbol{\lambda}, \tau)]u^2 = \\ \frac{1}{2}(m\omega^2 + 6a\boldsymbol{\lambda})u^2 + (m\omega^2\boldsymbol{\lambda} + 3a\boldsymbol{\lambda}^2 - b - A \sin(\Omega\tau))u. $$

Therefore differential master equation (0.2.46) is

$$ \frac{d^2 u(\tau, t, \boldsymbol{\lambda}, y, x)}{d\tau^2} = (m\omega^2 + 6a\boldsymbol{\lambda})u(\tau, t, \boldsymbol{\lambda}, y, x) + (m\omega^2\boldsymbol{\lambda} + 3a\boldsymbol{\lambda}^2 - b - A \sin(\Omega\tau)), \quad (0.2.58) $$

$$ u(0, t, \boldsymbol{\lambda}, y, x) = y, u(t, t, \boldsymbol{\lambda}, y, x) = x. $$

The corresponding master Lagrangian $\tilde{\mathcal{L}}(\dot{u}, u)$ is
$$ \mathcal{L}(\dot{x}(\tau), x(\tau), \tau) = \frac{m}{2} \dot{x}^2(\tau) - \frac{m\omega^2}{2} x^2(\tau) - ax^3(\tau) + bx(\tau) + [A \sin(\Omega\tau)]x(\tau). \quad (0.2.56) $$

From Eq.(0.2.48) and Eq.(0.2.50) using Eq.(0.2.55) we obtain

$$ \mathbf{V}'(\boldsymbol{\lambda}, \tau) = \left[ \frac{\partial V(x, \tau)}{\partial x} \right]_{x=\boldsymbol{\lambda}} = [m\omega^2 x + 3ax^2 - b - A \sin(\Omega\tau)]_{x=\boldsymbol{\lambda}} = \\ m\omega^2 \boldsymbol{\lambda} + 3a\boldsymbol{\lambda}^2 - b - A \sin(\Omega\tau), $$

$$ \mathbf{Hess}[V(\boldsymbol{\lambda}, \tau)] = \left[ \frac{\partial^2 V(x, \tau)}{\partial x^2} \right]_{x=\boldsymbol{\lambda}} = [m\omega^2 + 6ax]_{x=\boldsymbol{\lambda}} = m\omega^2 + 6a\boldsymbol{\lambda}, \quad (0.2.57) $$

$$ \hat{V}(u, \tau) = [V'(\boldsymbol{\lambda}, \tau)]u + \frac{1}{2} \mathbf{Hess}[V(\boldsymbol{\lambda}, \tau)]u^2 = \\ \frac{1}{2}(m\omega^2 + 6a\boldsymbol{\lambda})u^2 + (m\omega^2\boldsymbol{\lambda} + 3a\boldsymbol{\lambda}^2 - b - A \sin(\Omega\tau))u. $$

Therefore differential master equation (0.2.46) is

$$ \frac{d^2 u(\tau, t, \boldsymbol{\lambda}, y, x)}{d\tau^2} = (m\omega^2 + 6a\boldsymbol{\lambda})u(\tau, t, \boldsymbol{\lambda}, y, x) + (m\omega^2\boldsymbol{\lambda} + 3a\boldsymbol{\lambda}^2 - b - A \sin(\Omega\tau)), \quad (0.2.58) $$

$$ u(0, t, \boldsymbol{\lambda}, y, x) = y, u(t, t, \boldsymbol{\lambda}, y, x) = x. $$

The corresponding master Lagrangian $\tilde{\mathcal{L}}(\dot{u}, u)$ is
$$ \tilde{\mathcal{L}}(\dot{u}, u) = $$

$$ \tilde{\mathcal{L}}(\dot{u}, u) = \frac{m}{2}\dot{u}^2 - \frac{1}{2}(m\omega^2 + 6a\lambda)u^2 - (m\omega^2\lambda + 3a\lambda^2 - b - A\sin(\Omega t))u = \quad (0.2.59) $$

$$ \frac{m}{2}\dot{u}^2 - \frac{m}{2}\left(\omega^2 + \frac{6a\lambda}{m}\right)u^2 - (m\omega^2\lambda + 3a\lambda^2 - b - A\sin(\Omega t))u. $$

We assume now that $\omega^2 + \frac{6a\lambda}{m} \ge 0$ and rewrite master Lagrangian $\tilde{\mathcal{L}}(\dot{u}, u)$ given by Eq.(0.2.59) in the next form

$$ \tilde{\mathcal{L}}(\dot{u}, u) = \frac{m}{2}\dot{u}^2 - \frac{m\omega^2(\lambda)}{2}u^2 + g(\lambda, \tau)u, $$

$$ \omega(\lambda) = \left| \sqrt{\omega^2 + \frac{6a\lambda}{m}} \right|, \qquad (0.2.60) $$

$$ g(\lambda, \tau) = -(m\omega^2\lambda + 3a\lambda^2 - b - A\sin(\Omega\tau)). $$

The Euler-Lagrange equation of $\tilde{\mathcal{L}}(\dot{u}, u)$ with corresponding boundary conditions is

$$ \frac{d}{dt}\left(\frac{\partial\tilde{\mathcal{L}}}{\partial\dot{u}}\right) - \frac{\partial\tilde{\mathcal{L}}}{\partial u} = $$

$$ m\ddot{u} + m\omega^2(\lambda) - g(\lambda, \tau) = 0, \qquad (0.2.61) $$

$$ u(0, \tau, \lambda, y, x) = y, \quad u(t, t, \lambda, y, x) = x. $$

Corresponding master action $S(t, \lambda, y, x)$ given by Eq.(0.2.49) is:
$$ \tilde{\mathcal{L}}(\dot{u}, u) = $$

$$ \tilde{\mathcal{L}}(\dot{u}, u) = \frac{m}{2} \dot{u}^2 - \frac{1}{2} (m\omega^2 + 6a\lambda)u^2 - (m\omega^2\lambda + 3a\lambda^2 - b - A\sin(\Omega t))u = \quad (0.2.59) $$

$$ \frac{m}{2} \dot{u}^2 - \frac{m}{2} \left( \omega^2 + \frac{6a\lambda}{m} \right) u^2 - (m\omega^2\lambda + 3a\lambda^2 - b - A\sin(\Omega t))u. $$

We assume now that $\omega^2 + \frac{6a\lambda}{m} \ge 0$ and rewrite master Lagrangian $\tilde{\mathcal{L}}(\dot{u}, u)$ given by Eq.(0.2.59) in the next form

$$ \tilde{\mathcal{L}}(\dot{u}, u) = \frac{m}{2} \dot{u}^2 - \frac{m\omega^2(\lambda)}{2} u^2 + g(\lambda, \tau)u, $$

$$ \omega(\lambda) = \left| \sqrt{\omega^2 + \frac{6a\lambda}{m}} \right|, \qquad (0.2.60) $$

$$ g(\lambda, \tau) = -(m\omega^2\lambda + 3a\lambda^2 - b - A\sin(\Omega\tau)). $$

The Euler-Lagrange equation of $\tilde{\mathcal{L}}(\dot{u}, u)$ with corresponding boundary conditions is

$$ \frac{d}{dt} \left( \frac{\partial \tilde{\mathcal{L}}}{\partial \dot{u}} \right) - \frac{\partial \tilde{\mathcal{L}}}{\partial u} = $$

$$ m\ddot{u} + m\omega^2(\lambda) - g(\lambda, \tau) = 0, \qquad (0.2.61) $$

$$ u(0, \tau, \lambda, y, x) = y, \quad u(t, t, \lambda, y, x) = x. $$

Corresponding master action $S(t, \lambda, y, x)$ given by Eq.(0.2.49) is:
$$
\begin{equation}
\begin{aligned}
S(t, \lambda, y, x) = \frac{m\omega}{2\sin\omega T} & \left[ (\cos(\omega t))(y^2 + x^2) - 2yx + \frac{2x}{m\omega} \int_0^t g(\lambda, \tau) \sin(\omega\tau) d\tau - \right. \\
& \left. \qquad + \frac{2y}{m\omega} \int_0^t g(\lambda, \tau) \sin[\omega(t-\tau)] d\tau - \right. \\
& \left. \qquad - \frac{2}{m^2\omega^2} \int_0^t \int_0^\tau g(\lambda, \tau) g(\lambda, s) [\sin\omega(t-\tau)] \sin(\omega s) ds d\tau \right].
\end{aligned}
\tag{0.2.62}
\end{equation}
$$

System of the Eqs.(0.2.51) reduces to a single equation

$$
\frac{\partial S(t, \lambda, y, x)}{\partial x} = x_{cr}(t, \lambda, y) \cos(\omega t) - y + \frac{1}{m\omega} \int_{0}^{t} g(\lambda, \tau) \sin(\omega\tau) d\tau = 0. \quad (0.2.63)
$$

Therefore function $\hat{u}(t, \lambda, y) = x_{cr}(t, \lambda, y)$ is

$$
x_{cr}(t, \lambda, y) = \frac{1}{\cos(\omega t)} \left( y - \frac{1}{m\omega} \int_{0}^{t} g(\lambda, \tau) \sin(\omega\tau) d\tau \right) \quad (0.2.64)
$$

System of the Eqs.(0.2.52) reduces to a single equation

$$
\frac{1}{\cos(\omega t)} \left( y_{cr}(t, \lambda, x_0) - \frac{1}{m\omega} \int_0^t g(\lambda, \tau) \sin(\omega\tau) d\tau \right) + \lambda - x_0 = 0. \quad (0.2.65)
$$

I.0.3.Colombeau-Wiener measure $(\tilde{D}_x^\mathbb{R}[\omega(\tau), v_\varepsilon])_\varepsilon$
and regularized Colombeau-Feynman Path Integral
$$
\begin{equation}
\begin{aligned}
S(t, \lambda, y, x) = \frac{m\omega}{2\sin\omega T} & \left[ (\cos(\omega t))(y^2 + x^2) - 2yx + \frac{2x}{m\omega} \int_0^t g(\lambda, \tau) \sin(\omega\tau) d\tau - \right. \\
& \left. \qquad + \frac{2y}{m\omega} \int_0^t g(\lambda, \tau) \sin[\omega(t-\tau)] d\tau - \right. \\
& \left. \qquad - \frac{2}{m^2\omega^2} \int_0^t \int_0^\tau g(\lambda, \tau) g(\lambda, s) [\sin\omega(t-\tau)] \sin(\omega s) ds d\tau \right].
\end{aligned}
\tag{0.2.62}
\end{equation}
$$

System of the Eqs.(0.2.51) reduces to a single equation

$$
\frac{\partial S(t, \lambda, y, x)}{\partial x} = x_{cr}(t, \lambda, y) \cos(\omega t) - y + \frac{1}{m\omega} \int_{0}^{t} g(\lambda, \tau) \sin(\omega\tau) d\tau = 0. \quad (0.2.63)
$$

Therefore function $\hat{u}(t, \lambda, y) = x_{cr}(t, \lambda, y)$ is

$$
x_{cr}(t, \lambda, y) = \frac{1}{\cos(\omega t)} \left( y - \frac{1}{m\bar{\omega}} \int_{0}^{t} g(\lambda, \tau) \sin(\omega\tau) d\tau \right) \quad (0.2.64)
$$

System of the Eqs.(0.2.52) reduces to a single equation

$$
\frac{1}{\cos(\omega t)} \left( y_{cr}(t, \lambda, x_0) - \frac{1}{m\bar{\omega}} \int_0^t g(\lambda, \tau) \sin(\omega\tau) d\tau \right) + \lambda - x_0 = 0. \quad (0.2.65)
$$

I.0.3.Colombeau-Wiener measure $(\tilde{D}_x^\mathbb{R}[\omega(\tau), v_\varepsilon])_\varepsilon$
and regularized Colombeau-Feynman Path Integral
based on measure $(\tilde{D}_x^{\mathbb{R}}[\omega(\tau), v_\varepsilon])_\varepsilon$.

Let $x = (x_1, x_2, \dots, x_n)$ denote a point in $\mathbb{R}^d$ and $|x|$ denote its Euclidean length from the origin. Let us consider the initial-value problem for the Schrödinger equation (0.1.1) with $V \equiv 0$, i.e.

$$ \frac{\partial \psi(t,x)}{\partial t} = \frac{i}{2m} \Delta \psi(t,x), \psi(0) = \psi_0, x \in \mathbb{R}^n. \qquad (0.3.1) $$

On the domain $D(\Delta)$ of all square-integrable $\psi$, such that also $-|p|^2 \mathcal{F}\psi \in \mathcal{L}_2(\mathbb{R}^d)$ one can write

$$ \Delta\psi = \mathcal{F}^{-1}[(-|p|^2)[\mathcal{F}\psi](p)]. \qquad (0.3.2) $$

Here $p$ denotes the variable in momentum space and $|p|^2 = \sum_{i=1}^d p_i^2$. Therefore $\Delta$ is self-adjoint, and if we denote

$$ \Xi_m^t = \exp\left[\frac{it}{2m}\Delta\right], \qquad (0.3.3) $$

then $\psi(t,x) = \Xi_m^t \psi_0(x)$ is the solution of (0.3.1). Explicitly,

$$ \Xi_m^t \psi_0(x) = \left(\frac{2\pi it}{m}\right)^{-d/2} \int_{\mathbb{R}^n} d^d y \psi_0(y) \exp\left[i \frac{m}{2} \left(\frac{|x-y|^2}{t}\right)\right] \qquad (0.3.4) $$

The integral in (0.3.4) exists for all $\psi \in \mathcal{L}_1 \cap \mathcal{L}_2$ for a general $\psi_0 \in \mathcal{L}_2$ we define $\Xi_m^t \psi_0(x)$ as the limit $\Xi_{m,\Lambda}^t \psi_0(x)$ in $\mathcal{L}_2$ as $\Lambda \to \infty$ of the integral (0.3.4) restricted to $|y| \le \Lambda$. Suppose now that $1/2m$ is replaced by 0 in Eq. (0.1.1). The operator $V$ of multiplication by the function $V(t,x)$, on the domain $D(V)$ of all $\psi_0$ in $\mathcal{L}_2$ such that $V\psi_0$ is also in $\mathcal{L}_2$, is self-adjoint, and if
based on measure $\left( \tilde{D}_x^{\mathbb{R}} [\omega(\tau), v_\epsilon] \right)_\epsilon$.

Let $x = (x_1, x_2, \dots, x_n)$ denote a point in $\mathbb{R}^d$ and $|x|$ denote its Euclidean length from the origin. Let us consider the initial-value problem for the Schrödinger equation (0.1.1) with $V \equiv 0$, i.e.

$$ \frac{\partial \psi(t,x)}{\partial t} = \frac{i}{2m} \Delta \psi(t,x), \psi(0) = \psi_0, x \in \mathbb{R}^n. \qquad (0.3.1) $$

On the domain $D(\Delta)$ of all square-integrable $\psi$, such that also $-|p|^2 \mathcal{F}\psi \in \mathcal{L}_2(\mathbb{R}^d)$ one can write

$$ \Delta\psi = \mathcal{F}^{-1}[(-|p|^2)[\mathcal{F}\psi](p)]. \qquad (0.3.2) $$

Here $p$ denotes the variable in momentum space and $|p|^2 = \sum_{i=1}^d p_i^2$. Therefore $\Delta$ is self-adjoint, and if we denote

$$ \Xi_m^t = \exp\left[\frac{it}{2m}\Delta\right], \qquad (0.3.3) $$

then $\psi(t,x) = \Xi_m^t \psi_0(x)$ is the solution of (0.3.1). Explicitly,

$$ \Xi_m^t \psi_0(x) = \left(\frac{2\pi it}{m}\right)^{-d/2} \int_{\mathbb{R}^n} d^d y \psi_0(y) \exp\left[i \frac{m}{2} \left(\frac{|x-y|^2}{t}\right)\right] \qquad (0.3.4) $$

The integral in (0.3.4) exists for all $\psi \in \mathcal{L}_1 \cap \mathcal{L}_2$ for a general $\psi_0 \in \mathcal{L}_2$ we define $\Xi_m^t \psi_0(x)$ as the limit $\Xi_{m,\Lambda}^t \psi_0(x)$ in $\mathcal{L}_2$ as $\Lambda \to \infty$ of the integral (0.3.4) restricted to $|y| \le \Lambda$. Suppose now that $1/2m$ is replaced by 0 in Eq. (0.1.1). The operator $V$ of multiplication by the function $V(t,x)$, on the domain $D(V)$ of all $\psi_0$ in $\mathcal{L}_2$ such that $V\psi_0$ is also in $\mathcal{L}_2$, is self-adjoint, and if
$$ M_V^t = \exp\{-itV\}, \qquad (0.3.5) $$

then $\psi(t,x) = M_V^t \psi_0(x)$ is the solution of Eq.(0.3.1) with $1/2m$ replaced by 0.

Asume that Kato conditions (see Appendix V) under which the operator $(1/2m)\Delta - V$ is self-adjoint is satisfied.It well known [41] that under Kato conditions if we let

$$ U_{m,V}^t = \exp\left[it\left(\frac{1}{2m}\Delta - V\right)\right] \qquad (0.3.6) $$

then Trotter's theorem asserts that for all $\psi_0 \in \mathcal{L}_2$:

$$ U_{m,V}^t \psi_0 = \lim_{n \to \infty} \left( \Xi_m^{\frac{t}{n}} M_V^{\frac{t}{n}} \right)^n \psi_0. \qquad (0.3.7) $$

Using (0.3.4) and (0.3.5) one find that

$$ U_{m,V}^t \psi_0 = \lim_{n \to \infty} \left[ \left( \frac{2\pi i t}{nm} \right)^{-\frac{1}{2}dn} \int \dots \int \exp[iS(x_0, \dots, x_n)] \psi_0(x_n) dx_1 \dots dx_n \right]. \qquad (0.3.8) $$

Here we set $x_0 = x$ and

$$ S(x_0, \dots, x_n) = \left(\frac{t}{n}\right) \sum_{j=1}^{n} \left[ \frac{m}{2} \frac{(x_i - x_{j-1})^2}{(t/n)^2} - V(x_i) \right]. \qquad (0.3.9) $$

The limit in RHS of the Eq.(0.3.8) is well-defined, and according to Eq.(0.3.7) converges in $\mathcal{L}_2$ to the solution of the Schrödinger equation (0.1.1).Therefore using the results of Kato and Trotter one obtain a precise meaning to the Feynman integral (0.2.4) as limit in RHS of the Eq.(0.3.8), when the potential $V$ is sufficiently regular.
$$ M_V^t = \exp\{-itV\}, \qquad (0.3.5) $$

then $\psi(t,x) = M_V^t \psi_0(x)$ is the solution of Eq.(0.3.1) with $1/2m$ replaced by 0.

Asume that Kato conditions (see Appendix V) under which the operator $(1/2m)\Delta - V$ is self-adjoint is satisfied.It well known [41] that under Kato conditions if we let

$$ U_{m,V}^t = \exp\left[it\left(\frac{1}{2m}\Delta - V\right)\right] \qquad (0.3.6) $$

then Trotter's theorem asserts that for all $\psi_0 \in \mathcal{L}_2$:

$$ U_{m,V}^t \psi_0 = \lim_{n \to \infty} \left( \Xi_m^{\frac{t}{n}} M_V^{\frac{t}{n}} \right)^n \psi_0. \qquad (0.3.7) $$

Using (0.3.4) and (0.3.5) one find that

$$ U_{m,V}^t \psi_0 = \lim_{n \to \infty} \left[ \left( \frac{2\pi i t}{nm} \right)^{-\frac{1}{2}dn} \int \dots \int \exp[iS(x_0, \dots, x_n)] \psi_0(x_n) dx_1 \dots dx_n \right]. \qquad (0.3.8) $$

Here we set $x_0 = x$ and

$$ S(x_0, \dots, x_n) = \left(\frac{t}{n}\right) \sum_{j=1}^{n} \left[ \frac{m}{2} \frac{(x_i - x_{j-1})^2}{(t/n)^2} - V(x_i) \right]. \qquad (0.3.9) $$

The limit in RHS of the Eq.(0.3.8) is well-defined, and according to Eq.(0.3.7) converges in $\mathcal{L}_2$ to the solution of the Schrödinger equation (0.1.1).Therefore using the results of Kato and Trotter one obtain a precise meaning to the Feynman integral (0.2.4) as limit in RHS of the Eq.(0.3.8), when the potential $V$ is sufficiently regular.
Let $\omega(\tau)$ be a trajectory; that is, a function from $[0, t]$ to $\mathbb{R}^d$, with $\omega(0) = x$ and set $x_j = \omega(jt/n)$ for $j = 0, \dots, n$. Then (0.3.9) is, formally, the Riemann sum for the classical action

$$S_t(\dot{\omega}(\tau), \omega(\tau)) = \int_0^t \left[ \frac{m}{2} \dot{\omega}^2(\tau) - V(\omega(\tau)) \right] d\tau \quad (0.3.10)$$

of a particle of mass $m$ with the trajectory $\omega(\tau)$ in the presence of the potential $V$. Taking the limit formally in (0.3.8) yields the Feynman path integral (0.2.4)

$$N \times \left( \int_{\Omega_x} \exp \left[ \frac{i}{\hbar} S_t(\dot{\omega}(\tau), \omega(\tau)) \right] D[\omega(\tau)] \right), \quad (0.3.11)$$

$N = \text{constant}$,

where $\Omega_x \subset C[0, t]$ is the set of all continuous trajectories with $\omega(0) = x$. As is well-known, there are difficulties in interpreting formal expression (0.3.11) as well defined integral of a functional $\exp\left[\frac{i}{\hbar}S_t(\dot{\omega}(\tau), \omega(\tau))\right] \in (C^2[0, t])^*$ over subset $\Omega_x$:

(i) the "constant" is infinite (of infinite order),

(ii) the action (0.3.10) exists only if $\omega(\tau)$ and $V$ are sufficiently smooth, and

(iii) $D[\omega(\tau)] = \prod_{0 \le \tau \le t} dx_\tau$ has no meaning as measure on $C[0, t]$.

We wish to improve this difficulty using Colombeau approach. As a framework we use Colombeau algebra of generalized functionals, see Definition 0.3.1. We use the notation

$$(p_{\varepsilon}^{t}(x,y)dy)_{\varepsilon} = (4\pi t((v_{\varepsilon})_{\varepsilon}))^{-d/2} \exp\left(-\frac{|x-y|^2}{4t((v_{\varepsilon})_{\varepsilon})}\right)dy, \quad (0.3.12)$$

where $x$ are in $d$-dimensional Euclidean space $\mathbb{R}^d$, $\|x\| = \sqrt{x^2}$
Let $\omega(\tau)$ be a trajectory; that is, a function from $[0, t]$ to $\mathbb{R}^d$, with $\omega(0) = x$ and set $x_j = \omega(jt/n)$ for $j = 0, \dots, n$. Then (0.3.9) is, formally, the Riemann sum for the classical action

$$S_t(\dot{\omega}(\tau), \omega(\tau)) = \int_0^t \left[ \frac{m}{2} \dot{\omega}^2(\tau) - V(\omega(\tau)) \right] d\tau \quad (0.3.10)$$

of a particle of mass $m$ with the trajectory $\omega(\tau)$ in the presence of the potential $V$. Taking the limit formally in (0.3.8) yields the Feynman path integral (0.2.4)

$$N \times \left( \int_{\Omega_x} \exp \left[ \frac{i}{\hbar} S_t(\dot{\omega}(\tau), \omega(\tau)) \right] D[\omega(\tau)] \right), \quad (0.3.11)$$

$N = \text{constant}$,

where $\Omega_x \subset C[0, t]$ is the set of all continuous trajectories with $\omega(0) = x$. As is well-known, there are difficulties in interpreting formal expression (0.3.11) as well defined integral of a functional $\exp\left[\frac{i}{\hbar}S_t(\dot{\omega}(\tau), \omega(\tau))\right] \in (C^2[0, t])^*$ over subset $\Omega_x$:

(i) the "constant" is infinite (of infinite order),

(ii) the action (0.3.10) exists only if $\omega(\tau)$ and $V$ are sufficiently smooth, and

(iii) $D[\omega(\tau)] = \prod_{0 \le \tau \le t} dx_\tau$ has no meaning as measure on $C[0, t]$.

We wish to improve this difficulty using Colombeau approach. As a framework we use Colombeau algebra of generalized functionals, see Definition 0.3.1. We use the notation

$$(p_{\varepsilon}^t(x,y)dy)_{\varepsilon} = (4\pi t((v_{\varepsilon})_{\varepsilon}))^{-d/2} \exp\left(-\frac{|x-y|^2}{4t((v_{\varepsilon})_{\varepsilon})}\right)dy, \quad (0.3.12)$$

where $x$ are in $d$-dimensional Euclidean space $\mathbb{R}^d$, $\|x\| = \sqrt{x^2}$
$$dx = dx_1 \dots dx_d, x^2 = \sum_{i=1}^{d} x_i^2, \qquad (0.3.13)$$

and $v_\epsilon$, is a strictly positive Colombeau constant such that $v_\epsilon = O(\epsilon^r)$, if
$$\epsilon \to 0, r \in \mathbb{Z}.$$

**Colombeau-Wiener measure** $\left( \tilde{D}_x^{\mathbb{R}} [\omega(\tau), v_\epsilon] \right)_\epsilon$.

We wish to construct now real-valued Colombeau-Wiener measure $\left( \tilde{D}_x^{\mathbb{R}} [\omega(\tau), v_\epsilon] \right)_\epsilon$ on the space $\Omega_t$ of all trajectories. It is convenient to introduce the one-point compactification $\mathbb{R}^d$ of $\mathbb{R}^d$ by a point $\infty$, and let $\Omega_t$ be the Cartesian product space:

$$\Omega_t = \prod_{0 \le \tau \le t} \dot{\mathbb{R}}^d. \qquad (0.3.14)$$

Thus an element $\omega(\tau)$ of $\Omega_t$ is an entirely arbitrary function from time $[0, t]$ to space $\mathbb{R}^d$. Let us give $\Omega_t$ the product topology, so that by the Tychonoff theorem, $\Omega_t$ is a compact Hausdorff space.

**Definition 0.3.1.** Let $C(\Omega_t)$ be the set of all continuous functions on $\Omega_t$. Set $E(C(\Omega_t)) = (C(\Omega_t))^I, I = (0,1]$

$$E_{M,n}(C(\Omega_t)) =$$

$$\{ (\varphi_{\varepsilon}[\omega(\tau)])_{\varepsilon} \in E(C(\Omega_t)) | (\exists n \in \mathbb{N}) (\exists \tau_1 < \dots < \tau_n) (\exists p \in \mathbb{N}) [ ((\varphi_{\varepsilon}[\omega(\tau)])_{\varepsilon}) = \quad (0.3.15) $$

$$= (\mathcal{F}_{\varepsilon}[\omega(\tau_1), \dots, \omega(\tau_n)])_{\varepsilon} \wedge (\sup_{\omega \in \Omega_t} |\varphi_{\varepsilon}[\omega(\tau)]| = O(\varepsilon^{-p})) \},$$

where $p$ does not depend on $n$,
$$dx = dx_1 \dots dx_d, x^2 = \sum_{i=1}^{d} x_i^2, \qquad (0.3.13)$$

and $v_\epsilon$, is a strictly positive Colombeau constant such that $v_\epsilon = O(\epsilon^r)$, if $\epsilon \to 0, r \in \mathbb{Z}$.

**Colombeau-Wiener measure** $(\tilde{D}_x^{\mathbb{R}}[\omega(\tau), v_\epsilon])_\epsilon$.

We wish to construct now real-valued Colombeau-Wiener measure $(\tilde{D}_x^{\mathbb{R}}[\omega(\tau), v_\epsilon])_\epsilon$ on the space $\Omega_t$ of all trajectories. It is convenient to introduce the one-point compactification $\mathbb{R}^d$ of $\mathbb{R}^d$ by a point $\infty$, and let $\Omega_t$ be the Cartesian product space:

$$\Omega_t = \prod_{0 \le \tau \le t} \dot{\mathbb{R}}^d. \qquad (0.3.14)$$

Thus an element $\omega(\tau)$ of $\Omega_t$ is an entirely arbitrary function from time $[0, t]$ to space $\mathbb{R}^d$. Let us give $\Omega_t$ the product topology, so that by the Tychonoff theorem, $\Omega_t$ is a compact Hausdorff space.

**Definition 0.3.1.** Let $C(\Omega_t)$ be the set of all continuous functions on $\Omega_t$. Set $E(C(\Omega_t)) = (C(\Omega_t))^I, I = (0,1]$

$$E_{M,n}(C(\Omega_t)) =$$

$$\{(\varphi_{\varepsilon}[\omega(\tau)])_{\varepsilon} \in E(C(\Omega_t)) | (\exists n \in \mathbb{N})(\exists \tau_1 < \dots < \tau_n)(\exists p \in \mathbb{N}) [((\varphi_{\varepsilon}[\omega(\tau)])_{\varepsilon} = \qquad (0.3.15)$$

$$= (\mathcal{F}_{\varepsilon}[\omega(\tau_1), \dots, \omega(\tau_n)])_{\varepsilon} \wedge (\sup_{\omega \in \Omega_t} |\varphi_{\varepsilon}[\omega(\tau)]| = O(\varepsilon^{-p}))\},$$

where $p$ does not depend on $n$,
$$N_n(\mathbf{C}(\Omega_t)) =$$

$$\{(\varphi_{n,\varepsilon}[\omega(\tau)])_{\varepsilon} \in E(\mathbf{C}(\Omega_t)) | (\exists n \in \mathbb{N})(\exists \tau_1 < \dots < \tau_n)(\forall q \in \mathbb{N}) [((\varphi_{n,\varepsilon}[\omega(\tau)])_{\varepsilon} = \tag{0.3.16}$$

$$= (\mathcal{F}_{\varepsilon}[\omega(\tau_1), \dots, \omega(\tau_n)])_{\varepsilon} \wedge (\sup_{\omega \in \Omega_t} |\varphi_{n,\varepsilon}[\omega(\tau)]| = O(\varepsilon^q))]\},$$

$$G_{M,n}(\mathbf{C}(\Omega_t)) = E_{M,n}(\mathbf{C}(\Omega_t))/N_n(\mathbf{C}(\Omega_t)).$$

Elements of $E_{M,n}(\mathbf{C}(\Omega_t))$ and $N_n(\mathbf{C}(\Omega_t))$ are called moderate, resp. negligible simple functionals. Note that $E_{M,n}(\mathbf{C}(\Omega_t))$ is an algebra with pointwise operations and in which $N_n(\mathbf{C}(\Omega_t))$ is ideal. Thus, $G_{M,n}(\mathbf{C}(\Omega_t))$ is an associative, commutative algebra. If $(\varphi_{n,\varepsilon}[\omega(\tau)])_{\varepsilon} \in E(\mathbf{C}(\Omega_t))$ is a representative of $\varphi_n \in G_{M,n}(\mathbf{C}(\Omega_t))$, we write $\varphi_n = \mathbf{cl}[(\varphi_{n,\varepsilon}[\omega(\tau)])_{\varepsilon}]$ or simply $\varphi_n = [(\varphi_{n,\varepsilon}[\omega(\tau)])_{\varepsilon}]$.

**Definition 0.3.2.** We denote

$$E_M(\mathbf{C}(\Omega_t)) = \bigoplus_{n=1}^{\infty} E_{M,n}(\mathbf{C}(\Omega_t)),$$

$$N(\mathbf{C}(\Omega_t)) = \bigoplus_{n=1}^{\infty} N_n(\mathbf{C}(\Omega_t)), \tag{0.3.17}$$

$$G_M(\mathbf{C}(\Omega_t)) = E_M(\mathbf{C}(\Omega_t))/N(\mathbf{C}(\Omega_t)).$$

Elements of $E_M(\mathbf{C}(\Omega_t))$ and $N(\mathbf{C}(\Omega_t))$ are called moderate, resp. negligible functionals. Note that $E_M(\mathbf{C}(\Omega_t))$ is an algebra with pointwise operations and in which $N(\mathbf{C}(\Omega_t))$ is ideal. Thus, $G_M(\mathbf{C}(\Omega_t))$ is an associative, commutative algebra. Now we can define the Colombeau integral of Colombeau generalized functional $(\varphi_{\varepsilon}[\omega(\tau)])_{\varepsilon}$ i.e., Colombeau integral of Colombeau $\mathbb{R}$-valued generalized function $(\varphi_{\varepsilon}[\omega(\tau)])_{\varepsilon} \in I$ defined on $\Omega_t, I = (0,1]$.

Let's consider first Colombeau generalized functional $(\varphi_{\varepsilon}[\omega(\tau)])_{\varepsilon} : \Omega_t \to \mathbb{R}$ of the simple form, i.e.
$$N_n(\mathbf{C}(\Omega_t)) =$$

$$\{ (\varphi_{n,\varepsilon}[\omega(\tau)])_{\varepsilon} \in E(\mathbf{C}(\Omega_t)) | (\exists n \in \mathbb{N})(\exists \tau_1 < \dots < \tau_n)(\forall q \in \mathbb{N}) [ ((\varphi_{n,\varepsilon}[\omega(\tau)])_{\varepsilon} = \tag{0.3.16} \\ = (\mathcal{F}_{\varepsilon}[\omega(\tau_1), \dots, \omega(\tau_n)])_{\varepsilon}) \wedge (\sup_{\omega \in \Omega_t} |\varphi_{n,\varepsilon}[\omega(\tau)]| = O(\varepsilon^q))] \},$$

$$G_{M,n}(\mathbf{C}(\Omega_t)) = E_{M,n}(\mathbf{C}(\Omega_t))/N_n(\mathbf{C}(\Omega_t)).$$

Elements of $E_{M,n}(\mathbf{C}(\Omega_t))$ and $N_n(\mathbf{C}(\Omega_t))$ are called moderate, resp. negligible simple functionals. Note that $E_{M,n}(\mathbf{C}(\Omega_t))$ is an algebra with pointwise operations and in which $N_n(\mathbf{C}(\Omega_t))$ is ideal. Thus, $G_{M,n}(\mathbf{C}(\Omega_t))$ is an associative, commutative algebra. If $(\varphi_{n,\varepsilon}[\omega(\tau)])_{\varepsilon} \in E(\mathbf{C}(\Omega_t))$ is a representative of $\varphi_n \in G_{M,n}(\mathbf{C}(\Omega_t))$, we write $\varphi_n = \mathbf{cl}[(\varphi_{n,\varepsilon}[\omega(\tau)])_{\varepsilon}]$ or simply $\varphi_n = [(\varphi_{n,\varepsilon}[\omega(\tau)])_{\varepsilon}]$.

**Definition 0.3.2.** We denote

$$E_M(\mathbf{C}(\Omega_t)) = \bigoplus_{n=1}^{\infty} E_{M,n}(\mathbf{C}(\Omega_t)),$$

$$N(\mathbf{C}(\Omega_t)) = \bigoplus_{n=1}^{\infty} N_n(\mathbf{C}(\Omega_t)), \qquad (0.3.17)$$

$$G_M(\mathbf{C}(\Omega_t)) = E_M(\mathbf{C}(\Omega_t))/N(\mathbf{C}(\Omega_t)).$$

Elements of $E_M(\mathbf{C}(\Omega_t))$ and $N(\mathbf{C}(\Omega_t))$ are called moderate, resp. negligible functionals. Note that $E_M(\mathbf{C}(\Omega_t))$ is an algebra with pointwise operations and in which $N(\mathbf{C}(\Omega_t))$ is ideal. Thus, $G_M(\mathbf{C}(\Omega_t))$ is an associative, commutative algebra. Now we can define the Colombeau integral of Colombeau generalized functional $(\varphi_{\varepsilon}[\omega(\tau)])_{\varepsilon}$ i.e., Colombeau integral of Colombeau $\mathbb{R}$-valued generalized function $(\varphi_{\varepsilon}[\omega(\tau)])_{\varepsilon} \in I$ defined on $\Omega_t, I = (0,1]$.

Let's consider first Colombeau generalized functional $(\varphi_{\varepsilon}[\omega(\tau)])_{\varepsilon} : \Omega_t \to \mathbb{R}$ of the simple form, i.e.
$$ (\varphi_{\varepsilon}[\omega(\tau)])_{\varepsilon} = (\mathcal{F}_{\varepsilon}[\omega(\tau_1), \dots, \omega(\tau_n)])_{\varepsilon}, \qquad (0.3.18) \\ 0 = \tau_1 < \dots < \tau_n = t. $$

Let $x$ be a point in $\mathbb{R}^d$ and let

$$ (\mathbf{J}_{\varepsilon}^{\nu_{\varepsilon}}(\varphi_{\varepsilon}[\omega(\tau)]))_{\varepsilon} = \left( \int_{\Omega} \varphi_{\varepsilon}[\omega(\tau)] \widetilde{D}_{x}^{\mathbb{R}}[\omega(\tau), \nu_{\varepsilon}] \right)_{\varepsilon} = \\ \left( \int \dots \int p_{\nu_{\varepsilon}}^{\tau_1}(x, x_1) dx_1 p_{\nu_{\varepsilon}}^{\tau_2-\tau_1}(x_1, x_2) dx_2 \dots p_{\nu_{\varepsilon}}^{\tau_n-\tau_{n-1}}(x_{n-1}, x_n) dx_n \mathcal{F}_{\varepsilon}[\omega(\tau_1), \dots, \omega(\tau_n)] \right)_{\varepsilon} = \qquad (0.3.19) \\ \int \dots \int \left( (p_{\nu_{\varepsilon}}^{\tau_1}(x) p_{\nu_{\varepsilon}}^{\tau_2-\tau_1}(x_1) \dots p_{\nu_{\varepsilon}}^{\tau_n-\tau_{n-1}}(x_{n-1}, x_n) \mathcal{F}_{\varepsilon}[\omega(\tau_1), \dots, \omega(\tau_n)])_{\varepsilon} \right) dx_1 \dots dx_n $$

provided the integrals exist. Let $C(\Omega_t) = C^*[0,t]$ be the set of all continuous functions on $\Omega_t$ and let $C^*(\Omega_t)$ the set of all those of the form (0.3.15) where $\forall \varepsilon(\varepsilon \in (0,1]) \mathcal{F}_{\varepsilon}$, is continuous. Any function in $C(\Omega_t)$ may be uniformly approximated by functions in $C^*(\Omega_t)$, by the Stone-Weierstrass theorem. The mapping $\varphi_{\varepsilon} \to \mathbf{J}_{\varepsilon}^x(\varphi_{\varepsilon}[\omega(\tau)]) = \int_{\Omega_x} \varphi_{\varepsilon}[\omega(\tau)] \widetilde{D}_{x}^{\mathbb{R}}[\omega(\tau), \varepsilon]$, $\varepsilon \in (0,1]$ is a linear functional

on $C^*(\Omega_t)$, which is such that

$\mathbf{J}_{\varepsilon}^x(1) = N_{\varepsilon}$ and $\mathbf{J}_{\varepsilon}^x(\varphi_{\varepsilon}[\omega(\tau)]) \ge 0$ whenever $\varphi_{\varepsilon}[\omega(\tau)] \ge 0$. Consequently,

$$ \forall \varepsilon(\varepsilon \in (0,1]) \left| \mathbf{J}_{\varepsilon}^{x}(\varphi_{\varepsilon}[\omega(\tau)]) \right| \le c_{\varepsilon} \sup_{\omega(\tau)} \left| \varphi_{\varepsilon}[\omega(\tau)] \right|, \qquad (0.3.20) $$

i.e.

$$ (|\mathbf{J}_{\varepsilon}^{x}(\varphi_{\varepsilon}[\omega(\tau)])|)_{\varepsilon} \le (c_{\varepsilon})_{\varepsilon} (\sup_{\omega(\tau)} |\varphi_{\varepsilon}[\omega(\tau)]|)_{\varepsilon}, \qquad (0.3.21) $$

where $cl[(c_{\varepsilon})_{\varepsilon}] \in \tilde{\mathbb{R}}$ and so the mapping $\varphi_{\varepsilon} \to \mathbf{J}_{\varepsilon}^x(\varphi_{\varepsilon}[\omega(\tau)])$ has a unique extension
$$ (\varphi_{\varepsilon}[\omega(\tau)])_{\varepsilon} = (\mathcal{F}_{\varepsilon}[\omega(\tau_1), \dots, \omega(\tau_n)])_{\varepsilon}, \qquad (0.3.18) $$

$$ 0 = \tau_1 < \dots < \tau_n = t. $$

Let $x$ be a point in $\mathbb{R}^d$ and let

$$ (\mathbf{J}_{\varepsilon}^{\nu_{\varepsilon}}(\varphi_{\varepsilon}[\omega(\tau)]))_{\varepsilon} = \left( \int_{\Omega} \varphi_{\varepsilon}[\omega(\tau)] \tilde{D}_{x}^{\mathbb{R}}[\omega(\tau), \nu_{\varepsilon}] \right)_{\varepsilon} = $$

$$ \left( \int \dots \int p_{v_{\varepsilon}}^{\tau_1}(x, x_1) dx_1 p_{v_{\varepsilon}}^{\tau_2 - \tau_1}(x_1, x_2) dx_2 \dots p_{v_{\varepsilon}}^{\tau_n - \tau_{n-1}}(x_{n-1}, x_n) dx_n \mathcal{F}_{\varepsilon}[\omega(\tau_1), \dots, \omega(\tau_n)] \right)_{\varepsilon} = \qquad (0.3.19) $$

$$ \int \dots \int \left( (p_{v_{\varepsilon}}^{\tau_1}(x) p_{v_{\varepsilon}}^{\tau_2 - \tau_1}(x_1) \dots p_{v_{\varepsilon}}^{\tau_n - \tau_{n-1}}(x_{n-1}, x_n)) \mathcal{F}_{\varepsilon}[\omega(\tau_1), \dots, \omega(\tau_n)] \right)_{\varepsilon} dx_1 \dots dx_n $$

provided the integrals exist. Let $C(\Omega_t) = C^*[0, t]$ be the set of all continuous functions on $\Omega_t$ and let $C^*(\Omega_t)$ the set of all those of the form (0.3.15) where $\forall \varepsilon(\varepsilon \in (0,1]) \mathcal{F}_{\varepsilon}$, is continuous. Any function in $C(\Omega_t)$ may be uniformly approximated by functions in $C^*(\Omega_t)$, by the Stone-Weierstrass theorem. The mapping $\varphi_{\varepsilon} \to \mathbf{J}_{\varepsilon}^x(\varphi_{\varepsilon}[\omega(\tau)]) = \int_{\Omega_x} \varphi_{\varepsilon}[\omega(\tau)] \tilde{D}_{x}^{\mathbb{R}}[\omega(\tau), \varepsilon]$, $\varepsilon \in (0,1]$ is a linear functional

on $C^*(\Omega_t)$, which is such that

$\mathbf{J}_{\varepsilon}^x(1) = N_{\varepsilon}$ and $\mathbf{J}_{\varepsilon}^x(\varphi_{\varepsilon}[\omega(\tau)]) \ge 0$ whenever $\varphi_{\varepsilon}[\omega(\tau)] \ge 0$. Consequently,

$$ \forall \varepsilon(\varepsilon \in (0,1]) \left| \left| \mathbf{J}_{\varepsilon}^{x}(\varphi_{\varepsilon}[\omega(\tau)]) \right| \right| \le c_{\varepsilon} \sup_{\omega(\tau)} \left| \varphi_{\varepsilon}[\omega(\tau)] \right|, \qquad (0.3.20) $$

i.e.

$$ (|\mathbf{J}_{\varepsilon}^{x}(\varphi_{\varepsilon}[\omega(\tau)])|)_{\varepsilon} \le (c_{\varepsilon})_{\varepsilon} (\sup_{\omega(\tau)} |\varphi_{\varepsilon}[\omega(\tau)]|)_{\varepsilon}, \qquad (0.3.21) $$

where $\mathrm{cl}[(c_{\varepsilon})_{\varepsilon}] \in \tilde{\mathbb{R}}$ and so the mapping $\varphi_{\varepsilon} \to \mathbf{J}_{\varepsilon}^{x}(\varphi_{\varepsilon}[\omega(\tau)])$ has a unique extension
to a positive linear functional defined for all $\varphi_{\varepsilon}$ in $C(\Omega_t)$. The Riesz representation theorem asserts that $\forall \varepsilon(\varepsilon \in (0,1])$ there is exist a regular measure, denoted $\tilde{D}_x^R[\omega(\tau), v_\varepsilon]$ or for short $\tilde{D}_x[\omega(\tau), v_\varepsilon]$, such that

$$ \mathbf{J}_{\varepsilon}^{x}(\varphi_{\varepsilon}[\omega(\tau)]) = \int_{\Omega_t} \varphi_{\varepsilon}[\omega(\tau)] \tilde{D}_{x}^{R}[\omega(\tau), v_{\varepsilon}] \quad (0.3.22) $$

Therefore there is exist a regular Colombeau measure, denoted $(\tilde{D}_x^R[\omega(\tau), v_\varepsilon])_\varepsilon$, or for short $(\tilde{D}_x[\omega(\tau), v_\varepsilon])_\varepsilon$ such that

$$ \begin{align} (\mathbf{J}_{\varepsilon}^{x}(\varphi_{\varepsilon}[\omega(\tau)]))_{\varepsilon} &= \nonumber \\ & \left( \int_{\Omega_t} \varphi_{\varepsilon}[\omega(\tau)] \tilde{D}_{x}^{R}[\omega(\tau), v_{\varepsilon}] \right)_{\varepsilon} = \tag{0.3.23} \\ & \int_{\Omega_t} ((\varphi_{\varepsilon}[\omega(\tau)])_{\varepsilon}) (\tilde{D}_{x}^{R}[\omega(\tau), v_{\varepsilon}])_{\varepsilon}. \nonumber \end{align} $$

Note that the elements of $\Omega_t$ were allowed to be arbitrary trajectories, even taking the value $\infty$. However it easy to see that the Colombeau measure $(\tilde{D}_x^R[\omega(\tau), v_\varepsilon])_\varepsilon$ is concentrated on the continuous trajectories $\omega(\tau) \in C[0, t]$ taking values in $\mathbb{R}^d$. This is based on the fact (see [41]) that if $\varepsilon > 0, \varepsilon \in (0,1]$ and

$$ (\rho_{\varepsilon}(\varepsilon, \delta))_{\varepsilon} = \left( \sup_{t \le \delta(\varepsilon)} \int_{\|y-x\| \ge \varepsilon} p_{\varepsilon}^{t}(x, y) dy \right)_{\varepsilon} \quad (0.3.24) $$

then by Eq.(0.3.12) $(\rho_{\varepsilon}(\varepsilon, \delta))_{\varepsilon} = o(\delta(\varepsilon))$. We assume now that the function $V$ is regular or satisfy Nelson's assumption (see [41]). Then for almost every $\omega \in \Omega_t, V(\omega(\tau))$ is a continuous function of $\tau$ for $0 \le \tau < \infty$ and so we may form the Colombeau-Wiener integral:
to a positive linear functional defined for all $\varphi_{\varepsilon}$ in $C(\Omega_t)$. The Riesz representation theorem asserts that $\forall \varepsilon(\varepsilon \in (0,1])$ there is exist a regular measure, denoted $\tilde{D}_x^R[\omega(\tau), v_\varepsilon]$ or for short $\tilde{D}_x[\omega(\tau), v_\varepsilon]$, such that

$$ \mathbf{J}_{\varepsilon}^{x}(\varphi_{\varepsilon}[\omega(\tau)]) = \int_{\Omega_t} \varphi_{\varepsilon}[\omega(\tau)] \tilde{D}_{x}^{\mathbb{R}}[\omega(\tau), v_{\varepsilon}] \quad (0.3.22) $$

Therefore there is exist a regular Colombeau measure, denoted $(\tilde{D}_x^R[\omega(\tau), v_\varepsilon])_\varepsilon$, or for short $(\tilde{D}_x[\omega(\tau), v_\varepsilon])_\varepsilon$ such that

$$ \begin{gathered} (\mathbf{J}_{\varepsilon}^{x}(\varphi_{\varepsilon}[\omega(\tau)]))_{\varepsilon} = \\ \left( \int_{\Omega_t} \varphi_{\varepsilon}[\omega(\tau)] \tilde{D}_{x}^{\mathbb{R}}[\omega(\tau), v_{\varepsilon}] \right)_{\varepsilon} = \quad (0.3.23) \\ \int_{\Omega_t} ((\varphi_{\varepsilon}[\omega(\tau)])_{\varepsilon}) (\tilde{D}_{x}^{\mathbb{R}}[\omega(\tau), v_{\varepsilon}])_{\varepsilon}. \end{gathered} $$

Note that the elements of $\Omega_t$ were allowed to be arbitrary trajectories, even taking the value $\infty$. However it easy to see that the Colombeau measure $(\tilde{D}_x^R[\omega(\tau), v_\varepsilon])_\varepsilon$ is concentrated on the continuous trajectories $\omega(\tau) \in C[0, t]$ taking values in $\mathbb{R}^d$. This is based on the fact (see [41]) that if $\varepsilon > 0, \varepsilon \in (0,1]$ and

$$ (\rho_{\varepsilon}(\varepsilon, \delta))_{\varepsilon} = \left( \sup_{t \le \delta(\varepsilon)} \int_{\|y-x\| \ge \varepsilon} p_{\varepsilon}^{t}(x, y) dy \right)_{\varepsilon} \quad (0.3.24) $$

then by Eq.(0.3.12) $(\rho_{\varepsilon}(\varepsilon, \delta))_{\varepsilon} = o(\delta(\varepsilon))$. We assume now that the function $V$ is regular or satisfy Nelson's assumption (see [41]). Then for almost every $\omega \in \Omega_t, V(\omega(\tau))$ is a continuous function of $\tau$ for $0 \le \tau < \infty$ and so we may form the Colombeau-Wiener integral:
$$ (\mathbf{J}_{\varepsilon}^{x,t}[V(\omega(\tau)), \psi_{\varepsilon}(\omega(\tau))])_{\varepsilon} = $$

$$ \left( \int_{\Omega_t} \exp \left[ -i \int_0^t V(\omega(\tau)) d\tau \right] \psi_{\varepsilon}(\omega(t)) \tilde{D}_x^{\mathbb{R}}[\omega(\tau), \varepsilon] \right)_{\varepsilon} = \qquad (0.3.25) $$

$$ \int_{\Omega_t} \exp \left[ -i \int_0^t V(\omega(\tau)) d\tau \right] ((\psi_{\varepsilon}(\omega(t)))_{\varepsilon}) \left( \tilde{D}_x^{\mathbb{R}}[\omega(\tau), \varepsilon] \right)_{\varepsilon} $$

for any $(\psi_{\varepsilon}(\omega(\tau)))_{\varepsilon} \in G(\mathbb{R}^d)$ and any $t \ge 0$. In fact, the integrand is defined for almost every $\omega \in \Omega_t$, and it is bounded in absolute value by $|\psi_{\varepsilon}(\omega(t))|_{\varepsilon}$. But this is Colombeau-integrable, for by definition and (0.3.20) one obtain

$$ |(\mathbf{J}_{\varepsilon}^{x,t}[V(\omega(\tau)), \psi_{\varepsilon}(\omega(\tau))])_{\varepsilon}| \le (\mathbf{J}_{\varepsilon}^{x}[\psi_{\varepsilon}(\omega(\tau))])_{\varepsilon} \le ((c_{\varepsilon})_{\varepsilon}) (\sup_{\omega(\tau)} |\psi_{\varepsilon}(\omega(t))|)_{\varepsilon}. \quad (0.3.26) $$

Since $V(\omega(\tau))$ is continuous for almost every $\omega(\tau)$,

$$ \int_0^t V(\omega(\tau)) d\tau = \lim_{n \to \infty} \left( \frac{t}{n} \sum_{j=1}^n V\left(\omega\left(j \frac{t}{n}\right)\right) \right) \qquad (0.3.27) $$

for almost every $\omega$. By the Lebesgue dominated convergence theorem, this implies that

$$ (\mathbf{J}_{\varepsilon}^{x,t}[V(\omega(\tau)), \psi_{\varepsilon}(\omega(\tau))])_{\varepsilon} = $$

$$ \left( \lim_{n \to \infty} \int_{\Omega} \exp \left[ -i \sum_{j=1}^{n} \frac{t}{n} V\left(\omega\left(j \frac{t}{n}\right)\right) \right] \psi_{\varepsilon}(\omega(t)) \tilde{D}_x^{\mathbb{R}}[\omega(\tau), \varepsilon] \right)_{\varepsilon} \qquad (0.3.28) $$
$$ (\mathbf{J}_{\varepsilon}^{x,t}[V(\omega(\tau)), \psi_{\varepsilon}(\omega(\tau))])_{\varepsilon} = $$

$$ \left( \int_{\Omega_t} \exp \left[ -i \int_0^t V(\omega(\tau)) d\tau \right] \psi_{\varepsilon}(\omega(t)) \tilde{D}_x^{\mathbb{R}}[\omega(\tau), \varepsilon] \right)_{\varepsilon} = \qquad (0.3.25) $$

$$ \int_{\Omega_t} \exp \left[ -i \int_0^t V(\omega(\tau)) d\tau \right] ((\psi_{\varepsilon}(\omega(t)))_{\varepsilon}) (\tilde{D}_x^{\mathbb{R}}[\omega(\tau), \varepsilon])_{\varepsilon} $$

for any $(\psi_{\varepsilon}(\omega(\tau)))_{\varepsilon} \in G(\mathbb{R}^d)$ and any $t \ge 0$. In fact, the integrand is defined for almost every $\omega \in \Omega_t$, and it is bounded in absolute value by $|\psi_{\varepsilon}(\omega(t))|_{\varepsilon}$. But this is Colombeau-integrable, for by definition and (0.3.20) one obtain

$$ |(\mathbf{J}_{\varepsilon}^{x,t}[V(\omega(\tau)), \psi_{\varepsilon}(\omega(\tau))])_{\varepsilon}| \le (\mathbf{J}_{\varepsilon}^{x}[\psi_{\varepsilon}(\omega(\tau))])_{\varepsilon} \le ((c_{\varepsilon})_{\varepsilon}) (\sup_{\omega(\tau)} |\psi_{\varepsilon}(\omega(t))|)_{\varepsilon}. \quad (0.3.26) $$

Since $V(\omega(\tau))$ is continuous for almost every $\omega(\tau)$,

$$ \int_0^t V(\omega(\tau)) d\tau = \lim_{n \to \infty} \left( \frac{t}{n} \sum_{j=1}^n V\left(\omega\left(j \frac{t}{n}\right)\right) \right) \qquad (0.3.27) $$

for almost every $\omega$. By the Lebesgue dominated convergence theorem, this implies that

$$ (\mathbf{J}_{\varepsilon}^{x,t}[V(\omega(\tau)), \psi_{\varepsilon}(\omega(\tau))])_{\varepsilon} = $$

$$ \left( \lim_{n \to \infty} \int_{\Omega} \exp \left[ -i \sum_{j=1}^{n} \frac{t}{n} V\left(\omega\left(j \frac{t}{n}\right)\right) \right] \psi_{\varepsilon}(\omega(t)) \tilde{D}_{x}^{\mathbb{R}}[\omega(\tau), \varepsilon] \right)_{\varepsilon} \qquad (0.3.28) $$
for all $x \in \mathbb{R}^d \setminus F_V$, where $F_V$ is a closed set in $\mathbb{R}^d$ which depend only on $V$. By definition of the Colombeau-Wiener integral (0.3.25), this means that for all $x \notin F_V$

$$ \left( \left( U_{\nu_{\epsilon}, V}^{t} \right)_{\epsilon} \right) (\psi_{\epsilon}(x))_{\epsilon} = \left( \lim_{n \to \infty} \left( \Xi_{\nu_{\epsilon}}^{\frac{t}{n}} M_{V}^{\frac{t}{n}} \right)^{n} \psi_{\epsilon}(x) \right)_{\epsilon}, \quad (0.3.29) $$

where

$$ \begin{gathered} \left( U_{\nu_{\epsilon}, V}^{t} \right)_{\epsilon} = \exp[t(\nu_{\epsilon}\Delta - iV)], \quad (0.3.30) \\ \Xi_{\nu_{\epsilon}}^{t} = \exp(t\nu_{\epsilon}\Delta). \end{gathered} $$

Explicitly,

$$ (\Xi_{\nu_{\epsilon}}^{t} \varphi_{\epsilon}(x))_{\epsilon} = (2\pi t ((\nu_{\epsilon})_{\epsilon}))^{-d/2} \int_{\mathbb{R}^d} \exp\left(-\frac{|x-y|^2}{2t((\nu_{\epsilon})_{\epsilon})}\right) \varphi(y)dy, \quad (0.3.31) $$

A set $F$ of capacity 0 has Lebesgue measure 0, and so if $\psi_{n,\epsilon} = (\Xi_{\nu_{\epsilon}}^{\frac{t}{n}} M_{V}^{\frac{t}{n}})^n \psi_{\epsilon}$ then $\forall \epsilon \in (0,1], \psi_{n,\epsilon}$ converges almost everywhere in $\mathbb{R}^d$ to $U_{\nu_{\epsilon},V}^t \psi_{\epsilon}$. By (0.3.31)

$$ \left( \left| \Xi_{\nu_{\epsilon}}^{\frac{t}{n}} \varphi_{\epsilon}(x) \right| \right)_{\epsilon} \le \left( \Xi_{\nu_{\epsilon}}^{\frac{t}{n}} |\varphi_{\epsilon}(x)| \right)_{\epsilon}. \quad (0.3.32) $$

By definition $M_V^{\frac{t}{n}}$ is multiplication by a function of modulus 1. Consequently, $\forall \epsilon \in (0,1] : |\psi_{n,\epsilon}| \le \Xi_{\nu_{\epsilon}}^{t} |\psi_{n,\epsilon}|$, and so $|\psi_{n,\epsilon} - \psi_{m,\epsilon}|^2 \le 4(\Xi_{\nu_{\epsilon}}^{t} \psi_{\epsilon})^2$, which is in $\mathcal{L}_2(\mathbb{R}^n)$. Therefore, by the Lebesgue dominated convergence theorem, $\forall \epsilon \in (0,1]$, $\psi_{n,\epsilon}$ is a Cauchy sequence in $\mathcal{L}_2(\mathbb{R}^n)$, and so
for all $x \in \mathbb{R}^d \setminus F_V$, where $F_V$ is a closed set in $\mathbb{R}^d$ which depend only on $V$. By definition of the Colombeau-Wiener integral (0.3.25), this means that for all $x \notin F_V$

$$ \left( \left( U_{\nu_{\epsilon}, V}^{t} \right)_{\epsilon} \right) (\psi_{\epsilon}(x))_{\epsilon} = \left( \lim_{n \to \infty} \left( \Xi_{\nu_{\epsilon}}^{\frac{t}{n}} M_{V}^{\frac{t}{n}} \right)^{n} \psi_{\epsilon}(x) \right)_{\epsilon}, \quad (0.3.29) $$

where

$$ \left( U_{\nu_{\epsilon}, V}^{t} \right)_{\epsilon} = \exp[t(\nu_{\epsilon}\Delta - iV)], \quad (0.3.30) $$

$$ \Xi_{\nu_{\epsilon}}^{t} = \exp(t\nu_{\epsilon}\Delta). $$

Explicitly,

$$ (\Xi_{\nu_{\epsilon}}^{t} \varphi_{\epsilon}(x))_{\epsilon} = (2\pi t ((\nu_{\epsilon})_{\epsilon}))^{-d/2} \int_{\mathbb{R}^d} \exp\left(-\frac{|x-y|^2}{2t((\nu_{\epsilon})_{\epsilon})}\right) \varphi(y)dy, \quad (0.3.31) $$

A set $F$ of capacity 0 has Lebesgue measure 0, and so if $\psi_{n,\epsilon} = (\Xi_{\nu_{\epsilon}}^{\frac{t}{n}} M_{V}^{\frac{t}{n}})^n \psi_{\epsilon}$ then
$\forall \epsilon \in (0,1], \psi_{n,\epsilon}$ converges almost everywhere in $\mathbb{R}^d$ to $U_{\nu_{\epsilon},V}^t \psi_{\epsilon}$. By (0.3.31)

$$ \left( \left| \Xi_{\nu_{\epsilon}}^{\frac{t}{n}} \varphi_{\epsilon}(x) \right| \right)_{\epsilon} \le \left( \Xi_{\nu_{\epsilon}}^{\frac{t}{n}} |\varphi_{\epsilon}(x)| \right)_{\epsilon}. \quad (0.3.32) $$

By definition $M_V^{\frac{t}{n}}$ is multiplication by a function of modulus 1. Consequently, $\forall \epsilon \in (0,1] : |\psi_{n,\epsilon}| \le \Xi_{\nu_{\epsilon}}^{t} |\psi_{n,\epsilon}|$, and so $|\psi_{n,\epsilon} - \psi_{m,\epsilon}|^2 \le 4(\Xi_{\nu_{\epsilon}}^{t} \psi_{\epsilon})^2$, which is in $\mathcal{L}_2(\mathbb{R}^n)$. Therefore, by the Lebesgue dominated convergence theorem, $\forall \epsilon \in (0,1]$, $\psi_{n,\epsilon}$ is a Cauchy sequence in $\mathcal{L}_2(\mathbb{R}^n)$, and so
$$ ((U_{v_{\epsilon}, v}^t)_{\epsilon})(\psi_{\epsilon})_{\epsilon} = \left( \mathcal{L}_2 - \lim_{n \to \infty} \left( \Xi_{v_{\epsilon}}^{\frac{t}{n}} M_v^{\frac{t}{n}} \right)^n \psi_{\epsilon} \right)_{\epsilon}, \quad (0.3.33) $$

where the limit now being taken in $\mathcal{L}_2(\mathbb{R}^n)$.

### I.0.4. Regularized Colombeau-Feynman Path Integral

($\mathbf{J}_{\epsilon, V}^{x,t}[\mathcal{L}_{\epsilon}(\dot{\omega}_{\epsilon}(\tau), \omega(\tau)), \psi_0(\omega(t))]$)$_{\epsilon}$
based on Colombeau-Wiener measure

($\tilde{D}_x^{\mathbb{R}}[\omega(\tau), v_{\epsilon}]$)$_{\epsilon}$.

Let us consider now the regularised Schrödinger equation

$$ \frac{\partial \psi_{\epsilon}(t,x)}{\partial t} = \frac{1}{2}(i + v_{\epsilon})\Delta \psi_{\epsilon}(t,x) - iV(x)\psi_{\epsilon}(t,x), \quad (0.4.1) $$

$$ ((\psi_{\epsilon}(0,x)))_{\epsilon} = \psi_0(x), $$

where $v_{\epsilon} \to 0$ if $\epsilon \to 0$. If the operator $H = -\Delta/2 + V(x)$ is:

(i) self-adjoint and

(ii) bounded from below, by the spectral theorem one obtain

$$ \exp(-itH) = \lim_{\epsilon \to 0} \exp[-(i + v_{\epsilon})tH] \quad (0.4.2) $$

strongly for all positive $t$. Therefore
$$ ((U_{v_{\epsilon}, v}^t)_{\epsilon})(\psi_{\epsilon})_{\epsilon} = \left( \mathcal{L}_2 - \lim_{n \to \infty} \left( \Xi_{v_{\epsilon}}^{\frac{t}{n}} M_v^{\frac{t}{n}} \right)^n \psi_{\epsilon} \right)_{\epsilon}, \quad (0.3.33) $$

where the limit now being taken in $\mathcal{L}_2(\mathbb{R}^n)$.

### I.0.4. Regularized Colombeau-Feynman Path Integral

($\mathbf{J}_{\epsilon, V}^{x,t}[\mathcal{L}_{\epsilon}(\dot{\omega}_{\epsilon}(\tau), \omega(\tau)), \psi_0(\omega(t))]_{\epsilon}$ based on Colombeau-Wiener measure

($\tilde{D}_x^{\mathbb{R}}[\omega(\tau), v_{\epsilon}]$)$_{\epsilon}$.

Let us consider now the regularised Schrödinger equation

$$ \frac{\partial \psi_{\epsilon}(t, x)}{\partial t} = \frac{1}{2}(i + v_{\epsilon})\Delta \psi_{\epsilon}(t, x) - iV(x)\psi_{\epsilon}(t, x), \quad (0.4.1) $$

$$ ((\psi_{\epsilon}(0, x)))_{\epsilon} = \psi_{0}(x), $$

where $v_{\epsilon} \to 0$ if $\epsilon \to 0$. If the operator $H = -\Delta/2 + V(x)$ is:

(i) self-adjoint and

(ii) bounded from below, by the spectral theorem one obtain

$$ \exp(-itH) = \lim_{\epsilon \to 0} \exp[-(i + v_{\epsilon})tH] \quad (0.4.2) $$

strongly for all positive $t$. Therefore
$$ (U_{v_{\epsilon}, \nu}^{t})_{\epsilon} = \exp\{t[(i + (v_{\epsilon})_{\epsilon})\Delta - i\nu]\}, \qquad (0.4.3) $$

$$ \Xi_{v_{\epsilon}}^{t} = \exp(t(i + (v_{\epsilon})_{\epsilon})\Delta). $$

Explicitly,
$$ (U_{v_{\epsilon}, \nu}^{t})_{\epsilon} = \exp\{t[(i + (v_{\epsilon})_{\epsilon})\Delta - i\nu]\}, \qquad (0.4.3) $$

$$ \Xi_{v_{\epsilon}}^{t} = \exp(t(i + (v_{\epsilon})_{\epsilon})\Delta). $$

Explicitly,
$$((\Xi_{v_\varepsilon}^t \varphi_\varepsilon(x))_\varepsilon) = (2\pi t(i + (v_\varepsilon)_\varepsilon))^{-d/2} \int_{\mathbb{R}^d} \exp\left(-\frac{|x-y|^2}{2t(i + (v_\varepsilon)_\varepsilon)}\right) \varphi(y)dy =$$

$$(2\pi t(i + (v_\varepsilon)_\varepsilon))^{-d/2} \int_{\mathbb{R}^d} \exp\left(-\frac{|x-y|^2((v_\varepsilon)_\varepsilon - i)}{2t((v_\varepsilon)_\varepsilon + i)((v_\varepsilon)_\varepsilon - i)}\right) \varphi(y)dy =$$

$$(2\pi t(i + (v_\varepsilon)_\varepsilon))^{-d/2} \int_{\mathbb{R}^d} \exp\left(-\frac{|x-y|^2((v_\varepsilon)_\varepsilon - i)}{2t((v_\varepsilon)^2_\varepsilon + 1)}\right) \varphi(y)dy =$$

$$(2\pi t(i + (v_\varepsilon)_\varepsilon))^{-d/2} \times$$

$$\int_{\mathbb{R}^d} \exp\left(-\frac{((v_\varepsilon)_\varepsilon)|x-y|^2}{2t((v_\varepsilon)^2_\varepsilon + 1)}\right) \exp\left(\frac{i|x-y|^2}{2t((v_\varepsilon)^2_\varepsilon + 1)}\right) \varphi(y)dy =$$

$$(2\pi t(i + (v_\varepsilon)_\varepsilon))^{-d/2} \times \left[ \frac{((v_\varepsilon)_\varepsilon}{(v_\varepsilon)^2_\varepsilon + 1} \right]^{-d/2} \times \left[ \frac{((v_\varepsilon)_\varepsilon}{(v_\varepsilon)^2_\varepsilon + 1} \right]^{d/2} \times$$

$$\int_{\mathbb{R}^d} \exp\left(-\frac{((v_\varepsilon)_\varepsilon)|x-y|^2}{2t((v_\varepsilon)^2_\varepsilon + 1)}\right) \exp\left(\frac{i|x-y|^2}{2t((v_\varepsilon)^2_\varepsilon + 1)}\right) \varphi(y)dy =$$

$$(i + (v_\varepsilon)_\varepsilon)^{-d/2} \times \left[ \frac{((v_\varepsilon)_\varepsilon}{((v_\varepsilon)^2_\varepsilon + 1)} \right]^{-d/2} \times$$

$$\int_{\mathbb{R}^d} \exp\left(\frac{i|x-y|^2}{((v_\varepsilon)^2_\varepsilon + 1)}\right) \varphi(y)p(x,dy) =$$

$$\left[ \frac{((v_\varepsilon)_\varepsilon}{(i - (v_\varepsilon)^2_\varepsilon)} \right]^{-d/2} \int_{\mathbb{R}^d} \exp\left(\frac{i|x-y|^2}{((v_\varepsilon)^2_\varepsilon + 1)}\right) \varphi(y)p_{v_\varepsilon}(x,dy),$$

where
$$((\Xi_{v_\varepsilon}^t \varphi_\varepsilon(x))_\varepsilon) = (2\pi t(i + (v_\varepsilon)_\varepsilon))^{-d/2} \int_{\mathbb{R}^d} \exp\left(-\frac{|x-y|^2}{2t(i + (v_\varepsilon)_\varepsilon)}\right) \varphi(y) dy =$$

$$(2\pi t(i + (v_\varepsilon)_\varepsilon))^{-d/2} \int_{\mathbb{R}^d} \exp\left(-\frac{|x-y|^2((v_\varepsilon)_\varepsilon - i)}{2t((v_\varepsilon)_\varepsilon + i)((v_\varepsilon)_\varepsilon - i)}\right) \varphi(y) dy =$$

$$(2\pi t(i + (v_\varepsilon)_\varepsilon))^{-d/2} \int_{\mathbb{R}^d} \exp\left(-\frac{|x-y|^2((v_\varepsilon)_\varepsilon - i)}{2t((v_\varepsilon)^2_\varepsilon + 1)}\right) \varphi(y) dy =$$

$$(2\pi t(i + (v_\varepsilon)_\varepsilon))^{-d/2} \times$$

$$\int_{\mathbb{R}^d} \exp\left(-\frac{((v_\varepsilon)_\varepsilon)|x-y|^2}{2t((v_\varepsilon)^2_\varepsilon + 1)}\right) \exp\left(\frac{i|x-y|^2}{2t((v_\varepsilon)^2_\varepsilon + 1)}\right) \varphi(y) dy =$$

$$(2\pi t(i + (v_\varepsilon)_\varepsilon))^{-d/2} \times \left[ \frac{((v_\varepsilon)_\varepsilon}{(v_\varepsilon)^2_\varepsilon + 1} \right]^{-d/2} \times \left[ \frac{((v_\varepsilon)_\varepsilon}{(v_\varepsilon)^2_\varepsilon + 1} \right]^{d/2} \times \quad (0.4.4)$$

$$\int_{\mathbb{R}^d} \exp\left(-\frac{((v_\varepsilon)_\varepsilon)|x-y|^2}{2t((v_\varepsilon)^2_\varepsilon + 1)}\right) \exp\left(\frac{i|x-y|^2}{2t((v_\varepsilon)^2_\varepsilon + 1)}\right) \varphi(y) dy =$$

$$(i + (v_\varepsilon)_\varepsilon)^{-d/2} \times \left[ \frac{((v_\varepsilon)_\varepsilon}{((v_\varepsilon)^2_\varepsilon + 1)} \right]^{-d/2} \times$$

$$\int_{\mathbb{R}^d} \exp\left(\frac{i|x-y|^2}{((v_\varepsilon)^2_\varepsilon + 1)}\right) \varphi(y) p(x, dy) =$$

$$\left[ \frac{((v_\varepsilon)_\varepsilon}{(i - (v_\varepsilon)^2_\varepsilon)} \right]^{-d/2} \int_{\mathbb{R}^d} \exp\left(\frac{i|x-y|^2}{((v_\varepsilon)^2_\varepsilon + 1)}\right) \varphi(y) p_{v_\varepsilon}(x, dy),$$

where
$$p_{v_{\varepsilon}}(x, dy) = \left[ \frac{((v_{\varepsilon})_{\varepsilon})}{2\pi t((v_{\varepsilon}^2)_{\varepsilon} + 1)} \right]^{d/2} \exp\left( -\frac{((v_{\varepsilon})_{\varepsilon})|x-y|^2}{2t((v_{\varepsilon}^2)_{\varepsilon} + 1)} \right). \quad (0.4.5)$$

**Definition 0.4.1.** We define Colombeau-Feynman Path Integral $(J_{v_{\varepsilon},v}^{x,t}[\cdot])_{\varepsilon,(0,1]}$ based on real-valued Colombeau-Wiener measure $(\tilde{D}_{x}^{\mathbb{R}}[\omega(\tau),v_{\varepsilon}])_{\varepsilon}$ by formula

$$ (J_{v_{\varepsilon},v}^{x,t}[\mathcal{L}_{\varepsilon}(\dot{\omega}(\tau), \omega(\tau)), \psi_{\varepsilon}(\omega(\tau))])_{\varepsilon,(0,1]} = \\ \left( \lim_{n \to \infty} \left[ \frac{((v_{\varepsilon})_{\varepsilon})}{(i - (v_{\varepsilon}^2)_{\varepsilon})} \right]^{-nd/2} \right) \times \\ \int_{\Omega_t} \exp \left\{ i \frac{t}{n} \sum_{j=1}^{n} \left[ \frac{m}{2} \left( \frac{t}{n} \right)^{-2} (\omega_{j,n}(t) - \omega_{j-1,n}(t))^2 \right] - i \int_{0}^{t} V(\omega(\tau)) d\tau \right\} \times \\ \psi_0(\omega(t)) \tilde{D}_{x}^{\mathbb{R}}[\omega(\tau), v_{\varepsilon}] \}_{\varepsilon} \quad (0.4.6) $$

or by equivalent complete Colombeau formula:

$$ (J_{v_{\varepsilon},n_{\varepsilon},v}^{x,t}[\mathcal{L}_{\varepsilon}(\dot{\omega}(\tau), \omega(\tau)), \psi_0(\omega(t))])_{\varepsilon,(0,1]} = \\ \left( \left[ \frac{((v_{\varepsilon}^{n_{\varepsilon}})_{\varepsilon})}{((i - (v_{\varepsilon}^2)_{\varepsilon})^{n_{\varepsilon}})} \right]^{-d/2} \right) \times \\ \int_{\Omega_t} \exp \left\{ i \frac{t}{n_{\varepsilon}} \sum_{j=1}^{n_{\varepsilon}} \left[ \frac{m}{2} \left( \frac{t}{n_{\varepsilon}} \right)^{-2} (\omega_{j,n_{\varepsilon}}(t) - \omega_{j-1,n_{\varepsilon}}(t))^2 \right] - i \int_{0}^{t} V(\omega(\tau)) d\tau \right\} \times \\ \psi_{\varepsilon}(\omega(t)) \tilde{D}_{x}^{\mathbb{R}}[\omega(\tau), v_{\varepsilon}] \}_{\varepsilon} \quad (0.4.7) $$
$$p_{v_{\varepsilon}}(x, dy) = \left[ \frac{((v_{\varepsilon})_{\varepsilon})}{2\pi t((v_{\varepsilon}^2)_{\varepsilon} + 1)} \right]^{d/2} \exp\left( -\frac{((v_{\varepsilon})_{\varepsilon})|x-y|^2}{2t((v_{\varepsilon}^2)_{\varepsilon} + 1)} \right). \quad (0.4.5)$$

**Definition 0.4.1.** We define Colombeau-Feynman Path Integral $(J_{v_{\varepsilon},v}^{x,t}[\cdot])_{\varepsilon,(0,1]}$ based on real-valued Colombeau-Wiener measure $(\tilde{D}_{x}^{\mathbb{R}}[\omega(\tau),v_{\varepsilon}])_{\varepsilon}$ by formula

$$ (\mathbf{J}_{v_{\varepsilon},v}^{x,t}[\mathcal{L}_{\varepsilon}(\dot{\omega}(\tau), \omega(\tau)), \psi_{\varepsilon}(\omega(\tau))])_{\varepsilon,(0,1]} = \\ \left( \lim_{n \to \infty} \left[ \frac{((v_{\varepsilon})_{\varepsilon})}{(i - (v_{\varepsilon}^2)_{\varepsilon})} \right]^{-nd/2} \right) \times \\ \int_{\Omega_t} \exp \left\{ i \frac{t}{n} \sum_{j=1}^{n} \left[ \frac{m}{2} \left( \frac{t}{n} \right)^{-2} (\omega_{j,n}(t) - \omega_{j-1,n}(t))^2 \right] - i \int_{0}^{t} V(\omega(\tau)) d\tau \right\} \times \\ \psi_0(\omega(t)) \tilde{D}_{x}^{\mathbb{R}}[\omega(\tau), v_{\varepsilon}] \}_{\varepsilon} \quad (0.4.6) $$

or by equivalent complete Colombeau formula:

$$ (\mathbf{J}_{v_{\varepsilon},n_{\varepsilon},v}^{x,t}[\mathcal{L}_{\varepsilon}(\dot{\omega}(\tau), \omega(\tau)), \psi_0(\omega(t))])_{\varepsilon,(0,1]} = \\ \left( \left[ \frac{((v_{\varepsilon}^{n_{\varepsilon}})_{\varepsilon})}{((i - (v_{\varepsilon}^2)_{\varepsilon})^{n_{\varepsilon}})} \right]^{-d/2} \right) \times \\ \int_{\Omega_t} \exp \left\{ i \frac{t}{n_{\varepsilon}} \sum_{j=1}^{n_{\varepsilon}} \left[ \frac{m}{2} \left( \frac{t}{n_{\varepsilon}} \right)^{-2} (\omega_{j,n_{\varepsilon}}(t) - \omega_{j-1,n_{\varepsilon}}(t))^2 \right] - i \int_{0}^{t} V(\omega(\tau)) d\tau \right\} \times \\ \psi_{\varepsilon}(\omega(t)) \tilde{D}_{x}^{\mathbb{R}}[\omega(\tau), v_{\varepsilon}] \}_{\varepsilon} \quad (0.4.7) $$
where $n_{\epsilon} \to \infty$ if $\epsilon \to 0$.

We wish to expand now Colombeau-Wiener integral (0.3.25) on functionals given by

Eq.(0.3.10) using smoothing approach based on the mollification of the continuous

trajectories $\omega(\tau) \in \mathbb{C}[0,t]$. We use a net of mollifiers $\phi_{n,\epsilon}(\tau) \in S(\mathbb{R}_+), \epsilon \in (0,1]$ of the form

$$ \phi_{n,\epsilon}(\tau) = \frac{1}{\epsilon} \phi_n\left(\frac{\tau}{\epsilon}\right), \qquad (0.4.7) $$

where $\phi_n$ has the properties $\forall n \in \mathbb{N}$:

(i) $\int_0^t \phi_n(\tau) d\tau = 1,$

(ii) $\phi_n(\tau) \ge 0$ for all $\tau \in [0,t]$,

(iii) $\phi_n \in C^{\infty}$.

**Definition 0.4.1.**[46]. For $\epsilon > 0$ the $\epsilon$-mollification $\omega_{n,\epsilon}(\tau)$ of a locally integrable function $\omega: [0,t] \to \mathbb{R}^d, \omega(\tau) = (\omega_1(\tau), \dots, \omega_d(\tau))$ is the convolution on $\mathbb{R}$ such that

$$ \omega_{n,\epsilon}(\tau) = \phi_{n,\epsilon} * \omega(\tau), \qquad (0.4.8) $$

that is,

$$ \omega_{n,\epsilon}(\tau) = \int_{\mathbb{R}} \phi_{n,\epsilon}(\tau - z) \omega(z) dz. \qquad (0.4.9) $$

We remind that [46]: (i) for all $\epsilon > 0$, the $\epsilon$-mollification $\omega_{n,\epsilon}(\tau)$ in $C^{\infty}(\mathbb{R}_+, \mathbb{R}^d)$,

(ii) If $\omega$ is continuous on $\mathbb{R}_+$, then $\omega_{n,\epsilon}$ converges to $\omega$ uniformly on any compact sets for $\epsilon \to 0$.

**Definition 0.4.2.** We define smoothed classical action by formula
where $n_{\epsilon} \to \infty$ if $\epsilon \to 0$.

We wish to expand now Colombeau-Wiener integral (0.3.25) on functionals given by

Eq.(0.3.10) using smoothing approach based on the mollification of the continuous

trajectories $\omega(\tau) \in \mathbb{C}[0,t]$. We use a net of mollifiers $\phi_{n,\epsilon}(\tau) \in S(\mathbb{R}_+), \epsilon \in (0,1]$ of the form

$$ \phi_{n,\epsilon}(\tau) = \frac{1}{\epsilon} \phi_n\left(\frac{\tau}{\epsilon}\right), \qquad (0.4.7) $$

where $\phi_n$ has the properties $\forall n \in \mathbb{N}$:

(i) $\int_0^t \phi_n(\tau) d\tau = 1,$

(ii) $\phi_n(\tau) \ge 0$ for all $\tau \in [0,t]$,

(iii) $\phi_n \in C^{\infty}$.

**Definition 0.4.1.**[46]. For $\epsilon > 0$ the $\epsilon$-mollification $\omega_{n,\epsilon}(\tau)$ of a locally integrable function $\omega: [0,t] \to \mathbb{R}^d, \omega(\tau) = (\omega_1(\tau), \dots, \omega_d(\tau))$ is the convolution on $\mathbb{R}$ such that

$$ \omega_{n,\epsilon}(\tau) = \phi_{n,\epsilon} * \omega(\tau), \qquad (0.4.8) $$

that is,

$$ \omega_{n,\epsilon}(\tau) = \int_{\mathbb{R}} \phi_{n,\epsilon}(\tau - z)\omega(z)dz. \qquad (0.4.9) $$

We remind that [46]: (i) for all $\epsilon > 0$, the $\epsilon$-mollification $\omega_{n,\epsilon}(\tau)$ in $C^{\infty}(\mathbb{R}_+, \mathbb{R}^d)$,

(ii) If $\omega$ is continuous on $\mathbb{R}_+$, then $\omega_{n,\epsilon}$ converges to $\omega$ uniformly on any compact sets for $\epsilon \to 0$.

**Definition 0.4.2.** We define smoothed classical action by formula
$$
\begin{equation} \tag{0.4.10}
\begin{split}
S_{t,n,\epsilon}(\dot{\omega}_{\epsilon}(\tau), \omega(\tau)) &= \int_{0}^{t} \mathcal{L}_{n,\epsilon}(\dot{\omega}_{n,\epsilon}(\tau), \omega(\tau)) d\tau \\
&= \int_{0}^{t} \left[ \frac{m}{2} \dot{\omega}_{n,\epsilon}^{2}(\tau) - V(\omega(\tau)) \right] d\tau,
\end{split}
\end{equation}
$$

where $\dot{\omega}_{n,\epsilon}(\tau) = \frac{d\omega_{n,\epsilon}(\tau)}{d\tau} = \omega(\tau) * \frac{d\phi_{n,\epsilon}(\tau)}{d\tau}$, and so we may form the complex
Colombeau-Wiener integral:

$$
\begin{align*}
& (\mathbf{J}_{v_{\epsilon}, n_{\epsilon}}^{x,t} [\mathcal{L}_{\epsilon}(\dot{\omega}(\tau), \omega(\tau)), \psi_0(\omega(t))])_{\epsilon} = \\
& \quad \left( \left[ \frac{((v_{\epsilon}^{n_{\epsilon}})_{\epsilon})}{((i - (v_{\epsilon}^2))^{n_{\epsilon}})_{\epsilon}} \right]^{-d/2} \times \right. \\
& \quad \left. \int_{\Omega_t} \exp \left[ i \int_0^t \left[ \frac{m}{2} \dot{\omega}_{n_{\epsilon}, \epsilon}^2(\tau) - V(\omega(\tau)) \right] d\tau \right] \psi_0(\omega(t)) \tilde{D}_x^{\mathbb{R}}[\omega(\tau), v_{\epsilon}] \right)_{\epsilon} = \\
& \quad \left[ \frac{((v_{\epsilon}^{n_{\epsilon}})_{\epsilon})}{((i - (v_{\epsilon}^2))^{n_{\epsilon}})_{\epsilon}} \right]^{-d/2} \times \int_{\Omega_t} \exp \left[ i \left( \int_0^t \left[ \frac{m}{2} \dot{\omega}_{n_{\epsilon}, \epsilon}^2(\tau) - V(\omega(\tau)) \right] d\tau \right)_{\epsilon} \right] \times \\
& \quad ((\psi_0(\omega(t))))_{\epsilon} (\tilde{D}_x^{\mathbb{R}}[\omega(\tau), v_{\epsilon}])_{\epsilon}
\end{align*}
$$

and in more general case
$$
\begin{equation}
\begin{aligned}
S_{t,n,\epsilon}(\dot{\omega}_{\epsilon}(\tau), \omega(\tau)) &= \int_{0}^{t} \mathcal{L}_{n,\epsilon}(\dot{\omega}_{n,\epsilon}(\tau), \omega(\tau)) d\tau = \\
&\qquad \int_{0}^{t} \left[ \frac{m}{2} \dot{\omega}_{n,\epsilon}^{2}(\tau) - V(\omega(\tau)) \right] d\tau,
\end{aligned}
\tag{0.4.10}
\end{equation}
$$

where $\dot{\omega}_{n,\epsilon}(\tau) = \frac{d\omega_{n,\epsilon}(\tau)}{d\tau} = \omega(\tau) * \frac{d\phi_{n,\epsilon}(\tau)}{d\tau}$, and so we may form the complex
Colombeau-Wiener integral:

$$
\begin{align*}
& (\mathbf{J}_{v_{\epsilon}, n_{\epsilon}}^{x,t} [\mathcal{L}_{\epsilon}(\dot{\omega}(\tau), \omega(\tau)), \psi_0(\omega(t))])_{\epsilon} = \\
& \qquad \left( \left[ \frac{((v_{\epsilon}^{n_{\epsilon}})_{\epsilon})}{((i - (v_{\epsilon}^2))^{n_{\epsilon}})_{\epsilon}} \right]^{-d/2} \times \right. \\
& \qquad \left. \int_{\Omega_t} \exp \left[ i \int_0^t \left[ \frac{m}{2} \dot{\omega}_{n_{\epsilon}, \epsilon}^2(\tau) - V(\omega(\tau)) \right] d\tau \right] \psi_0(\omega(t)) \tilde{D}_x^{\mathbb{R}}[\omega(\tau), v_{\epsilon}] \right)_{\epsilon} = \\
& \qquad \left[ \frac{((v_{\epsilon}^{n_{\epsilon}})_{\epsilon})}{((i - (v_{\epsilon}^2))^{n_{\epsilon}})_{\epsilon}} \right]^{-d/2} \times \int_{\Omega_t} \exp \left[ i \left( \int_0^t \left[ \frac{m}{2} \dot{\omega}_{n_{\epsilon}, \epsilon}^2(\tau) - V(\omega(\tau)) \right] d\tau \right)_{\epsilon} \right] \times \\
& \qquad \qquad ((\psi_0(\omega(t))))_{\epsilon} (\tilde{D}_x^{\mathbb{R}}[\omega(\tau), v_{\epsilon}])_{\epsilon}
\end{align*}
$$

and in more general case
$$ (\mathbf{J}_{v_{\varepsilon}, n_{\delta}, v}^{x,t}[\mathcal{L}_{\varepsilon, \delta}(\dot{\omega}_{n, \delta}(\tau), \omega(\tau)), \psi_0(\omega(t))])_{\varepsilon \in (0,1], \delta \in (0,1]} = $$

$$ \left( \left[ \frac{((v_{\varepsilon}^{n_{\delta}})_{\varepsilon})}{((i - (v_{\varepsilon}^2))_{\varepsilon}^{n_{\delta}})} \right]^{-d/2} \right)_{\varepsilon(0,1], \delta(0,1]} \times $$

$$ \left( \int_{\Omega_t} \exp \left[ i \int_0^t \left[ \frac{m}{2} \dot{\omega}_{n_{\delta}, \delta}^2(\tau) - V(\omega(\tau)) \right] d\tau \right] \psi_0(\omega(t)) \tilde{D}_x^{\mathbb{R}}[\omega(\tau), v_{\varepsilon}] \right)_{\varepsilon(0,1], \delta(0,1]} = \quad (0.4.12) $$

$$ \int_{\Omega_t} \exp \left[ i \left( \int_0^t \left[ \frac{m}{2} \dot{\omega}_{n_{\delta}, \delta}^2(\tau) - V(\omega(\tau)) \right] d\tau \right)_{\delta \in (0,1]} \right] \times $$

$$ ((\psi_{\varepsilon}(\omega(t)))_{\varepsilon \in (0,1]}) (\tilde{D}_{x}^{\mathbb{R}}[\omega(\tau), v_{\varepsilon}])_{\varepsilon \in (0,1]} $$

**Definition 0.4.3.** We define smoothed Colombeau-Feynman Path Integral based on real-valued Colombeau-Wiener measure $(\tilde{D}_x^{\mathbb{R}}[\omega(\tau), v_\varepsilon])_\varepsilon$ by formula

$$ \int_{\Omega_t} \exp \left[ i \int_0^t \left[ \frac{m}{2} \dot{\omega}^2(\tau) - V(\omega(\tau)) \right] d\tau \right] ((\psi_0(\omega(t)))_\varepsilon) (\tilde{D}_x^{\mathbb{R}}[\omega(\tau), v_\varepsilon])_\varepsilon \triangleq \quad (0.4.13) $$

$$ (\mathbf{J}_{v_{\varepsilon}, n_{\varepsilon}, v}^{x,t}[\mathcal{L}_{\varepsilon}(\dot{\omega}_{n_{\varepsilon}, \varepsilon}(\tau), \omega(\tau)), \psi_0(\omega(t))])_{\varepsilon} $$

and in more general case
$$ (\mathbf{J}_{v_{\varepsilon}, n_{\delta}, v}^{x,t}[\mathcal{L}_{\varepsilon, \delta}(\dot{\omega}_{n, \delta}(\tau), \omega(\tau)), \psi_0(\omega(t))])_{\varepsilon \in (0,1], \delta \in (0,1]} = $$

$$ \left( \left[ \frac{((v_{\varepsilon}^{n_{\delta}})_{\varepsilon})}{((i - (v_{\varepsilon}^2))_{\varepsilon}^{n_{\delta}})} \right]^{-d/2} \right)_{\varepsilon(0,1], \delta(0,1]} \times $$

$$ \left( \int_{\Omega_t} \exp \left[ i \int_0^t \left[ \frac{m}{2} \dot{\omega}_{n_{\delta}, \delta}^2(\tau) - V(\omega(\tau)) \right] d\tau \right] \psi_0(\omega(t)) \tilde{D}_x^{\mathbb{R}}[\omega(\tau), v_{\varepsilon}] \right)_{\varepsilon(0,1], \delta(0,1]} = \quad (0.4.12) $$

$$ \int_{\Omega_t} \exp \left[ i \left( \int_0^t \left[ \frac{m}{2} \dot{\omega}_{n_{\delta}, \delta}^2(\tau) - V(\omega(\tau)) \right] d\tau \right)_{\delta \in (0,1]} \right] \times $$

$$ ((\psi_{\varepsilon}(\omega(t)))_{\varepsilon \in (0,1]}) (\tilde{D}_x^{\mathbb{R}}[\omega(\tau), v_{\varepsilon}])_{\varepsilon \in (0,1]} $$

**Definition 0.4.3.** We define smoothed Colombeau-Feynman Path Integral based on real-valued Colombeau-Wiener measure $(\tilde{D}_x^{\mathbb{R}}[\omega(\tau), v_{\varepsilon}])_{\varepsilon}$ by formula

$$ \int_{\Omega_t} \exp \left[ i \int_0^t \left[ \frac{m}{2} \dot{\omega}^2(\tau) - V(\omega(\tau)) \right] d\tau \right] ((\psi_0(\omega(t)))_{\varepsilon}) (\tilde{D}_x^{\mathbb{R}}[\omega(\tau), v_{\varepsilon}])_{\varepsilon} \triangleq \quad (0.4.13) $$

$$ (\mathbf{J}_{v_{\varepsilon}, n_{\varepsilon}, v}^{x,t}[\mathcal{L}_{\varepsilon}(\dot{\omega}_{n_{\varepsilon}, \varepsilon}(\tau), \omega(\tau)), \psi_0(\omega(t))])_{\varepsilon} $$

and in more general case
$$
\begin{align}
& \int_{\Omega_t} \exp \left[ i \int_0^t \left[ \frac{m}{2} \dot{\omega}^2(\tau) - V(\omega(\tau)) \right] d\tau \right] \times \nonumber \\
& \qquad \left( (\psi_0(\omega(t)))_{\varepsilon \in (0,1]} \right) \left( \tilde{D}_x^{\mathbb{R}}[\omega(\tau), v_\varepsilon] \right)_{\varepsilon \in (0,1]_{\cdot}, \delta \in (0,1]} \triangleq \tag{0.4.14} \\
& \qquad (\mathbf{J}_{v_\varepsilon, n_\delta, \nu}^{x,t} [\mathcal{L}_\varepsilon(\dot{\omega}_{n_\delta, \delta}(\tau), \omega(\tau)), \psi_0(\omega(t))])_{\varepsilon \in (0,1]_{\cdot}, \delta \in (0,1]} \nonumber
\end{align}
$$

Since $\forall \varepsilon \in (0,1]$

$$
\mathcal{L}_{\varepsilon}(\dot{\omega}_{n_{\varepsilon}, \varepsilon}(\tau), \omega(\tau)) = \frac{m}{2} \dot{\omega}_{n_{\varepsilon}, \varepsilon}^{2}(\tau) - V(\omega(\tau)) \quad (0.4.15)
$$

is continuous for almost every $\omega(\tau)$, one obtain

$$
\int_0^t \mathcal{L}_\varepsilon(\dot{\omega}_{n_\varepsilon, \varepsilon}(\tau), \omega(\tau)) d\tau = \lim_{n_\varepsilon \to \infty} \left( \frac{t}{n_\varepsilon} \sum_{j=1}^{n_\varepsilon} \left[ \frac{m}{2} \dot{\omega}_{n_\varepsilon, \varepsilon}^2 \left( j \frac{t}{n_\varepsilon} \right) - V \left( \omega \left( j \frac{t}{n_\varepsilon} \right) \right) \right] \right) \quad (0.4.16)
$$

for almost every $\omega \in \Omega_t$. By the Lebesgue dominated convergence theorem, this
implies that

$$
(\mathbf{J}_{v_{\varepsilon}, n_{\varepsilon}, \nu}^{x,t} [\mathcal{L}_{\varepsilon}(\dot{\omega}_{n_{\varepsilon}, \varepsilon}(\tau), \omega(\tau)), \psi_0(\omega(t))])_{\varepsilon} = \\
\left[ \frac{((v_{\varepsilon}^{n_{\varepsilon}})_{\varepsilon})}{((i - (v_{\varepsilon}^2)^{n_{\varepsilon}})_{\varepsilon})} \right]^{-d/2} \qquad (0.4.17)
$$

$$
\left( \int_{\Omega_t} \exp \left[ i \frac{t}{n} \sum_{j=1}^{n} \left[ \frac{m}{2} \dot{\omega}_{n_{\varepsilon}, \varepsilon}^2 \left( j \frac{t}{n} \right) - V \left( \omega \left( j \frac{t}{n} \right) \right) \right] \right] \psi_{\varepsilon}(\omega(t)) \tilde{D}_x^{\mathbb{R}} [\omega(\tau), v_{\varepsilon}] \right)_{\varepsilon}
$$
$$
\begin{align}
& \int_{\Omega_t} \exp \left[ i \int_0^t \left[ \frac{m}{2} \dot{\omega}^2(\tau) - V(\omega(\tau)) \right] d\tau \right] \times \nonumber \\
& \qquad \left( (\psi_0(\omega(t)))_{\varepsilon \in (0,1]} \right) \left( \tilde{D}_x^{\mathbb{R}}[\omega(\tau), v_\varepsilon] \right)_{\varepsilon \in (0,1]_{\cdot}, \delta \in (0,1]} \triangleq \tag{0.4.14} \\
& \qquad (\mathbf{J}_{v_\varepsilon, n_\delta, \nu}^{x,t} [\mathcal{L}_\varepsilon(\dot{\omega}_{n_\delta, \delta}(\tau), \omega(\tau)), \psi_0(\omega(t))])_{\varepsilon \in (0,1]_{\cdot}, \delta \in (0,1]} \nonumber
\end{align}
$$

Since $\forall \varepsilon \in (0,1]$

$$
\mathcal{L}_{\varepsilon}(\dot{\omega}_{n_{\varepsilon}, \varepsilon}(\tau), \omega(\tau)) = \frac{m}{2} \dot{\omega}_{n_{\varepsilon}, \varepsilon}^2(\tau) - V(\omega(\tau)) \quad (0.4.15)
$$

is continuous for almost every $\omega(\tau)$, one obtain

$$
\int_0^t \mathcal{L}_\varepsilon(\dot{\omega}_{n_\varepsilon, \varepsilon}(\tau), \omega(\tau)) d\tau = \lim_{n_\varepsilon \to \infty} \left( \frac{t}{n_\varepsilon} \sum_{j=1}^{n_\varepsilon} \left[ \frac{m}{2} \dot{\omega}_{n_\varepsilon, \varepsilon}^2 \left(j \frac{t}{n_\varepsilon}\right) - V\left(\omega\left(j \frac{t}{n_\varepsilon}\right)\right) \right] \right) \quad (0.4.16)
$$

for almost every $\omega \in \Omega_t$. By the Lebesgue dominated convergence theorem, this
implies that

$$
\begin{align}
& (\mathbf{J}_{v_\epsilon, n_\epsilon, \nu}^{x,t} [\mathcal{L}_\epsilon(\dot{\omega}_{n_\epsilon, \epsilon}(\tau), \omega(\tau)), \psi_0(\omega(t))])_\epsilon = \nonumber \\
& \quad \left[ \frac{((v_\epsilon^{n_\epsilon})_\epsilon)}{((i - (v_\epsilon^2))^{n_\epsilon})_\epsilon} \right]^{-d/2} \tag{0.4.17} \\
& \quad \left( \int_{\Omega_t} \exp \left[ i \frac{t}{n} \sum_{j=1}^n \left[ \frac{m}{2} \dot{\omega}_{n_\epsilon, \epsilon}^2 \left(j \frac{t}{n}\right) - V\left(\omega\left(j \frac{t}{n}\right)\right) \right] \right] \psi_\epsilon(\omega(t)) \tilde{D}_x^{\mathbb{R}}[\omega(\tau), v_\epsilon] \right)_\epsilon \nonumber
\end{align}
$$
for all $x \in \mathbb{R}^d \setminus F_V$, where $F_V$ is a closed set in $\mathbb{R}^d$ which depend only on $V$. By definition of the Colombeau-Wiener integral (0.3.25), this means that for all $x \notin F_V$

$$ \left( \left( U_{v_{\epsilon}, n_{\epsilon}, \nu}^{t} \psi_{0}(x) \right)_{\epsilon} \right) = \left( \left( \Xi_{v_{\epsilon}, n_{\epsilon}}^{\frac{t}{n_{\epsilon}}} M_{\nu}^{\frac{t}{n_{\epsilon}}} \right)^{n_{\epsilon}} \psi_{0}(x) \right)_{\epsilon}. \quad (0.4.18) $$

Here

$$ (U_{v_{\epsilon}, n_{\epsilon}, \nu}^{t})_{\epsilon} = \exp\{t[(i + (v_{\epsilon})_{\epsilon})(\Delta[\phi_{n_{\epsilon}, \epsilon}])_{\epsilon} - iV]\}, \quad (0.4.19) $$

$$ \Xi_{v_{\epsilon}, n_{\epsilon}}^{t} = \exp(t(i + v_{\epsilon})\Delta[\phi_{n_{\epsilon}, \epsilon}]), $$

where we denote by $\Delta[\phi_{n_{\epsilon}, \epsilon}]$ smoothed Laplace operator, which defined by obvious way.

### I.0.5. Colombeau-Feynman Path Integral
($\mathbf{J}_{v_{\epsilon}, n_{\epsilon}, \nu}^{x,t}[\mathcal{L}_{\epsilon}(\dot{\omega}_{\epsilon}(\tau), \omega(\tau)), \psi_{\epsilon}(\omega(t))]$ based on
Colombeau-Wiener measure $(\tilde{D}_{x}^{\mathbb{R}}[\omega(\tau), v_{\epsilon}])_{\epsilon}$.

In our subsequent consideration, the potential $V(t,x)$ does not satisfy Kato conditions. We remove it by using cut-off or by another type of the regularization. As a framework we use Colombeau's algebra of generalized functions, i.e. we change potential $V(t,x)$ by Colombeau generalized function $(V_{\epsilon}(t,x))_{\epsilon \in (0,1]}$ such that $V_0(t,x) = V(t,x)$. Therefore classical action is

$$ (S_{t,\epsilon}(\dot{\omega}(\tau), \omega(\tau)))_{\epsilon} = \left( \int_{0}^{t} \left[ \frac{m}{2} \dot{\omega}^{2}(\tau) - V_{\epsilon}(\omega(\tau)) \right] d\tau \right)_{\epsilon} \quad (0.5.1) $$
for all $x \in \mathbb{R}^d \setminus F_V$, where $F_V$ is a closed set in $\mathbb{R}^d$ which depend only on $V$. By definition of the Colombeau-Wiener integral (0.3.25), this means that for all $x \notin F_V$

$$ \left( \left( U_{v_{\epsilon}, n_{\epsilon}, \nu}^{t} \psi_{0}(x) \right)_{\epsilon} \right) = \left( \left( \Xi_{v_{\epsilon}, n_{\epsilon}}^{\frac{t}{n_{\epsilon}}} M_{\nu}^{\frac{t}{n_{\epsilon}}} \right)^{n_{\epsilon}} \psi_{0}(x) \right)_{\epsilon}. \quad (0.4.18) $$

Here

$$ (U_{v_{\epsilon}, n_{\epsilon}, \nu}^{t})_{\epsilon} = \exp\{t[(i + (v_{\epsilon})_{\epsilon})(\Delta[\phi_{n_{\epsilon}, \epsilon}])_{\epsilon} - iV]\}, \quad (0.4.19) $$

$$ \Xi_{v_{\epsilon}, n_{\epsilon}}^{t} = \exp(t(i + v_{\epsilon})\Delta[\phi_{n_{\epsilon}, \epsilon}]), $$

where we denote by $\Delta[\phi_{n_{\epsilon}, \epsilon}]$ smoothed Laplace operator, which defined by obvious way.

### 1.0.5. Colombeau-Feynman Path Integral
($\mathbf{J}_{v_{\epsilon}, n_{\epsilon}, \nu}^{x,t}[\mathcal{L}_{\epsilon}(\dot{\omega}_{\epsilon}(\tau), \omega(\tau)), \psi_{\epsilon}(\omega(t))]$ based on Colombeau-Wiener measure ($\tilde{D}_{x}^{\mathbb{R}}[\omega(\tau), v_{\epsilon}]$) .

In our subsequent consideration, the potential $V(t,x)$ does not satisfy Kato conditions. We remove it by using cut-off or by another type of the regularization. As a framework we use Colombeau's algebra of generalized functions, i.e. we change potential $V(t,x)$ by Colombeau generalized function $(V_{\epsilon}(t,x))_{\epsilon \in (0,1]}$ such that $V_0(t,x) = V(t,x)$. Therefore classical action is

$$ (S_{t,\epsilon}(\dot{\omega}(\tau), \omega(\tau)))_{\epsilon} = \left( \int_{0}^{t} \left[ \frac{m}{2} \dot{\omega}^{2}(\tau) - V_{\epsilon}(\omega(\tau)) \right] d\tau \right)_{\epsilon} \quad (0.5.1) $$
We assume that $\forall \epsilon \in (0,1]$ the potential $V_{\epsilon}(t,x)$ is satisfy Kato conditions.Therefore the operator $V_{\epsilon}$ of multiplication by the function $V_{\epsilon}(t,x)$, on the domain $D(V_{\epsilon})$ of all $\psi_{0,\epsilon}$ in $\mathcal{L}_2$ such that $\forall \epsilon \in (0,1] : V_{\epsilon}\psi_{0,\epsilon}$ is also in $\mathcal{L}_2$, operator $V_{\epsilon}$ is self-adjoint, and if

$$ (M_{V_{\epsilon}}^{t})_{\epsilon} = (\exp\{-itV_{\epsilon}\})_{\epsilon} = \exp\{-it(V_{\epsilon})_{\epsilon}\}, \qquad (0.5.2) $$

then $(\psi_{\epsilon}(t,x))_{\epsilon} = (M_{V_{\epsilon}}^{t}\psi_{0,\epsilon}(x))_{\epsilon}$ is the Colombeau solution of Eq.(0.2.21) with $1/2m$ replaced by 0. If we let now

$$ (U_{m,V_{\epsilon}}^{t})_{\epsilon} = \exp\left[it\left(\frac{1}{2m}\Delta - (V_{\epsilon})_{\epsilon}\right)\right] \qquad (0.5.3) $$

then Trotter's theorem asserts that for all $(\psi_{0,\epsilon})_{\epsilon}$ such that $\psi_{0,\epsilon} \in \mathcal{L}_2$:

$$ ((U_{m,V_{\epsilon}}^{t})_{\epsilon})(\psi_{0,\epsilon})_{\epsilon} = (U_{m,V_{\epsilon}}^{t}\psi_{0,\epsilon})_{\epsilon} = \left(\lim_{n\to\infty} \left(\Xi_{m}^{\frac{t}{n}} M_{V_{\epsilon}}^{\frac{t}{n}}\right)^{n} \psi_{0,\epsilon}\right)_{\epsilon}. \qquad (0.5.4) $$

Using (0.5.2)-(0.5.4) we find that

$$ ((U_{m,V_{\epsilon}}^{t})_{\epsilon})(\psi_{0,\epsilon})_{\epsilon} = \left(\lim_{n\to\infty} \left(\frac{2\pi i t}{nm}\right)^{-\frac{1}{2}dn} \int \dots \int \exp[iS_{\epsilon}(x_0, \dots, x_n)] \psi_{0,\epsilon}(x_n) dx_1 \dots dx_n\right)_{\epsilon}. \qquad (0.5.5) $$

Here we set $x_0 = x$ and
We assume that $\forall \epsilon \in (0,1]$ the potential $V_{\epsilon}(t,x)$ is satisfy Kato conditions.Therefore the operator $V_{\epsilon}$ of multiplication by the function $V_{\epsilon}(t,x)$, on the domain $D(V_{\epsilon})$ of all $\psi_{0,\epsilon}$ in $\mathcal{L}_2$ such that $\forall \epsilon \in (0,1] : V_{\epsilon}\psi_{0,\epsilon}$ is also in $\mathcal{L}_2$, operator $V_{\epsilon}$ is self-adjoint, and if

$$ (M_{V_{\epsilon}}^{t})_{\epsilon} = (\exp\{-itV_{\epsilon}\})_{\epsilon} = \exp\{-it(V_{\epsilon})_{\epsilon}\}, \qquad (0.5.2) $$

then $(\psi_{\epsilon}(t,x))_{\epsilon} = (M_{V_{\epsilon}}^{t}\psi_{0,\epsilon}(x))_{\epsilon}$ is the Colombeau solution of Eq.(0.2.21) with $1/2m$ replaced by 0. If we let now

$$ (U_{m,V_{\epsilon}}^{t})_{\epsilon} = \exp\left[it\left(\frac{1}{2m}\Delta - (V_{\epsilon})_{\epsilon}\right)\right] \qquad (0.5.3) $$

then Trotter's theorem asserts that for all $(\psi_{0,\epsilon})_{\epsilon}$ such that $\psi_{0,\epsilon} \in \mathcal{L}_2$:

$$ ((U_{m,V_{\epsilon}}^{t})_{\epsilon})(\psi_{0,\epsilon})_{\epsilon} = (U_{m,V_{\epsilon}}^{t}\psi_{0,\epsilon})_{\epsilon} = \left(\lim_{n\to\infty} \left(\Xi_{m}^{\frac{t}{n}} M_{V_{\epsilon}}^{\frac{t}{n}}\right)^{n} \psi_{0,\epsilon}\right)_{\epsilon}. \qquad (0.5.4) $$

Using (0.5.2)-(0.5.4) we find that

$$ ((U_{m,V_{\epsilon}}^{t})_{\epsilon})(\psi_{0,\epsilon})_{\epsilon} = \left(\lim_{n\to\infty} \left(\frac{2\pi i t}{nm}\right)^{-\frac{1}{2}dn} \int \dots \int \exp[iS_{\epsilon}(x_0, \dots, x_n)] \psi_{0,\epsilon}(x_n) dx_1 \dots dx_n\right)_{\epsilon}. \qquad (0.5.5) $$

Here we set $x_0 = x$ and
$$S_{\varepsilon}(x_0, \dots, x_n) = \left(\frac{t}{n}\right) \sum_{j=1}^{n} \left[ \frac{m}{2} \frac{(x_i - x_{j-1})^2}{(t/n)^2} - V_{\varepsilon}(x_i) \right]. \quad (0.5.6)$$

$x \in \bigcup_{\varepsilon \in (0,1]} \mathbb{R}^d \setminus F_{V_{\varepsilon}}$, where $F_{V_{\varepsilon}}$ is a closed set in $\mathbb{R}^d$ which depend on $V_{\varepsilon}$.

We assume now that $\forall \varepsilon \in (0,1]$, the function $V_{\varepsilon}(x)$ is regular or satisfy Nelson's assumption (see [41]). Then for almost every $\omega \in \Omega_t$ and $\forall \varepsilon \in (0,1]$, $V_{\varepsilon}(\omega(\tau))$ is a continuous function of $\tau$ for $0 \le \tau < \infty$ and so (as above in section 1.0.4) we may form the Colombeau-Wiener integral:

$$(\mathbf{J}_{V_{\varepsilon}}^{x,t}[V_{\varepsilon}(\omega(\tau)), \psi_{\varepsilon}(\omega(\tau))])_{\varepsilon} =$$

$$\left( \int_{\Omega_t} \exp \left[ -i \int_0^t V_{\varepsilon}(\omega(\tau)) d\tau \right] \psi_{\varepsilon}(\omega(t)) \widetilde{D}_x^{\mathbb{R}}[\omega(\tau), v_{\varepsilon}] \right)_{\varepsilon} = \quad (0.5.7)$$

$$\int_{\Omega_t} \exp \left[ -i \left( \int_0^t V_{\varepsilon}(\omega(\tau)) d\tau \right)_{\varepsilon} \right] \left( (\psi_{\varepsilon}(\omega(t)))_{\varepsilon} \right) \left( \widetilde{D}_x^{\mathbb{R}}[\omega(\tau), v_{\varepsilon}] \right)_{\varepsilon}$$

for any $(\psi_{\varepsilon}(\omega(\tau)))_{\varepsilon} \in G(\mathbb{R}^d)$ and any $t \ge 0$. In fact, the integrand is defined $\forall \varepsilon \in (0,1]$ for almost every $\omega \in \Omega_t$, and it is bounded in absolute value by $|\psi_{\varepsilon}(\omega(t))|_{\varepsilon}$. But this is Colombeau-integrable, for by definition and (0.5.7) one obtain

$$| (\mathbf{J}_{V_{\varepsilon}}^{x,t}[V_{\varepsilon}(\omega(\tau)), \psi_{\varepsilon}(\omega(\tau))])_{\varepsilon} | \le (\mathbf{J}_{V_{\varepsilon}}^{x,t}[\psi_{\varepsilon}(\omega(\tau))])_{\varepsilon} \le ((c_{\varepsilon})_{\varepsilon}) (\sup_{\omega(\tau)} |\psi_{\varepsilon}(\omega(t))|)_{\varepsilon}. \quad (0.5.8)$$

Since $\forall \varepsilon \in (0,1]$, $V_{\varepsilon}(\omega(\tau))$ is continuous for almost every $\omega(\tau)$,
$$S_{\varepsilon}(x_0, \dots, x_n) = \left(\frac{t}{n}\right) \sum_{j=1}^{n} \left[ \frac{m}{2} \frac{(x_i - x_{j-1})^2}{(t/n)^2} - V_{\varepsilon}(x_i) \right]. \quad (0.5.6)$$

$x \in \bigcup_{\varepsilon \in (0,1]} \mathbb{R}^d \setminus F_{V_{\varepsilon}}$, where $F_{V_{\varepsilon}}$ is a closed set in $\mathbb{R}^d$ which depend on $V_{\varepsilon}$.

We assume now that $\forall \varepsilon \in (0,1]$, the function $V_{\varepsilon}(x)$ is regular or satisfy Nelson's assumption (see [41]). Then for almost every $\omega \in \Omega_t$ and $\forall \varepsilon \in (0,1]$, $V_{\varepsilon}(\omega(\tau))$ is a continuous function of $\tau$ for $0 \le \tau < \infty$ and so (as above in section 1.0.4) we may form the Colombeau-Wiener integral:

$$ \begin{gathered} (\mathbf{J}_{V_{\varepsilon}}^{x,t}[V_{\varepsilon}(\omega(\tau)), \psi_{\varepsilon}(\omega(\tau))])_{\varepsilon} = \\ \left( \int_{\Omega_t} \exp \left[ -i \int_0^t V_{\varepsilon}(\omega(\tau)) d\tau \right] \psi_{\varepsilon}(\omega(t)) \widetilde{D}_x^{\mathbb{R}}[\omega(\tau), v_{\varepsilon}] \right)_{\varepsilon} = \\ \int_{\Omega_t} \exp \left[ -i \left( \int_0^t V_{\varepsilon}(\omega(\tau)) d\tau \right)_{\varepsilon} \right] ((\psi_{\varepsilon}(\omega(t)))_{\varepsilon}) \left( \widetilde{D}_x^{\mathbb{R}}[\omega(\tau), v_{\varepsilon}] \right)_{\varepsilon} \end{gathered} \quad (0.5.7) $$

for any $(\psi_{\varepsilon}(\omega(\tau)))_{\varepsilon} \in G(\mathbb{R}^d)$ and any $t \ge 0$. In fact, the integrand is defined $\forall \varepsilon \in (0,1]$ for almost every $\omega \in \Omega_t$, and it is bounded in absolute value by $|\psi_{\varepsilon}(\omega(t))|_{\varepsilon}$. But this is Colombeau-integrable, for by definition and (0.5.7) one obtain

$$ |(\mathbf{J}_{V_{\varepsilon}}^{x,t}[V_{\varepsilon}(\omega(\tau)), \psi_{\varepsilon}(\omega(\tau))])_{\varepsilon}| \le (\mathbf{J}_{V_{\varepsilon}}^{x,t}[\psi_{\varepsilon}(\omega(\tau))])_{\varepsilon} \le ((c_{\varepsilon})_{\varepsilon}) (\sup_{\omega(\tau)} |\psi_{\varepsilon}(\omega(t))|)_{\varepsilon}. \quad (0.5.8) $$

Since $\forall \varepsilon \in (0,1]$, $V_{\varepsilon}(\omega(\tau))$ is continuous for almost every $\omega(\tau)$,
$$ \int_0^t V_\varepsilon(\omega(\tau)) d\tau = \lim_{n \to \infty} \left( \frac{t}{n} \sum_{j=1}^n V_\varepsilon\left(\omega\left(j \frac{t}{n}\right)\right) \right) \quad (0.5.9) $$

for almost every $\omega$. By the Lebesgue dominated convergence theorem, this implies that

$$ (\mathbf{J}_{\varepsilon}^{x,t}[V(\omega(\tau)), \psi_{\varepsilon}(\omega(\tau))])_{\varepsilon} = \left( \lim_{n \to \infty} \int_{\Omega} \exp \left[ -i \sum_{j=1}^{n} \frac{t}{n} V_{\varepsilon}\left(\omega\left(j \frac{t}{n}\right)\right) \right] \psi_{\varepsilon}(\omega(t)) \tilde{D}_{x}^{\mathbb{R}}[\omega(\tau), \varepsilon] \right)_{\varepsilon} \quad (0.5.10) $$

for all $x \in \mathbb{R}^d \setminus \bigcup_{\varepsilon \in (0,1]} F_{V_\varepsilon}$, where $F_{V_\varepsilon}$ is a closed set in $\mathbb{R}^d$ which depend only on $V_\varepsilon$. By definition of the Colombeau-Wiener integral (0.3.25), this means that for all $x \notin \bigcup_{\varepsilon \in (0,1]} F_{V_\varepsilon}$

$$ \left( \left( U_{V_\varepsilon, V_\varepsilon}^t \right)_\varepsilon \right) (\psi_\varepsilon(x))_\varepsilon = \left( \lim_{n \to \infty} \left( \Xi_{V_\varepsilon}^{\frac{t}{n}} M_{V_\varepsilon}^{\frac{t}{n}} \right)^n \psi_\varepsilon(x) \right)_\varepsilon, \quad (0.5.11) $$

where

$$ \left( U_{V_\varepsilon, V_\varepsilon}^t \right)_\varepsilon = \exp[t(v_\varepsilon \Delta - iV_\varepsilon)], \quad (0.5.12) $$

$$ \Xi_{V_\varepsilon}^t = \exp(tv_\varepsilon \Delta). $$

Explicitly,
$$ \int_0^t V_{\varepsilon}(\omega(\tau)) d\tau = \lim_{n \to \infty} \left( \frac{t}{n} \sum_{j=1}^n V_{\varepsilon}\left(\omega\left(j \frac{t}{n}\right)\right) \right) \quad (0.5.9) $$

for almost every $\omega$. By the Lebesgue dominated convergence theorem, this implies that

$$ (\mathbf{J}_{\varepsilon}^{x,t}[V(\omega(\tau)), \psi_{\varepsilon}(\omega(\tau))])_{\varepsilon} = \left( \lim_{n \to \infty} \int_{\Omega} \exp \left[ -i \sum_{j=1}^{n} \frac{t}{n} V_{\varepsilon} \left( \omega \left( j \frac{t}{n} \right) \right) \right] \psi_{\varepsilon}(\omega(t)) \tilde{D}_{x}^{\mathbb{R}}[\omega(\tau), \varepsilon] \right)_{\varepsilon} \quad (0.5.10) $$

for all $x \in \mathbb{R}^d \setminus \bigcup_{\varepsilon \in (0,1]} F_{V_{\varepsilon}}$, where $F_{V_{\varepsilon}}$ is a closed set in $\mathbb{R}^d$ which depend only on $V_{\varepsilon}$. By definition of the Colombeau-Wiener integral (0.3.25), this means that for all $x \notin \bigcup_{\varepsilon \in (0,1]} F_{V_{\varepsilon}}$

$$ \left( \left( U_{V_{\varepsilon}, V_{\varepsilon}}^{t} \right)_{\varepsilon} \right) (\psi_{\varepsilon}(x))_{\varepsilon} = \left( \lim_{n \to \infty} \left( \Xi_{V_{\varepsilon}}^{\frac{t}{n}} M_{V_{\varepsilon}}^{\frac{t}{n}} \right)^{n} \psi_{\varepsilon}(x) \right)_{\varepsilon}, \quad (0.5.11) $$

where

$$ \left( U_{V_{\varepsilon}, V_{\varepsilon}}^{t} \right)_{\varepsilon} = \exp[t(V_{\varepsilon}\Delta - iV_{\varepsilon})], \quad (0.5.12) $$

$$ \Xi_{V_{\varepsilon}}^{t} = \exp(tV_{\varepsilon}\Delta). $$

Explicitly,
$$ (\Xi_{v_\varepsilon}^t \varphi_\varepsilon(x))_\varepsilon = (2\pi t((v_\varepsilon)_\varepsilon))^{-d/2} \int_{\mathbb{R}^d} \exp\left(-\frac{|x-y|^2}{2t((v_\varepsilon)_\varepsilon)}\right) \varphi(y)dy, \quad (0.5.13) $$

A set $F_{v_\varepsilon}$ of capacity 0 has Lebesgue measure 0, and so if $\psi_{n,\varepsilon} = (\Xi_{v_\varepsilon}^t M_{v_\varepsilon}^t)^n \psi_\varepsilon$ then $\forall \varepsilon \in (0,1], \psi_{n,\varepsilon}$ converges almost everywhere in $\mathbb{R}^d$ to $U_{v_\varepsilon, v_\varepsilon}^t \psi_\varepsilon$. By (0.5.13)

$$ (|\Xi_{v_\varepsilon}^t \varphi_\varepsilon(x)|)_\varepsilon \le (\Xi_{v_\varepsilon}^t |\varphi_\varepsilon(x)|)_\varepsilon. \quad (0.5.14) $$

By definition $M_{v_\varepsilon}^t$ is multiplication by a function of modulus 1. Consequently, $\forall \varepsilon \in (0,1] : |\psi_{n,\varepsilon}| \le \Xi_{v_\varepsilon}^t |\psi_{n,\varepsilon}|$, and so $|\psi_{n,\varepsilon} - \psi_{m,\varepsilon}|^2 \le 4(\Xi_{v_\varepsilon}^t \psi_\varepsilon)^2$, which is in $\mathcal{L}_2(\mathbb{R}^n)$. Therefore, by the Lebesgue dominated convergence theorem, $\forall \varepsilon \in (0,1]$, $\psi_{n,\varepsilon}$ is a Cauchy sequence in $\mathcal{L}_2(\mathbb{R}^n)$, and so

$$ \left( \left( U_{v_\varepsilon, v_\varepsilon}^t \right)_\varepsilon \right) (\psi_\varepsilon)_\varepsilon = \left( \mathcal{L}_2 - \lim_{n \to \infty} \left( \Xi_{v_\varepsilon}^t M_{v_\varepsilon}^t \right)^n \psi_\varepsilon \right)_\varepsilon, \quad (0.5.15) $$

where the limit now being taken in $\mathcal{L}_2(\mathbb{R}^n)$.

Let us consider now the regularised Colombeau-Schrödinger equation

$$ \frac{\partial \psi_{\varepsilon}(t,x)}{\partial t} = \frac{1}{2}(i + v_{\varepsilon})\Delta \psi_{\varepsilon}(t,x) - iV_{\varepsilon}(x)\psi_{\varepsilon}(t,x), \quad (0.5.16) $$

$$ ((\psi_{\varepsilon}(0,x)))_{\varepsilon} = (\psi_{0,\varepsilon}(x))_{\varepsilon}, $$
$$ (\Xi_{v_\epsilon}^t \varphi_\epsilon(x))_\epsilon = (2\pi t((v_\epsilon)_\epsilon))^{-d/2} \int_{\mathbb{R}^d} \exp\left(-\frac{|x-y|^2}{2t((v_\epsilon)_\epsilon)}\right) \varphi(y)dy, \quad (0.5.13) $$

A set $F_{v_\epsilon}$ of capacity 0 has Lebesgue measure 0, and so if $\psi_{n,\epsilon} = (\Xi_{v_\epsilon}^t M_{v_\epsilon}^t)^n \psi_\epsilon$ then $\forall \epsilon \in (0,1], \psi_{n,\epsilon}$ converges almost everywhere in $\mathbb{R}^d$ to $U_{v_\epsilon, v_\epsilon}^t \psi_\epsilon$. By (0.5.13)

$$ (|\Xi_{v_\epsilon}^t \varphi_\epsilon(x)|)_\epsilon \le (\Xi_{v_\epsilon}^t |\varphi_\epsilon(x)|)_\epsilon. \quad (0.5.14) $$

By definition $M_{v_\epsilon}^t$ is multiplication by a function of modulus 1. Consequently, $\forall \epsilon \in (0,1] : |\psi_{n,\epsilon}| \le \Xi_{v_\epsilon}^t |\psi_{n,\epsilon}|$, and so $|\psi_{n,\epsilon} - \psi_{m,\epsilon}|^2 \le 4(\Xi_{v_\epsilon}^t \psi_\epsilon)^2$, which is in $\mathcal{L}_2(\mathbb{R}^n)$. Therefore, by the Lebesgue dominated convergence theorem, $\forall \epsilon \in (0,1]$, $\psi_{n,\epsilon}$ is a Cauchy sequence in $\mathcal{L}_2(\mathbb{R}^n)$, and so

$$ \left( \left( U_{v_\epsilon, v_\epsilon}^t \right)_\epsilon \right) (\psi_\epsilon)_\epsilon = \left( \mathcal{L}_2 - \lim_{n \to \infty} \left( \Xi_{v_\epsilon}^t M_{v_\epsilon}^t \right)^n \psi_\epsilon \right)_\epsilon, \quad (0.5.15) $$

where the limit now being taken in $\mathcal{L}_2(\mathbb{R}^n)$.

Let us consider now the regularised Colombeau-Schrödinger equation

$$ \frac{\partial \psi_{\epsilon}(t, x)}{\partial t} = \frac{1}{2}(i + v_{\epsilon})\Delta \psi_{\epsilon}(t, x) - iV_{\epsilon}(x)\psi_{\epsilon}(t, x), \quad (0.5.16) $$

$$ ((\psi_{\epsilon}(0, x)))_{\epsilon} = (\psi_{0, \epsilon}(x))_{\epsilon}, $$
where $v_ε \to 0$ if $ε \to 0$. If the operator $H_ε = -Δ/2 + V_ε(x)$ is:

(i) $\forall ε \in (0, 1]$, self-adjoint and

(ii) $\forall ε \in (0, 1]$, bounded from below, by the spectral theorem one obtain

$$ \exp(-itH) = \lim_{\epsilon \to 0} \exp[-(i + v_\epsilon)tH] \qquad (0.5.17) $$

strongly for all positive $t$. Therefore

$$ (U_{v_\epsilon, V_\epsilon}^t)_\epsilon = \exp\{t[(i + (v_\epsilon)_\epsilon)\Delta - iV_\epsilon]\}, \qquad (0.5.18) $$

$$ \Xi_{v_\epsilon}^t = \exp(t(i + (v_\epsilon)_\epsilon)\Delta). $$

Explicitly,
where $v_ε \to 0$ if $ε \to 0$. If the operator $H_ε = -Δ/2 + V_ε(x)$ is:

(i) $\forall ε \in (0, 1]$, self-adjoint and

(ii) $\forall ε \in (0, 1]$, bounded from below, by the spectral theorem one obtain

$$ \exp(-itH) = \lim_{\epsilon \to 0} \exp[-(i + v_\epsilon)tH] \qquad (0.5.17) $$

strongly for all positive $t$. Therefore

$$ (U_{v_\epsilon, V_\epsilon}^t)_\epsilon = \exp\{t[(i + (v_\epsilon)_\epsilon)\Delta - iV_\epsilon]\}, \qquad (0.5.18) $$

$$ \Xi_{v_\epsilon}^t = \exp(t(i + (v_\epsilon)_\epsilon)\Delta). $$

Explicitly,
$$((\Xi_{v_\varepsilon}^t \varphi_\varepsilon(x))_\varepsilon) = (2\pi t(i + (v_\varepsilon)_\varepsilon))^{-d/2} \int_{\mathbb{R}^d} \exp\left(-\frac{|x-y|^2}{2t(i + (v_\varepsilon)_\varepsilon)}\right) \varphi(y)dy =$$

$$(2\pi t(i + (v_\varepsilon)_\varepsilon))^{-d/2} \int_{\mathbb{R}^d} \exp\left(-\frac{|x-y|^2((v_\varepsilon)_\varepsilon - i)}{2t((v_\varepsilon)_\varepsilon + i)((v_\varepsilon)_\varepsilon - i)}\right) \varphi(y)dy =$$

$$(2\pi t(i + (v_\varepsilon)_\varepsilon))^{-d/2} \int_{\mathbb{R}^d} \exp\left(-\frac{|x-y|^2((v_\varepsilon)_\varepsilon - i)}{2t((v_\varepsilon)^2_\varepsilon + 1)}\right) \varphi(y)dy =$$

$$(2\pi t(i + (v_\varepsilon)_\varepsilon))^{-d/2} \times$$

$$\int_{\mathbb{R}^d} \exp\left(-\frac{((v_\varepsilon)_\varepsilon)|x-y|^2}{2t((v_\varepsilon)^2_\varepsilon + 1)}\right) \exp\left(\frac{i|x-y|^2}{2t((v_\varepsilon)^2_\varepsilon + 1)}\right) \varphi(y)dy =$$

$$(2\pi t(i + (v_\varepsilon)_\varepsilon))^{-d/2} \times \left[ \frac{((v_\varepsilon)_\varepsilon}{(v_\varepsilon)^2_\varepsilon + 1}} \right]^{-d/2} \times \left[ \frac{((v_\varepsilon)_\varepsilon}{(v_\varepsilon)^2_\varepsilon + 1}} \right]^{d/2} \times \quad (0.5.19)$$

$$\int_{\mathbb{R}^d} \exp\left(-\frac{((v_\varepsilon)_\varepsilon)|x-y|^2}{2t((v_\varepsilon)^2_\varepsilon + 1)}\right) \exp\left(\frac{i|x-y|^2}{2t((v_\varepsilon)^2_\varepsilon + 1)}\right) \varphi(y)dy =$$

$$(i + (v_\varepsilon)_\varepsilon)^{-d/2} \times \left[ \frac{((v_\varepsilon)_\varepsilon}{((v_\varepsilon)^2_\varepsilon + 1)} \right]^{-d/2} \times$$

$$\int_{\mathbb{R}^d} \exp\left(\frac{i|x-y|^2}{((v_\varepsilon)^2_\varepsilon + 1)}\right) \varphi(y)p(x, dy) =$$

$$\left[ \frac{((v_\varepsilon)_\varepsilon}{(i - (v_\varepsilon)^2_\varepsilon)} \right]^{-d/2} \int_{\mathbb{R}^d} \exp\left(\frac{i|x-y|^2}{((v_\varepsilon)^2_\varepsilon + 1)}\right) \varphi(y)p_{v_\varepsilon}(x, dy),$$

where
$$((\Xi_{v_{\varepsilon}}^t \varphi_{\varepsilon}(x))_{\varepsilon}) = (2\pi t(i + (v_{\varepsilon})_{\varepsilon}))^{-d/2} \int_{\mathbb{R}^d} \exp\left(-\frac{|x-y|^2}{2t(i + (v_{\varepsilon})_{\varepsilon})}\right) \varphi(y)dy =$$

$$(2\pi t(i + (v_{\varepsilon})_{\varepsilon}))^{-d/2} \int_{\mathbb{R}^d} \exp\left(-\frac{|x-y|^2((v_{\varepsilon})_{\varepsilon} - i)}{2t((v_{\varepsilon})_{\varepsilon} + i)((v_{\varepsilon})_{\varepsilon} - i)}\right) \varphi(y)dy =$$

$$(2\pi t(i + (v_{\varepsilon})_{\varepsilon}))^{-d/2} \int_{\mathbb{R}^d} \exp\left(-\frac{|x-y|^2((v_{\varepsilon})_{\varepsilon} - i)}{2t((v_{\varepsilon})_{\varepsilon} + 1)}\right) \varphi(y)dy =$$

$$(2\pi t(i + (v_{\varepsilon})_{\varepsilon}))^{-d/2} \times$$

$$\int_{\mathbb{R}^d} \exp\left(-\frac{((v_{\varepsilon})_{\varepsilon})|x-y|^2}{2t((v_{\varepsilon})_{\varepsilon} + 1)}\right) \exp\left(\frac{i|x-y|^2}{2t((v_{\varepsilon})_{\varepsilon} + 1)}\right) \varphi(y)dy =$$

$$(2\pi t(i + (v_{\varepsilon})_{\varepsilon}))^{-d/2} \times \left[ \frac{((v_{\varepsilon})_{\varepsilon})}{(v_{\varepsilon}^2)_{\varepsilon} + 1} \right]^{-d/2} \times \left[ \frac{((v_{\varepsilon})_{\varepsilon})}{(v_{\varepsilon}^2)_{\varepsilon} + 1} \right]^{d/2} \times \quad (0.5.19)$$

$$\int_{\mathbb{R}^d} \exp\left(-\frac{((v_{\varepsilon})_{\varepsilon})|x-y|^2}{2t((v_{\varepsilon})_{\varepsilon} + 1)}\right) \exp\left(\frac{i|x-y|^2}{2t((v_{\varepsilon})_{\varepsilon} + 1)}\right) \varphi(y)dy =$$

$$(i + (v_{\varepsilon})_{\varepsilon})^{-d/2} \times \left[ \frac{((v_{\varepsilon})_{\varepsilon})}{((v_{\varepsilon}^2)_{\varepsilon} + 1)} \right]^{-d/2} \times$$

$$\int_{\mathbb{R}^d} \exp\left(\frac{i|x-y|^2}{((v_{\varepsilon})_{\varepsilon} + 1)}\right) \varphi(y)p(x, dy) =$$

$$\left[ \frac{((v_{\varepsilon})_{\varepsilon})}{(i - (v_{\varepsilon}^2)_{\varepsilon})} \right]^{-d/2} \int_{\mathbb{R}^d} \exp\left(\frac{i|x-y|^2}{((v_{\varepsilon}^2)_{\varepsilon} + 1)}\right) \varphi(y)p_{v_{\varepsilon}}(x, dy),$$

where
$$p_{v_{\varepsilon}}(x, dy) = \left[ \frac{((v_{\varepsilon})_{\varepsilon})}{2\pi t((v_{\varepsilon}^2)_{\varepsilon} + 1)} \right]^{d/2} \exp\left( -\frac{((v_{\varepsilon})_{\varepsilon})|x-y|^2}{2t((v_{\varepsilon}^2)_{\varepsilon} + 1)} \right). \quad (0.5.20)$$

**Definition 0.5.1.** We define Colombeau-Feynman Path Integral $(J_{v_{\varepsilon},v_{\varepsilon}}^{x,t}[\cdot])_{\varepsilon,(0,1]}$ based on real-valued Colombeau-Wiener measure $(\tilde{D}_{x}^{\mathbb{R}}[\omega(\tau),v_{\varepsilon}])_{\varepsilon}$ by formula

$$ (J_{v_{\varepsilon},v_{\varepsilon}}^{x,t}[\mathcal{L}_{\varepsilon}(\dot{\omega}(\tau),\omega(\tau)),\psi_{\varepsilon}(\omega(t))])_{\varepsilon,(0,1]} = \\ \left( \lim_{n \to \infty} \left[ \frac{((v_{\varepsilon})_{\varepsilon})}{(i-(v_{\varepsilon}^2)_{\varepsilon})} \right]^{-nd/2} \right) \times \\ \int_{\Omega_t} \exp \left\{ i \frac{t}{n} \sum_{j=1}^{n} \left[ \frac{m}{2} \left(\frac{t}{n}\right)^{-2} (\omega_{j,n}(t) - \omega_{j-1,n}(t))^2 \right] - i \int_{0}^{t} V_{\varepsilon}(\omega(\tau)) d\tau \right\} \times \\ \psi_{0,\varepsilon}(\omega(t)) \tilde{D}_{x}^{\mathbb{R}}[\omega(\tau),v_{\varepsilon}] \Big|_{\varepsilon} \quad (0.5.21) $$

or by equivalent complete Colombeau formula:
$$p_{v_{\varepsilon}}(x, dy) = \left[ \frac{((v_{\varepsilon})_{\varepsilon})}{2\pi t((v_{\varepsilon}^2)_{\varepsilon} + 1)} \right]^{d/2} \exp\left( - \frac{((v_{\varepsilon})_{\varepsilon})|x-y|^2}{2t((v_{\varepsilon}^2)_{\varepsilon} + 1)} \right). \quad (0.5.20)$$

**Definition 0.5.1.** We define Colombeau-Feynman Path Integral $(J_{v_{\varepsilon},v_{\varepsilon}}^{x,t}[\cdot])_{\varepsilon,(0,1]}$ based on real-valued Colombeau-Wiener measure $(\tilde{D}_{x}^{\mathbb{R}}[\omega(\tau),v_{\varepsilon}])_{\varepsilon}$ by formula

$$ (J_{v_{\varepsilon},v_{\varepsilon}}^{x,t}[\mathcal{L}_{\varepsilon}(\dot{\omega}(\tau),\omega(\tau)),\psi_{\varepsilon}(\omega(t))])_{\varepsilon,(0,1]} = \\ \left( \lim_{n \to \infty} \left[ \frac{((v_{\varepsilon})_{\varepsilon})}{(i-(v_{\varepsilon}^2)_{\varepsilon})} \right]^{-nd/2} \right) \times \\ \int_{\Omega_t} \exp \left\{ i \frac{t}{n} \sum_{j=1}^{n} \left[ \frac{m}{2} \left(\frac{t}{n}\right)^{-2} (\omega_{j,n}(t) - \omega_{j-1,n}(t))^2 \right] - i \int_{0}^{t} V_{\varepsilon}(\omega(\tau)) d\tau \right\} \times \\ \psi_{0,\varepsilon}(\omega(t)) \tilde{D}_{x}^{\mathbb{R}}[\omega(\tau),v_{\varepsilon}] \Big|_{\varepsilon} \quad (0.5.21) $$

or by equivalent complete Colombeau formula:
$$ (\mathbf{J}_{v_{\varepsilon}, n_{\delta}, v_{\varepsilon}}^{x,t} [\mathcal{L}_{\varepsilon}(\dot{\omega}(\tau), \omega(\tau)), \psi_{0}(\omega(t))])_{\varepsilon \in (0,1], \delta \in (0,1]} = $$

$$ \left( \left[ \frac{((v_{\varepsilon}^{n_{\delta}})_{\varepsilon})}{((i - (v_{\varepsilon}^{2})^{n_{\delta}})_{\varepsilon}} \right]^{-d/2} \times $$

$$ \int_{\Omega_t} \exp \left\{ i \frac{t}{n_{\varepsilon}} \sum_{j=1}^{n_{\delta}} \left[ \frac{m}{2} \left( \frac{t}{n_{\delta}} \right)^{-2} (\omega_{j,n_{\delta}}(t) - \omega_{j-1,n_{\delta}}(t))^2 \right] - i \int_0^t V_{\varepsilon}(\omega(\tau)) d\tau \right\} \times $$

$$ \psi_{\varepsilon}(\omega(t)) \tilde{D}_{x}^{\mathbb{R}}[\omega(\tau), v_{\varepsilon}] \Big|_{\varepsilon, \delta}, $$

where $n_{\delta} \to \infty$ if $\delta \to 0$.

We wish to expand now Colombeau-Wiener integral (0.6.22) on functionals given by Eq.(0.3.10) using smoothing approach based on the mollification of the

continuous trajectories $\omega(\tau) \in C[0,t]$. We use a net of mollifiers $\phi_{n,\delta}(\tau) \in S(\mathbb{R}_+), \delta \in (0,1]$ of the form

$$ \phi_{n,\delta}(\tau) = \frac{1}{\delta} \phi_n\left(\frac{\tau}{\delta}\right), \qquad (0.5.23) $$

where $\phi_n$ has the properties $\forall n \in \mathbb{N}$:

(i) $\int_0^t \phi_n(\tau) d\tau = 1,$

(ii) $\phi_n(\tau) \ge 0$ for all $\tau \in [0, t]$,

(iii) $\phi_n \in C^\infty$.

**Definition 0.5.2.**[46].For $\delta > 0$ the $\delta$-mollification $\omega_{n,\delta}(\tau)$ of a locally integrable function $\omega : [0,t] \to \mathbb{R}^d, \omega(\tau) = (\omega_1(\tau), \dots, \omega_d(\tau))$ is the convolution on $\mathbb{R}$ such that

$$ \omega_{n,\delta}(\tau) = \phi_{n,\delta} * \omega(\tau), \qquad (0.5.24) $$

that is,
$$ (\mathbf{J}_{v_{\varepsilon}, n_{\delta}, v_{\varepsilon}}^{x,t} [\mathcal{L}_{\varepsilon}(\dot{\omega}(\tau), \omega(\tau)), \psi_{0}(\omega(t))])_{\varepsilon \in (0,1], \delta \in (0,1]} = $$

$$ \left( \left[ \frac{((v_{\varepsilon}^{n_{\delta}})_{\varepsilon})}{((i - (v_{\varepsilon}^{2})^{n_{\delta}})_{\varepsilon}} \right]^{-d/2} \times $$

$$ \int_{\Omega_t} \exp \left\{ i \frac{t}{n_{\varepsilon}} \sum_{j=1}^{n_{\delta}} \left[ \frac{m}{2} \left( \frac{t}{n_{\delta}} \right)^{-2} (\omega_{j,n_{\delta}}(t) - \omega_{j-1,n_{\delta}}(t))^2 \right] - i \int_0^t V_{\varepsilon}(\omega(\tau)) d\tau \right\} \times $$

$$ \psi_{\varepsilon}(\omega(t)) \tilde{D}_{x}^{\mathbb{R}}[\omega(\tau), v_{\varepsilon}] \Big|_{\varepsilon, \delta}, $$

where $n_{\delta} \to \infty$ if $\delta \to 0$.

We wish to expand now Colombeau-Wiener integral (0.6.22) on functionals given by Eq.(0.3.10) using smoothing approach based on the mollification of the

continuous trajectories $\omega(\tau) \in C[0,t]$. We use a net of mollifiers $\phi_{n,\delta}(\tau) \in S(\mathbb{R}_+), \delta \in (0,1]$ of the form

$$ \phi_{n,\delta}(\tau) = \frac{1}{\delta} \phi_n\left(\frac{\tau}{\delta}\right), \qquad (0.5.23) $$

where $\phi_n$ has the properties $\forall n \in \mathbb{N}$:

(i) $\int_0^t \phi_n(\tau) d\tau = 1,$

(ii) $\phi_n(\tau) \ge 0$ for all $\tau \in [0, t]$,

(iii) $\phi_n \in C^{\infty}$.

**Definition 0.5.2.**[46].For $\delta > 0$ the $\delta$-mollification $\omega_{n,\delta}(\tau)$ of a locally integrable function $\omega : [0,t] \to \mathbb{R}^d, \omega(\tau) = (\omega_1(\tau), \dots, \omega_d(\tau))$ is the convolution on $\mathbb{R}$ such that

$$ \omega_{n,\delta}(\tau) = \phi_{n,\delta} * \omega(\tau), \qquad (0.5.24) $$

that is,
$$ \omega_{n,\delta}(\tau) = \int_{\mathbb{R}} \phi_{n,\delta}(\tau-z)\omega(z)dz. \qquad (0.5.25) $$

We remind that [46]: (i) for all $\delta > 0$, the $\delta$-mollification $\omega_{n,\delta}(\tau)$ in $C^\infty(\mathbb{R}_+, \mathbb{R}^d)$, (ii) If $\omega$ is continuous on $\mathbb{R}_+$, then $\omega_{n,\delta}$ converges to $\omega$ uniformly on any compact sets for $\delta \to 0$.

**Definition 0.5.3.** We define smoothed classical action by formula

$$ S_{t,n,\epsilon,\delta}(\dot{\omega}_{\delta}(\tau), \omega(\tau)) = \int_{0}^{t} \mathcal{L}_{n,\epsilon,\delta}(\dot{\omega}_{n,\delta}(\tau), \omega(\tau))d\tau = \int_{0}^{t} \left[ \frac{m}{2} \dot{\omega}_{n,\delta}^{2}(\tau) - V_{\epsilon}(\omega(\tau)) \right] d\tau, \qquad (0.5.26) $$

where $\dot{\omega}_{n,\delta}(\tau) = \frac{d\omega_{n,\delta}(\tau)}{d\tau} = \omega(\tau) * \frac{d\phi_{n,\delta}(\tau)}{d\tau}$, and so we may form the complex

Colombeau-Wiener path integral $(J_{V_\epsilon, n_\epsilon V_\epsilon}^{x,t}[\cdot])_{\epsilon \in (0,1], \delta \in (0,1]}$:
$$ \omega_{n,\delta}(\tau) = \int_{\mathbb{R}} \phi_{n,\delta}(\tau-z)\omega(z)dz. \qquad (0.5.25) $$

We remind that [46]: (i) for all $\delta > 0$, the $\delta$-mollification $\omega_{n,\delta}(\tau)$ in $C^\infty(\mathbb{R}_+, \mathbb{R}^d)$, (ii) If $\omega$ is continuous on $\mathbb{R}_+$, then $\omega_{n,\delta}$ converges to $\omega$ uniformly on any compact sets for $\delta \to 0$.

**Definition 0.5.3.** We define smoothed classical action by formula

$$ S_{t,n,\epsilon,\delta}(\dot{\omega}_{\delta}(\tau), \omega(\tau)) = \int_{0}^{t} \mathcal{L}_{n,\epsilon,\delta}(\dot{\omega}_{n,\delta}(\tau), \omega(\tau))d\tau = \\ \int_{0}^{t} \left[ \frac{m}{2} \dot{\omega}_{n,\delta}^{2}(\tau) - V_{\epsilon}(\omega(\tau)) \right] d\tau, \qquad (0.5.26) $$

where $\dot{\omega}_{n,\delta}(\tau) = \frac{d\omega_{n,\delta}(\tau)}{d\tau} = \omega(\tau) * \frac{d\phi_{n,\delta}(\tau)}{d\tau}$, and so we may form the complex

Colombeau-Wiener path integral $(J_{V_\epsilon, n_\epsilon V_\epsilon}^{x,t}[\cdot])_{\epsilon \in (0,1], \delta \in (0,1]}$:
$$
\begin{align}
& (\mathbf{J}_{V_{\varepsilon}, n_{\varepsilon} V_{\varepsilon}}^{x,t} [\mathcal{L}_{\varepsilon, \delta}(\dot{\omega}(\tau), \omega(\tau)), \psi_0(\omega(t))])_{\varepsilon \in (0,1], \delta \in (0,1]} = \nonumber \\
& \qquad \left( \left[ \frac{V_{\varepsilon}^{n_{\delta}}}{(i - (V_{\varepsilon}^2))^{n_{\delta}}} \right]^{-d/2} \times \right. \nonumber \\
& \qquad \int_{\Omega_t} \exp \left[ i \int_0^t \left[ \frac{m}{2} \dot{\omega}_{n_{\delta}, \delta}^2(\tau) - V(\omega(\tau)) \right] d\tau \right] \psi_{\varepsilon}(\omega(t)) \tilde{D}_{x}^{\mathbb{R}}[\omega(\tau), v_{\varepsilon}] \Bigg|_{\varepsilon, \delta} = \tag{0.5.27} \\
& \qquad \left. \left[ \frac{(V_{\varepsilon}^{n_{\delta}})_{\varepsilon, \delta}}{((i - (V_{\varepsilon}^2))^{n_{\delta}})_{\varepsilon, \delta}} \right]^{-d/2} \times \int_{\Omega_t} \exp \left[ i \left( \int_0^t \left[ \frac{m}{2} \dot{\omega}_{n_{\delta}, \varepsilon}^2(\tau) - V_{\varepsilon}(\omega(\tau)) \right] d\tau \right)_{\varepsilon, \delta} \right] \times \right. \nonumber \\
& \qquad \left. ((\psi_{\varepsilon}(\omega(t)))_{\varepsilon}) (\tilde{D}_{x}^{\mathbb{R}}[\omega(\tau), v_{\varepsilon}])_{\varepsilon} \right] \nonumber
\end{align}
$$

**Definition 0.5.4.** We define smoothed Colombeau-Feynman Path Integral based on real-valued Colombeau-Wiener measure $(\tilde{D}_x^{\mathbb{R}}[\omega(\tau), v_\epsilon])_\epsilon$ by formula

$$
\begin{align}
& \int_{\Omega_t} \exp \left[ i \int_0^t \left[ \frac{m}{2} \dot{\omega}^2(\tau) - V_{\varepsilon}(\omega(\tau)) \right] d\tau \right] ((\psi_{\varepsilon}(\omega(t)))_{\varepsilon}) (\tilde{D}_x^{\mathbb{R}}[\omega(\tau), v_{\varepsilon}])_{\varepsilon} \triangleq \tag{0.5.28} \\
& (\mathbf{J}_{V_{\varepsilon}, n_{\varepsilon}, V_{\varepsilon}}^{x,t} [\mathcal{L}_{\varepsilon}(\dot{\omega}_{n_{\varepsilon}, \varepsilon}(\tau), \omega(\tau)), \psi_0(\omega(t))])_{\varepsilon} \nonumber
\end{align}
$$

and in more general case by formula
$$ (\mathbf{J}_{v_{\varepsilon}, n_{\varepsilon} v_{\varepsilon}}^{x,t} [\mathcal{L}_{\varepsilon, \delta}(\dot{\omega}(\tau), \omega(\tau)), \psi_0(\omega(t))])_{\varepsilon \in (0,1], \delta \in (0,1]} = $$

$$ \left( \left[ \frac{v_{\varepsilon}^{n_{\delta}}}{(i - (v_{\varepsilon}^2))^{n_{\delta}}} \right]^{-d/2} \times \right. $$

$$ \int_{\Omega_t} \exp \left[ i \int_0^t \left[ \frac{m}{2} \dot{\omega}_{n_{\delta}, \delta}^2(\tau) - V(\omega(\tau)) \right] d\tau \right] \psi_{\varepsilon}(\omega(t)) \tilde{D}_x^{\mathbb{R}}[\omega(\tau), v_{\varepsilon}] \Bigg|_{\varepsilon, \delta} = \qquad (0.5.27) $$

$$ \left[ \frac{(v_{\varepsilon}^{n_{\delta}})_{\varepsilon, \delta}}{((i - (v_{\varepsilon}^2))^{n_{\delta}})_{\varepsilon, \delta}} \right]^{-d/2} \times \int_{\Omega_t} \exp \left[ i \left( \int_0^t \left[ \frac{m}{2} \dot{\omega}_{n_{\delta}, \varepsilon}^2(\tau) - V_{\varepsilon}(\omega(\tau)) \right] d\tau \right)_{\varepsilon, \delta} \right] \times \\ ((\psi_{\varepsilon}(\omega(t)))_{\varepsilon}) (\tilde{D}_x^{\mathbb{R}}[\omega(\tau), v_{\varepsilon}])_{\varepsilon} $$

**Definition 0.5.4.** We define smoothed Colombeau-Feynman Path Integral based on real-valued Colombeau-Wiener measure $(\tilde{D}_x^{\mathbb{R}}[\omega(\tau), v_{\varepsilon}])_{\varepsilon}$ by formula

$$ \int_{\Omega_t} \exp \left[ i \int_0^t \left[ \frac{m}{2} \dot{\omega}^2(\tau) - V_{\varepsilon}(\omega(\tau)) \right] d\tau \right] ((\psi_{\varepsilon}(\omega(t)))_{\varepsilon}) (\tilde{D}_x^{\mathbb{R}}[\omega(\tau), v_{\varepsilon}])_{\varepsilon} \triangleq \qquad (0.5.28) $$

$$ (\mathbf{J}_{v_{\varepsilon}, n_{\varepsilon}, v_{\varepsilon}}^{x,t} [\mathcal{L}_{\varepsilon}(\dot{\omega}_{n_{\varepsilon}, \varepsilon}(\tau), \omega(\tau)), \psi_0(\omega(t))])_{\varepsilon} $$

and in more general case by formula
$$ \int_{\Omega_t} \exp\left[i \int_0^t \left[\frac{m}{2} \dot{\omega}^2(\tau) - V_{\varepsilon}(\omega(\tau))\right] d\tau\right] \times \\ \left( (\psi_{\varepsilon}(\omega(t)))_{\varepsilon \in (0,1]} \right) \left( \tilde{D}_x^{\mathbb{R}}[\omega(\tau), v_{\varepsilon}] \right)_{\varepsilon \in (0,1], \delta \in (0,1]} \triangleq \\ (\mathbf{J}_{v_{\varepsilon}, n_{\delta}, V_{\varepsilon}}^{x,t}[\mathcal{L}_{\varepsilon, \delta}(\dot{\omega}_{n_{\delta}, \delta}(\tau), \omega(\tau)), \psi_{\varepsilon}(\omega(t))])_{\varepsilon \in (0,1], \delta \in (0,1]} $$

Since $\forall \varepsilon, \delta \in (0,1]$

$$ \mathcal{L}_{\varepsilon, \delta}(\dot{\omega}_{n_{\delta}, \delta}(\tau), \omega(\tau)) = \frac{m}{2} \dot{\omega}_{n_{\delta}, \delta}^2(\tau) - V_{\varepsilon}(\omega(\tau)) \quad (0.5.30) $$

is continuous for almost every $\omega(\tau)$, one obtain

$$ \int_0^t \mathcal{L}_{\varepsilon}(\dot{\omega}_{n_{\delta}, \delta}(\tau), \omega(\tau)) d\tau = \\ \lim_{n_{\delta} \to \infty, n_{\varepsilon} \to \infty} \left( \frac{t}{n_{\delta}} \sum_{j=1}^{n_{\delta}} \left[ \frac{m}{2} \dot{\omega}_{n_{\delta}, \delta}^2 \left(j \frac{t}{n_{\delta}}\right) - V_{\varepsilon}\left(\omega\left(j \frac{t}{n_{\delta}}\right)\right) \right] \right) \quad (0.5.31) $$

for almost every $\omega \in \Omega_t$. By the Lebesgue dominated convergence theorem, this implies that
$$
\begin{gather}
\int_{\Omega_t} \exp\left[i \int_0^t \left[\frac{m}{2} \dot{\omega}^2(\tau) - V_{\varepsilon}(\omega(\tau))\right] d\tau\right] \times \nonumber \\
((\psi_{\varepsilon}(\omega(t)))_{\varepsilon \in (0,1]}) \left(\tilde{D}_x^{\mathbb{R}}[\omega(\tau), v_{\varepsilon}]\right)_{\varepsilon \in (0,1], \delta \in (0,1]} \triangleq \tag{0.5.29} \\
(\mathbf{J}_{v_{\varepsilon}, n_{\delta}, V_{\varepsilon}}^{x,t}[\mathcal{L}_{\varepsilon, \delta}(\dot{\omega}_{n_{\delta}, \delta}(\tau), \omega(\tau)), \psi_{\varepsilon}(\omega(t))])_{\varepsilon \in (0,1], \delta \in (0,1]} \nonumber
\end{gather}
$$

Since $\forall \varepsilon, \delta \in (0,1]$

$$
\mathcal{L}_{\varepsilon, \delta}(\dot{\omega}_{n_{\delta}, \delta}(\tau), \omega(\tau)) = \frac{m}{2} \dot{\omega}_{n_{\delta}, \delta}^2(\tau) - V_{\varepsilon}(\omega(\tau)) \quad (0.5.30)
$$

is continuous for almost every $\omega(\tau)$, one obtain

$$
\begin{gather}
\int_0^t \mathcal{L}_\varepsilon(\dot{\omega}_{n_\delta, \delta}(\tau), \omega(\tau)) d\tau = \tag{0.5.31} \\
\lim_{n_\delta \to \infty, n_\varepsilon \to \infty} \left( \frac{t}{n_\delta} \sum_{j=1}^{n_\delta} \left[ \frac{m}{2} \dot{\omega}_{n_\delta, \delta}^2 \left(j \frac{t}{n_\delta}\right) - V_\varepsilon \left(\omega \left(j \frac{t}{n_\delta}\right)\right) \right] \right) \nonumber
\end{gather}
$$

for almost every $\omega \in \Omega_t$. By the Lebesgue dominated convergence theorem, this
implies that
$$ (\mathbf{J}_{v_{\varepsilon}, n_{\delta}, v_{\varepsilon}}^{x,t} [\mathcal{L}_{\varepsilon}(\dot{\omega}_{n_{\delta}, \varepsilon}(\tau), \omega(\tau)), \psi_{0}(\omega(t))])_{\varepsilon \in (0,1], \delta \in (0,1]} = \\ \left( \left( \left[ \frac{v_{\varepsilon}^{n_{\delta}}}{((i - (v_{\varepsilon}^2))^{n_{\delta}})} \right]^{-d/2} \right)_{\varepsilon \in (0,1], \delta \in (0,1]} \times \\ \left( \int_{\Omega_t} \exp \left[ i \frac{t}{n_{\delta}} \sum_{j=1}^{n_{\delta}} \left[ \frac{m}{2} \dot{\omega}_{n_{\delta}, \varepsilon}^2 \left(j \frac{t}{n_{\delta}}\right) - V_{\varepsilon} \left(\omega\left(j \frac{t}{n_{\delta}}\right)\right) \right] \right] \times \\ \psi_{\varepsilon}(\omega(t)) \tilde{D}_{x}^{\mathbb{R}}[\omega(\tau), v_{\varepsilon}] \right)_{\varepsilon \in (0,1], \delta \in (0,1]} \quad (0.5.32) $$

for all $x \in \mathbb{R}^d \setminus \cup_{\varepsilon \in (0,1]} F_{v_{\varepsilon}}$, where $F_{v_{\varepsilon}}$ is a closed set in $\mathbb{R}^d$ which depend only on $V_{\varepsilon}$. By definition of the Colombeau-Wiener integral (0.3.25), this means that for all $x \notin \cup_{\varepsilon \in (0,1]} F_{v_{\varepsilon}}$

$$ \left( \left( U_{v_{\varepsilon}, n_{\delta}, v_{\varepsilon}}^{t} \psi_{\varepsilon}(x) \right)_{\varepsilon \in (0,1], \delta \in (0,1]} \right) = \left( \left( \Xi_{v_{\varepsilon}, n_{\delta}, v_{\varepsilon}}^{\frac{t}{n_{\delta}}, \frac{t}{n_{\delta}}} M_{v_{\varepsilon}}^{n_{\delta}} \psi_{\varepsilon}(x) \right)_{\varepsilon \in (0,1], \delta \in (0,1]} \right). \quad (0.5.33) $$

Here

$$ \left( U_{v_{\varepsilon}, n_{\delta}, v_{\varepsilon}}^{t} \right)_{\varepsilon \in (0,1], \delta \in (0,1]} = \left( \exp\{t[(i + v_{\varepsilon})\Delta[\phi_{n_{\delta}, \delta}] - iV_{\varepsilon}]\} \right)_{\varepsilon \in (0,1], \delta \in (0,1]}, \quad (0.5.34) $$ $$ \Xi_{v_{\varepsilon}, n_{\delta}}^{t} = \exp(t(i + v_{\varepsilon})\Delta[\phi_{n_{\delta}, \delta}]), $$

where we denote by $\Delta[\phi_{n_{\delta}, \delta}]$ smoothed Laplace operator, which defined by obvious way.

### I.0.6. Colombeau-Feynman Path Integral based on complex Colombeau-Wiener measure $\tilde{D}^{\mathbb{C}}[x(\tau), v_{\varepsilon}]$.