数理解析研究所講究録
第 1797 巻 2012 年 167-186

167

Phase Space Feynman Path Integrals-Calculation Examples
via Piecewise Bicharacteristic Paths
By

Naoto KUMANO-GO *Abstract
This survey of [19] is base on the introductory talk at RIMS. Because the RIMS K\^oky\^uroku gives us achance for introducing the examples which are not suited for publication in ordinary joumal, we introduce
the calculation examples of the fundamental solutions for some equations, using the phase space path
integral [19] via piecewise bicharacteristic paths.

\S 1. Introduction
Let $T>0$ and $x\in R^{d}$ . Let $U(T,0)$ be the fundamental solution for the Schr\"odinger equation(1.1)$(i \hslash\partial_{T}-H(T,x, \frac{\hslash}{i}\partial_{x}))U(T,0)=0,$$U(O,0)=I$ ,with the Planck parameter $0<\hslash<1$ . By the Fourier transform with respect toinverse Fourier transform with respect to, the identity operator is given by$x_{0}\in R^{d}$$\xi_{0}\in R^{d}$$Iv(x)=v(x)=( \frac{1}{2\pi\hslash})^{d}\int_{R^{2d}}e^{\frac{i}{\hslash}(x-x_{0})\cdot\xi_{0}}v(x_{0})dx_{0}d\xi_{0}$and the Hamilton operator$H(T,x, \frac{\hslash}{i}\partial_{x})$and the$I$,

is given by.

$H(T,x, \frac{\hslash}{i}\partial_{x})v(x)=(\frac{1}{2\pi\hslash})^{d}\int_{R^{2d}}e^{\frac{i}{\hslash}(x-x_{0})\cdot\xi_{0}}H(T,x,\xi_{0})u(x_{0})dx_{0}d\xi_{0}$

When
(1.2)

$T$

is small, we consider the function $U(T,0,x,\xi_{0})$ satisfying$U(T,0)v(x)=( \frac{1}{2\pi\hslash})^{d}\int_{R^{2d}}e^{\frac{i}{h}(x-x_{0})\cdot\xi_{0}}U(T,0,x,\xi_{0})v(xo)dx0d\xi 0$.

2010 Mathematics Subject Classification(s): SIS40; $35S30;81Q20;58D30$
Key Words: Path integrals, Fourier integral operators, Semiclassical approximation
Supported by JSPS KAKENHI(C)21540196*Division of Liberal Arts, Kogakuin University, 1-24-2 Nishi-shinjuku, Shinjuku-ku, Tokyo 163-8677, Japan.

168

What is the position path$q$

What is the momentum path

?

$p$

?

Figure 1.

Using the phase space path integral in R. P Feynman [8, Appendix
(1.3)

$B$

$e^{i} \hslash^{10}x-x)\cdot\xi_{0}U(T,0,x,\xi_{0})=\int e^{i}\hslash^{\phi[q,p]}\mathcal{D}[q,p]$], we formally write.

Here $q;[0, T]arrow R^{d}$ is a position path with $q(O)=x0$ and $q(T)=x,$ $p:[0,T)arrow R^{d}$ is a
momentum path with $p(O)=\xi_{0},$ $\phi[q,p]$ is the action of Hamltonian type along the phase spacepath $(q.p)$ defined by$\phi[q,p]=\int_{[0,T)}p(t)\cdot dq(t)-\int_{[0,T)}H(t,q(t),p(t))dt$,

is a sum over all the paths $(q,p)$ (Figure 1).
and the phase space path integralof the path integral (1.3) does not
However, in the sense of mathematics, the measureexist. Furthennore, in the sense of the uncertain principle, we can not have the position $q(t)$ andthe momentum $p(t)$ at the same time .
In [19], when the time interval $[0,T]$ is small, using piecewise bicharacteristic paths, we
proved the existence of the phase space Feynman path integrals$\int\sim \mathcal{D}[q,p]$$\mathcal{D}[q,p]$

$t$

(1.4)

$\int\hslash^{\phi[q,p]}$with general functional $F[q,p]$ as integrand. More precisely, we gave a fairly general classof functionals $F[q,p]$ such that for any $F[q,p]\in \mathcal{F}$, the time slicing approximation of (1.4). Our apconverges uniformly on compact subsets with respect toproach via piecewise bicharacteristic paths is a little different from the known approaches. In
this article, as a merit of piecewise bicharacteristic paths, we give the calculation examples ofthe fundamental solutions for some equations.
$\mathcal{F}$$(x,\xi_{0},xo)\in R^{d}\cross R^{d}\cross R^{d}$Remark Using Fourier integral operators, H. Kitada-H. Kumano-go [17] proved the convergence of the time slicing approximation of (1.3). We regard (1.3) as a particular case of (1.4)
with $F[q,p]=1$ . In this sense, [19] has its origin in [17].

169
Remark Using broken line paths of position and piecewise constant paths of momentum,
W. Ichinose [14] gave some functionals $F[q,p]= \prod_{k=1}^{K}B_{k}(q(\tau_{k}),p(\tau_{k})),$ $0<\tau_{1}<\tau_{2}<\cdots<$$\tau_{K}<T$ for which the time slicing approximations of (1.4) diverge as an operator. We exclude
these functional from our class .
$\mathcal{F}$Remark. Inspired by the forward and backward approach of K. L. Chung-J.-C. Zambrini[4, \S 2.4], we use left-continuous paths and right-continuous paths. Furthermore, inspired by L.
S. Shulman [24, \S 31], we pay attention to the operations which are valid in the phase space path
integrals.

Since [8, Appendix ], the phase space path integral (1.3) has been rediscovered repeatedly(cf. W. Tobocman [25], H. Davies [6], C. Garrod [10]) and developed in various forms (cf. L. S.
Schulman [24, \S 31], H. Kleinert [21], C. Grosche-F. Steiner [12], P. Cartier-C. DeWitt-Morette[3, \S 3.4], J. R. Klauder [20, \S 6.2] . For giving a well-defined mathematical meaning, various
approaches have been proposed. C. DeWitt-Morette-A. Maheshwari-B. Nelson [7] and M. M.
Mizrahi [23] introduced the formulation without limiting procedure. K. Gawedzki [11] used
the technique analogous to that used by K. It\^o [15]. I. Daubechies-J. R. Klauder [5] presented
the phase space path integral via analytic continuation from Wiener measure. Furthermore, S.
Albeverio-G. Guatteri-S. Mazzucchi [2] (cf. [1, \S 10.5.3], [22, \S 3.3]) realized the phase spacepath integral as an infinite dimensional oscillatory integral. O. G. Smolyanov-A. G. Tokarev-A.
Tmman [26] formulated the phase space path integral via Chemoff formula. For the main partof [8], G. W. Johnson-M. Lapidus [16] and T. L. Gill-W. W. Zachary [13] developed Feynman‘soperational calculus.
$B$

$)$

\S 2. Existence of Phase Space Path Integrals
We explain the existence of the phase space path integrals (1.4) step by step.

\S 2.1. Assumption for Hamiltonian
Our assumption for the Hamiltonian function $H(t,x,\xi)$ of (1.1) are the following.
Assumption 1 (Hamiltonian function). $H(t,x,\xi)$ is a real-valued function of $(t,x,\xi)$ in, and for any multi-indicesis continuous. For any non-negative
integer , there exists a positive constant such that$R\cross$$R^{d}\cross R^{d}$$\alpha,$$k$$\partial_{x}^{\alpha}\partial_{\xi}^{\beta}H(t,x,\xi)$$K_{k}$(2.1)

for any multi-indices$\beta,$$|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}H(t,x,\xi)|\leq\kappa_{k}(1+|x|+|\xi|)^{\max(2-|x+\beta|,0)}$$\alpha,$$\beta$,with $|\alpha+\beta|=k$ .

A typical example of the Hamiltonian operator$H(t,x, \frac{\hslash}{i}\partial_{x})$of (1.1) is the following.

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Example 1 (Hamiltonian operator).
$H(t,x, \frac{\hslash}{i}\partial_{x})=\sum_{j,k=1}^{d}(a_{j,k}(t)\frac{\hslash}{i}\partial_{x_{j}}\frac{\hslash}{i}\partial_{x_{k}}+b_{j,k}(t)x_{j}\frac{\hslash}{i}\partial_{x_{k}}+c_{j,k}(t)x_{j}x_{k})$$+ \sum_{j=1}^{d}(a_{j}(t)\frac{\hslash}{i}\partial_{x_{j}}+b_{j}(t)x_{j})+c(t,x)$Heretions.

$a_{j,k}(t),$$b_{j,k}(t),$$c_{j,k}(t),$ $a_{j}(t),$$b_{j}(t)$and$\partial_{x}^{\alpha}c(t,x)$.

are real-valued continuous $boundedfi\ell nc-$

\S 2.2. We can produce many functionals
Typical examples of the functionals $F[q,p]$ in our class$\mathcal{F}$$F[q,p]\in \mathcal{F}$are the following.

Example 2 $(F[q,p]\in \mathcal{F})$ .
(1) Let $m\geq 0$.

Let $B(t,x)$ be a function of $(t,x)\in R\cross R^{d}$ such that for any multi-index
is continuous and $satisfies|\partial_{x^{t}}(B(t,x)|\leq C_{\alpha}(1+|x|)^{m}$ with a positive constantThen, the value at time $0\leq t\leq T$ ,$C_{\alpha}$$\partial_{x}^{\alpha}B(t,x)$,$\alpha$.

$t,$$F[q,p]=B(t,q(t))\in \mathcal{F}$.

In particular, $F[q,p]=1\in \mathcal{F}$.
(2) Let $m\geq 0$ and $0\leq T’\leq T’’\leq T$. Let $B(t,x,\xi)$ be afimction of $(t,x,\xi)\in R\cross R^{d}\cross R^{d}$ suchis continuous and satisfies
that for any multi-indices$C_{\alpha\beta}(1+|x|+|\xi|)^{m}$ with a positive constant. Then$\alpha,$$\beta,$$|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}B(t,x,\xi)|\leq$$\partial_{\chi}^{\alpha}\partial_{\xi}^{\beta}B(t,x,\xi)$$C_{\alpha\beta}$$F[q,p]= \int_{[T’,T’’)}B(t,q(t),p(t))dt\in \mathcal{F}$

Furthermore,$\iota f|\partial_{x}^{\alpha}\partial_{\xi}^{\beta}B(t,x,\xi)|\leq C_{\alpha\beta}$.

, then$F[q,p]=e^{\int_{IT’,T’’)}B(t,q(t),p(t))dt}\in \mathcal{F}$.

For simplicity, we will state the definition of the class in \S 6. Even if we do not state thedefinition of here, we can produce many functionals $F[q,p]\in \mathcal{F}$, applying Theorem 1 to
Example 2.
$\mathcal{F}$$\mathcal{F}$

Theorem 1 (Algebra). If $F[q,p]\in \mathcal{F}$ and $G[q,p]\in \mathcal{F}$, then $F[q,p]+G[q,p]\in \mathcal{F}$ and.

$F[q,p]G[q,p]\in \mathcal{F}$\S 2.3. Bicharacteristic paths
Let
(2.2)

$\Delta_{T,0}=(T_{J+1}, T_{J}, \ldots, T_{1}, T_{0})$be any division of the interval $[0, T]$ given by$\Delta_{T,0}:T=T_{J+1}>T_{J}>\cdots>T_{1}>T_{0}=0$

.

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The bicharacteristic path

The bicharacteristic path$\overline{q}_{T_{j},T_{j-1}}$$\overline{p}_{T_{j},T_{j-1}}$

Figure 2.

Let $t_{j}=T_{j}-T_{j-1}$ and$j=1,2,$. Assume that

. Set $x_{J+1}=x$ . Let
andfor. Then we can define the bicharacteristic pathsby the canonical$x_{j}\in R^{d}$$| \Delta_{T,0}|=\max_{1\leq l\leq J+1}t_{j}$$\ldots,J$$\overline{q}\tau,\cdot,\tau_{j-1}=\overline{q}_{T_{j},T_{j-1}}(t,x_{j},\xi_{j-1})$$\kappa 2d|\Delta_{\tau,0}|<1/2$

and$\overline{p}_{T_{j},T_{j-1}}=\overline{p}\tau_{j},\tau_{;-1}(t,x_{j},\xi_{j-1}),$$\xi_{j}\in R^{d}$$T_{j-1}\leq t\leq T_{j}$equation,(2.3)$\partial_{t}\overline{q}_{T_{j},T_{j-1}}(t)=(\partial_{\xi}H)(t,\overline{q}\tau_{j},\tau_{j-1},\overline{p}_{T_{j},T_{j-1}})$$\partial_{t}\overline{p}_{T_{j},T_{j-1}}(t)=-(\partial_{x}H)(t,\overline{q}\tau_{j},\tau_{j-1},\overline{p}_{T_{j},\tau_{-1}},\cdot),$withindependent ofandand (Figure 2).

$\overline{q}_{T_{j},T_{j-1}}(T_{j})=x_{j}$$x_{j-1}$$\overline{p}_{T_{j},T_{j-1}}(T_{j-1})=\xi_{j-1}$. Note that$T_{j-1}\leq t\leq T_{j}$$\overline{q}_{T_{j},T_{j-I}}(T_{j-1})$,and$\overline{p}_{T_{j},T_{\dot{j}}-1}(T_{j})$are$\xi_{j}$\S 2.4. Piecewise bicharacteristic paths
Using the bicharacteristic paths
acteristic paths$\overline{q}\tau_{j},\tau_{j-1}$

and$\overline{p}_{T_{j},T_{j-1}}$of (2.3), we define the piecewise bicharand$q_{\Delta_{T,0}}=q_{\Delta_{T,0}}(t,x_{J+1},\xi_{J},x_{J}, \ldots,\xi_{11}x,\xi_{0},x_{0})$$p_{\Delta_{T,0}}=p(t,x,\xi_{J},x_{J}, \ldots,\xi_{1^{X}1},\xi_{0})$

(2.4)

by

$q_{\Delta_{T,0}}(t)=\overline{q}\tau_{j},\tau_{-1}(t,x_{j},\xi_{j-1}),$$T_{j-1}<t\leq T_{j},$$p_{\Delta_{T,0}}(t)=\overline{p}_{T_{j},T_{j-1}}(t,x_{j},\xi_{j-1}),$$T_{j-1}\leq t<T_{j}$for $j=1,2,$ $\ldots,J,J+1$ (Figure 3). Then the functionals
the functions
given by$\phi_{\Delta_{T,0}},$$q_{\Delta_{T,0}}(0)=x\circ$$\phi[q_{\Delta_{T,0}},p_{\Delta_{T,0}}],$,$F[q_{\Delta_{T,0}},p_{\Delta_{T,0}}]$$F_{\Delta_{T,0}}$(2.5)$\phi[q_{\Delta_{T,0}},p_{\Delta_{T,0}}]=\phi_{\Delta_{T,0}}(x_{J+1},\xi_{J},x_{J}, \ldots,\xi_{1},x_{1},\xi_{0},x_{0})$

,

(2.6)

$F[q_{\Delta_{T,0}},p_{\Delta_{T,0}}]=F_{\Delta_{T,0}}(x_{J+1},\xi_{J},x_{J}, \ldots,\xi_{1},x_{1},\xi_{0},x_{0})$.

\S 2.5. Phase space path integrals exist
Our result about the existence of phase space path integrals is the following.

become

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The piecewise bicharacteristic path$q_{\Delta_{T,0}}$$)$The piecewise bicharacteristic path

$p_{\Delta_{T,0}}$Figure 3.

Theorem 2 (Existence of phase space path integrals). Let
,
any

$T$

be sufficiently small. Then, for$F[q,p]\in \mathcal{F}$(2.7)$\int e^{\frac{i}{\hslash}\phi[q,p]}F[q,p]\mathcal{D}[q,p]$$\equiv\lim_{|\Delta_{T,0}|arrow 0}(\frac{1}{2\pi\hslash}I^{dJ}.1_{R^{2dJ}}e^{\hslash^{i_{\phi[q_{\Delta_{T,0}},p_{\Delta_{T,0}}]}}}F[q_{\Delta_{T,0}},p_{\Delta_{T,0}}]\prod_{j=1}^{J}dx_{j}d\xi_{j}$converges unifomly on compact sets ofintegral (2.7) is well-defined.

$R^{3d}$with respect to $(x,\xi_{0},xo)$ , i. e., the phase space path

Remark. Even when $F[q,p]=1$ , each integral of the right hand side of (2.7) does not converge absolutely, i.e.,$\int_{R^{2d}}d\xi_{j}dx_{j}=\infty$

Furthermore, the number$J$.

of integrals (division points) tends to$\infty\cross\infty\cross\infty\cross\infty\cross\cdots\cdots\cdots$,$\infty$$Jarrow\infty$, i.e.,.

We treat the multiple integral of (2.7) directly as an oscillatory integral (cf. H. Kumano-go [18,.
and\S 1.6] to keep the paths$)$$q_{\Delta_{T,0}}$$p_{\Delta_{T,0}}$\S 3. Calculation Examples
As examples of (2.7), we calculate the fundamental solutions for some equations.

\S 3.1. Example: $d=1,$ $H(t,x,\xi)=\xi^{2}/2,$ $F[q,p]=1$
Note$(\partial_{\xi}H)=\xi$

and$(\partial_{x}H)=0$. By the canonical equation$\partial_{t}\overline{q}_{T_{j},T_{j-1}}(t)=\overline{p}_{T_{j},T_{j-1}}(t),$$\partial_{t}\overline{p}_{T_{j},T_{j-1}}(t)=0,$$T_{j-1}\leq t\leq T_{j}$

173

$0$

$T$

$T_{j-1}T_{j}$The path$0$$T_{j-1}T_{j}$The path$q_{\Delta_{T,0}}$$p_{\Delta_{T,0}}$Figure 4.
with$\overline{q}\tau_{j},\tau_{;-1}(T_{j})=x_{j}$and$\overline{p}_{T_{j},T_{j-1}}(T_{j-1})=\xi_{j-1}$, we have the bicharacteristic paths$\overline{q}_{T_{j},T_{j-1}}(t)=x_{j}-(T_{j}-t)\xi_{j-1}$$\overline{p}_{T_{j},T_{j-1}}(t)=\xi_{j-1}$

Let

,

.

be the piecewise bicharacteristic paths of (2.4) (Figure 4).
Then the functional
becomes the function$q_{\Delta_{T,0}},$$p_{\Delta_{T,0}}$$\phi[q_{\Delta_{T,0}},p_{\Delta_{T,0}}]$$\phi_{\Delta_{T,0}}=\phi[q_{\Delta_{T,0}},p_{\Delta_{T,0}}]=\int_{l0,T)}p_{\Delta_{T,0}}\cdot dq_{\Delta_{T,0}}-\int_{l0,T)}H(t,q_{\Delta_{T,0}},p_{\Delta_{T,0}})$$= \sum_{j=1}^{J+1}\phi_{T_{j},\tau_{-1}},\cdot(x_{j},\xi_{j-1},x_{j-1})$

,

where
$\phi_{T_{j},T_{j-1}}(x_{j},\xi_{j-1},x_{j-1})$$= \int_{[T_{j-1},T_{j})}p_{\Delta_{T,0}}\cdot dq_{\Delta_{T,0}}-\int_{[T_{j-1},T_{j})}H(q_{\Delta_{T,0}},p_{\Delta_{T,0}})dt$$=(\overline{q}_{T_{j},T_{j-1}}(T_{j-1})-x_{j-1})\cdot\xi_{j-1}$$+ \int_{T_{j-1}}^{T_{j_{\overline{p}_{T_{j},T_{j-1}}\cdot\partial_{t}\overline{q}_{T_{j},T_{j-1}}}}}dt-\int_{T_{j-1}}^{T_{j}}H(t,\overline{q}\tau_{j},\tau_{j-1},\overline{p}_{T_{j},T_{j-1}})dt$$=(x_{j}-(T_{j}-T_{j-1}) \xi_{j-1}-x_{j-1})\cdot\xi_{j-1}+\int_{T_{j-1}}^{T_{j}}\xi_{j-1}^{2}dt-\int_{T_{j-1}}^{T_{j}}\xi_{j-1}^{2}/2dt$$=(x_{j}-x_{j-1}) \cdot\xi_{j-1}-\frac{t_{j}}{2}\xi_{j-1}^{2}$.

First we consider the oscillatory integral$( \frac{1}{2\pi\hslash}).\int_{R^{2}}e^{\frac{i}{\hslash}\phi_{T_{2},T_{1}}(x_{2},\xi_{1},x_{1})+\frac{i}{\hslash}\phi_{T_{1},0}(x_{1},\xi_{0},x_{0})}dx_{1}d\xi_{1}$

.

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$0$

$T_{1}$The path$0$$T$$T_{2}$$q(\Delta_{T,T_{2}},0)$

and$T_{1}$The path$x_{1}^{*}$$T$$T_{2}$$p_{(\Delta_{T,T_{2}},0)}$and$\xi_{1}^{*}$

Figure 5.

Letbe the solution ofThen we have$(\xi_{1}^{*},x_{1}^{*})$(Figure 5).

$\partial_{(\xi_{1},x_{1})}(\phi_{T_{2},T_{1}}+\phi_{T_{1},0})(x2,\xi_{1}^{*},x_{1}^{*},\xi_{0})=0$$\phi\tau_{2},\tau_{1}(x_{2},\xi_{1},x_{1})+\phi\tau_{1},o(x_{1},\xi_{0},x_{0})$,$= \phi\tau_{2},o(x_{2},\xi 0,x_{0})+\frac{1}{2}\partial_{(\xi_{1},x_{1})}^{2}(\phi r_{2},\tau_{1}+\phi_{T_{1},0})\{\begin{array}{l}\xi_{l}-\xi_{1}^{*}x_{l}-x_{l}^{*}\end{array}\}\cdot\{\begin{array}{l}\xi_{l}-\xi_{l}^{*}x_{1}-x_{l}^{*}\end{array}\}$$(-1)\det\partial_{(\xi_{1},x_{1})}^{2}(\phi_{T_{2},T_{1}}+\phi_{T_{1},0})=(-1)\det\{\begin{array}{l}-t_{2}-1-10\end{array}\}=1$

Lemma 3.1. For any$2\cross 2$

real symmetric matrix$A$.

, we have$\int_{R^{2}}e^{i1}\pi z^{Ax\cdot x}dx=\sqrt{\frac{(2\pi\hslash i)^{2}}{\det A}}=\frac{2\pi\hslash}{\sqrt{(-1)\det A}}$.

By lemma 3.1, we have.

$( \frac{1}{2\pi\hslash})\int_{R^{2}}^{i_{\phi_{T_{2},T}(x_{2},\xi_{1},x_{1})+^{i}\emptyset r_{1},o(x_{1},\xi_{0},x_{0})}}e^{\hslash\iota\hslash}dx\iota d\xi_{1}=e^{i}\hslash^{\phi_{T_{2},0}(x_{2},\xi_{0},x_{0})}$Using this relation inductively, we have$e^{\frac{i}{\hslash}(x-x)\cdot\xi_{0}}0U(T,0,x, \xi_{0})=\int e^{\frac{i}{h}\phi[q,p]}\mathcal{D}[q,p]$$= \lim_{|\Delta_{T,0}|arrow 0}(\frac{1}{2\pi\hslash}I^{J}\int_{R^{y}}^{i_{\Sigma_{j=\iota}^{J+1}\phi_{T_{j},T_{j-1}}(x_{j},\xi_{j-1},x_{j-1})}}e^{\hslash}\prod_{j=1}^{J}dx_{j}d\xi_{j}$$= \lim_{|\Delta_{T,0}|arrow 0}e^{\frac{i}{\hslash}\phi_{T,0}(x,\xi_{0^{X}0})}$$= \exp\frac{i}{\hslash}((x-xo)\cdot\xi_{0}-\frac{T}{2}\xi_{0}^{2})$.

The operator $U(T,0)$ of (1.2) satisfies the Schr\"odinger equation$(i\hslash\partial_{T}+\hslash^{2}\triangle/2)U(T,0)=0,$$U(O,O)=I$ .

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The path

The path$q_{\Delta_{T,0}}$$p_{\Delta_{T,0}}$Figure 6.

\S 3.2. Example: $d=1,$ $H(t,x,\xi)=x^{2}/2+\xi^{2}/2,$ $F[q,p]=1$

Note$(\partial_{\xi}H)=\xi$

and $(\partial_{x}H)=x$. By the canonical equation$\partial_{t}\overline{q}_{T_{j},T_{j-1}}(t)=\overline{p}_{T_{j},T_{j-1}}(t),$with$\overline{q}_{T_{j},T_{j-1}}(T_{j})=x_{j}$and$\partial_{\overline{p}_{T_{j},T_{j-1}}}f(t)=-\overline{q}_{T_{j},T_{j-1}}(t),$$\overline{p}_{T_{j},T_{j-1}}(T_{j-1})=\xi_{j-1}$$T_{j-1}\leq t\leq T_{j}$, we have the bicharacteristic paths$\overline{q}_{T_{j},T_{j-1}}(t)=\frac{x_{j}\cos(t-T_{j-1})-\xi_{j-1}\sin(T_{j}-t)}{\cos(T_{j}-T_{j-1})}$,$\overline{p}_{T_{j},T_{j-1}}(t)=\frac{-x_{j}\sin(t-T_{j-1})+\xi_{j-1}\cos(T_{j}-t)}{\cos(T_{j}-T_{j-1})}$Let.

be the piecewise bicharacteristic paths of (2.4) (Figure 6).
Then the functional
becomes the function$q_{\Delta_{T,0}},$$p_{\Delta_{T,0}}$$\phi[q_{\Delta_{T,0}},p_{\Delta_{T,0}}]$$\phi_{\Delta_{T,0}}=\phi[q_{\Delta_{T,0}},p_{\Delta_{T,0}}]=\int_{[0,T)}p_{\Delta_{T,0}}\cdot dq_{\Delta_{T,0}}-\int_{[0,T)}H(t,q_{\Delta_{T,0}},p_{\Delta_{T,0}})$$= \sum_{j=1}^{J+1}\phi_{T_{j},T_{j-1}}(x_{j},\xi_{j-1},x_{j-1})$

,

176

$0$

$T_{1}$The path$0$$T$$T_{2}$$q_{(\Delta_{T,T_{2}},0)}$and$T_{1}$The path$x_{1}^{*}$$T$$T_{2}$$p_{(\Delta_{T,T_{2}},0)}$and$\xi_{1}^{*}$

Figure 7.
where$\phi_{T_{j},T_{j-1}}(x_{j},\xi_{j-1},x_{j-1})$$= \int_{[T_{j-1},T_{j})}p_{\Delta_{T,0}}\cdot dq_{\Delta_{T,0}}-\int_{[T_{j-1},T_{j})}H(q_{\Delta_{T,0}},p_{\Delta_{T,0}})dt$$=(\overline{q}_{T_{j},T_{j-1}}(T_{j-1})-x_{j-1})\cdot\xi_{j-1}$$+ \frac{1}{2}\int_{T_{j-1}}^{T_{j}}\overline{p}_{T_{j},T_{j-1}}\cdot\partial_{t}\overline{q}_{T_{j},T_{j-1}}dt+\frac{1}{2}[\overline{p}\tau_{j},\tau_{j-1}\cdot\overline{q}_{T_{j},T_{j-1}]_{T_{j-1}}^{T_{j}}}$$- \frac{1}{2}\int_{T_{j-1}}\tau_{j_{\overline{q}_{T_{j},T_{j-1}}\cdot\overline{p}_{T_{j},T_{j-1}}}}\partial_{t}dt-\frac{1}{2}\int_{T_{j-1}}^{T_{j}}(\overline{q}_{\tau_{j},\tau_{j-1}+\overline{p}_{T_{j},T_{j-1}}^{2})dt}^{2}$$=-x_{j-1} \cdot\xi_{j-1}+\frac{\overline{p}_{T_{j},T_{j-1}}(T_{j})x_{j}+\overline{q}_{T_{j},T_{j-1}}(T_{j-1})\xi_{j-1}}{2}$$=-x_{j-1} \cdot\xi_{j-1}+\frac{2x_{j}\cdot\xi_{j-1}-(x_{j}^{2}+\xi_{j-1}^{2})\sin(T_{j}-T_{j-1})}{2\cos(T_{j}-T_{j-1})}$.

First we consider the oscillatory integral$( \frac{1}{2\pi\hslash})\int_{R^{2}}^{i_{\phi r_{2},r_{1}(x_{2},\xi_{1},x_{1})+^{i}}}e^{\pi\pi^{\phi_{T_{1},0}(x_{1},\xi_{0^{X}0})}\prime}d_{X1}d\xi_{1}$Letbe the solution ofThen we have$(\xi_{1}^{*},x_{1}^{*})$$\partial_{(\xi_{1},x_{1})}(\phi_{T_{2},T_{1}}+\phi)(x,\xi_{1}^{*},x_{1}^{*},\xi_{0})=0$.

(Figure 7).

$\phi\tau_{2},\tau_{1}(x_{2},\xi_{1},x_{1})+\phi\tau_{1},o(x_{1},\xi_{0},x_{0})$$= \phi(x,\xi 0,xo)+\frac{1}{2}\partial_{(\xi_{1},x_{1})}^{2}(\phi r_{2},\tau_{1}+\phi_{T_{1},0})\{\begin{array}{l}\xi_{l}-\xi_{1}^{*}x_{l}-x_{l}^{*}\end{array}\}\cdot\{\begin{array}{l}\xi_{l}-\xi_{l}^{*}x_{l}-x_{1}^{*}\end{array}\}$$(-1)\det\partial_{(\xi_{1},x_{1})}^{2}(\phi_{T_{2},T_{1}}+\phi_{T_{1},0})$$=(-1) \det[^{-}\hat{\cos(T_{2}-T_{1})}\sin(T-T)-1-\frac{\sin(T_{1}-0)-1}{\cos(T_{1}-0)}]=\frac{\cos T_{2}}{\cos t_{2}\cos t_{1}}$

.

,

177
By Lemma 3.1, we have$( \frac{1}{2\pi\hslash})\int_{R^{2}}e^{\frac{i}{\hslash}\psi_{T_{2},T_{1}(x_{2},\xi,x_{1})+\frac{i}{\hslash}\phi_{T_{1},0}(x_{1},\xi_{0)}x_{0})}}[dx_{1}d\xi_{1}$$=e^{\frac{i}{\hslash}\phi_{T_{2},0}(x_{2},\xi_{0^{X()}},)}( \frac{\cos t_{2}\cos t_{1}}{\cos T_{2}}I^{1/2}$

Using this relation inductively and taking$| \Delta_{T,0}|=\max_{1\leq J\leq J+1}t_{j}arrow 0$, we have$e^{\frac{i}{\hslash}(x-x_{0})\cdot\xi_{0}}U(T,0,x, \xi_{0})=\int e^{\frac{i}{\hslash}\phi[q,p]}\mathcal{D}[q,p]$$= \lim_{|\Delta_{T,0}|arrow 0}(\frac{1}{2\pi\hslash})^{J}\int_{R^{2J}}e^{\frac{i}{\hslash}\Sigma_{j=1}^{J+1}\phi_{T_{j},T_{j-1}}(x_{j},\xi_{j-1},x_{j-1})}\prod_{j=1}^{J}dx_{j}d\xi_{j}$$= \lim_{|\Delta_{T,0}|arrow 0}e^{\frac{i}{\hslash}\emptyset\tau,o(x,\xi_{0},x_{0})}(\frac{\prod_{j=1}^{J+1}\cos t_{j}}{\cos T}I^{1/2}$.

$= \frac{1}{(\cos T)^{1/2}}\exp\frac{i}{\hslash}(-x_{0}\cdot\xi_{0}+\frac{2x\cdot\xi_{0}-(+\xi_{0}^{2})\sin T}{2\cos T})$

The operator $U(T,0)$ of (1.2) satisfies the Schr\"odinger equation$(i\hslash\partial_{T}+\hslash^{2}\triangle/2-f/2)U(T,0)=0,$$U(O,0)=I$ .

\S 3.3. Example: $d=1,$ $H(t,x,\xi)=x^{2}/2+x\cdot\xi+\xi^{2}/2,$ $F[q,p]=1$
Note $(\partial_{\xi}H)=x+\xi=(\partial_{x}H)$. By the canonical equation$\partial_{t\overline{q}_{T_{j},T_{j-1}}}(t)=\overline{q}_{T_{j},T_{j-1}}(t)+\overline{p}_{T_{j},T_{j-1}}(t)=-\partial_{t}\overline{p}\tau_{;},\tau_{j-1}(t),$with$\overline{q}_{T_{j},T_{j-1}}(T_{j})=x_{j}$and$\overline{p}_{T_{j},T_{j-1}}(T_{j-1})=\xi_{j-1}$$T_{j-1}\leq t\leq T_{j}$, we have the bicharacteristic paths$\overline{q}_{T_{j},T_{j-1}}(t)=\frac{x_{j}(1+t-T_{j-1})-\xi_{j-1}(T_{j}-t)}{1+T_{j}-T_{j-1}}$,$\overline{p}_{T_{j},T_{j-1}}(t)=\frac{-x_{j}(t-T_{j-1})+\xi_{j-1}(1+T_{j}-t)}{1+T_{j}-T_{j-1}}$Let.

be the piecewise bicharacteristic paths of (2.4) (Figure 8).
Then the functional
becomes the function$q_{\Delta_{T,0}},$$p_{\Delta_{T,0}}$$\phi[q_{\Delta_{T,0}},p_{\Delta_{T,0}}]$$\phi_{\Delta_{T,0}}=\phi[q_{\Delta_{T,0}},p_{\Delta_{T,0}}]=\sum_{j=1}^{J+1}\phi_{T_{j},T_{j-1}}(x_{j},\xi_{j-1},x_{j-1})$

,

where
$\phi_{T_{j},T_{j-1}}(x_{j},\xi_{j-1},x_{j-1})$$=-x_{j-1} \cdot\xi_{j-1}+\frac{2x_{j}\cdot\xi_{j-1}-(x_{j}^{2}+\xi_{j-1}^{2})(T_{j}-T_{j-1})}{2(1+T_{j}-T_{j-1})}$

.

178

The path

The path$q_{\Delta_{T,0}}$$p_{\Delta_{T,0}}$Figure 8.

Let$(\xi_{1}^{*},x_{1}^{*})$

be the solution of $\partial_{(\xi_{1},x_{1})}(\phi_{T_{2},T_{1}}+\phi_{T_{1},0})2,=0$ (Figure 9).

The path$q_{(\Delta_{T,T_{2}},0)}$and

The path$x_{1}^{*}$$p(\Delta_{T,T_{2}},0)$and$\xi_{1}^{*}$

Figure 9.
Then we have$\phi\tau_{2},r_{1}(x_{2},\xi_{1},x_{1})+\phi\tau_{1},o(x_{1},\xi_{0^{X}0})$$= \phi\tau_{2},o(x2,\xi 0,xo)+\frac{1}{2}\partial_{(\xi_{1},x_{1})}^{2}(\phi\tau_{2},\tau_{1}+\phi_{T_{1},0})\{\begin{array}{l}\xi_{l}-\xi_{1}^{*}x_{l}-x_{l}^{*}\end{array}\}\cdot\{\begin{array}{l}\xi_{1}-\xi_{1}^{*}x_{l}-x_{l}^{*}\end{array}\}$$(-1)\det\partial_{(\xi_{1},x_{1})}^{2}(\phi_{T_{2},T_{1}}+\phi_{T_{1},0})$$=(-1) \det\{\begin{array}{ll}-\frac{t_{2}}{1+t_{2}} -1-1 -\frac{t_{1}}{l+t_{1}}\end{array}\}= \frac{1+T_{2}}{(1+t_{2})(1+t_{1})}$.

By Lemma 3.1, we have$( \frac{1}{2\pi\hslash})\int_{R^{2}}0,0$$=e^{\frac{i}{h}\phi_{T_{2},0}(x_{2},\xi_{0^{X}0})}( \frac{(1+t_{2})(1+t_{1})}{1+T_{2}})^{1/2}$,

179
Using this relation inductively and taking, we have$| \Delta_{T,0}|=\max_{1\leq J\leq J+1}t_{j}arrow 0$$e^{\frac{i}{\hslash}(x-x_{0})\cdot\xi_{0}}U(T,0,x, \xi_{0})=\int e^{\frac{i}{\hslash}\phi[q,p]}\mathcal{D}[q,p]$$= \lim_{|\Delta_{T,0}|arrow 0}(\frac{1}{2\pi\hslash})^{J}\int_{R^{2J}}e^{\hslash^{i_{\Sigma_{;=1}^{J+1}\phi_{T_{j},T_{j-1}}(x_{j},\xi_{j-1},x_{j-1})}}}\prod_{j=1}^{J}dx_{j}d\xi_{j}$$= \lim_{|\Delta_{T,0}|arrow 0}e^{\frac{i}{\hslash}\emptyset\tau,o(x,\xi_{0},x_{0})}(\frac{\prod_{j=1}^{J+1}(1+t_{j})}{1+T}I^{1/2}$.

$=( \frac{e^{T}}{\cos T})^{1/2}\exp\frac{i}{\hslash}(-x_{0}\cdot\xi_{0}+\frac{2x\cdot\xi_{0}-(+\xi_{0}^{2})T}{2(1+T)})$

The operator $U(T,0)$ of (1.2) satisfies the equation$(i \hslash\partial_{T}+\hslash^{2}\triangle/2-x\frac{\hslash}{i}\partial_{x}-x^{2}/2)U(T,0)=0,$$U(O,0)=I$ .

\S 3.4. Example: $d=1,$ $H(t,x,\xi)=x^{2}/2-\xi^{2}/2,$ $F[q,p]=1$
Note

and $(\partial_{x}H)=x$ . By the canonical equation$(\partial_{\xi}H)=-\xi$$\partial_{t}\overline{q}_{T_{j},T_{j-1}}(t)=-\overline{p}_{T_{j},T_{j-1}}(t),$with$\overline{q}_{T_{j},T_{j-1}}(T_{j})=x_{j}$and$\partial_{t}\overline{p}_{T_{j},T_{j-1}}(t)=-\overline{q}_{T,\cdot,T_{j-1}}(t),$$\overline{p}_{T_{j},T_{j-1}}(T_{j-1})=\xi_{j-1}$$T_{j-1}\leq t\leq T_{j}$, we have the bicharacteristic paths$\overline{q}_{T_{j},T_{j-1}}(t)=\frac{x_{j}\cosh(t-T_{j-1})+\xi_{j-1}\sinh(T_{j}-t)}{\cosh(T_{j}-T_{j-1})}$,$\overline{p}_{T_{j},T_{j-1}}(t)=\frac{-x_{j}\sinh(t-T_{j-1})+\xi_{j-1}\cosh(T_{j}-t)}{\cosh(T_{j}-T_{j-1})}$Let.

be the piecewise bicharacteristic paths of (2.4) (Figure 6).
Then the functional
becomes the function$q_{\Delta_{T,0}},$$p_{\Delta_{T,0}}$$\phi[q_{\Delta_{T,0}},p_{\Delta_{T,0}}]$$\phi_{\Delta_{T,0}}=\phi[q_{\Delta_{T,0}},p_{\Delta_{T,0}}]=\sum_{j=1}^{J+1}\phi_{T,T_{j-1}};(x_{j},\xi_{j-1},x_{j-1})$

,

where
$\phi_{T_{j},T_{\dot{j}}-1}(x_{j},\xi_{j-1},x_{j-1})$$=-x_{j-1} \cdot\xi_{j-1}+\frac{2x_{j}\cdot\xi_{j-1}-(x_{j}^{2}-\xi_{j-1}^{2})\sinh(T_{j}-T_{j-1})}{2\cosh(T_{j}-T_{j-1})}$Let$(\xi_{1}^{*},x_{1}^{*})$

be the solution of$\partial_{(\xi_{1},x_{1})}(\phi_{T_{2},T_{1}}+\phi)(x,\xi_{1}^{*},x_{1}^{*},\xi_{0})=0$.

(Figure 7).

180
Then we have$\phi\tau_{2},\tau_{1}(x2,\xi_{1},x_{1})+\phi\tau_{1},o(x_{1},\xi_{0^{X}0})$$= \phi\tau_{2},o(x2,\xi_{0^{X}0})+\frac{1}{2}\partial_{(\xi_{1},x_{1})}^{2}(\phi\tau_{2},r_{1}+\phi_{T_{1},0})\{\begin{array}{l}\xi_{l}-\xi_{1}^{*}x_{l}-x_{l}^{*}\end{array}\}\cdot\{\begin{array}{l}\xi_{1}-\xi_{1}^{*}x_{l}-x_{l}^{*}\end{array}\}$,$(-1)\det\partial_{(\xi_{1},x_{1})}^{2}(\phi_{T_{2},T_{1}}+\phi_{T_{1},0})$$=(-1) \det[^{\frac{\sinh(T_{2}-T_{1})}{\cosh(T_{2}-T_{1})-1}}-\frac{\sinh(T_{1}-0)-1}{\cosh(T_{1}-0)}]=\frac{\cosh T_{2}}{\cosh t_{2}\cosh t_{1}}$.

By Lemma 3.1, we have$( \frac{1}{2\pi\hslash}I\int_{R^{2}}e^{\hslash^{i_{\phi_{T_{2},T_{1}}(x_{2},\xi_{1},x_{1})+\frac{i}{\hslash}\phi_{T_{1},0(x_{1},\xi_{0},x_{0})}}}}dx1d\xi_{1}$$=e^{\hslash^{\phi_{T_{2},0}(x_{2},\xi_{0^{X}i)})}}i,( \frac{\cosh t_{2}\cosh t_{1}}{\cosh T_{2}})^{1/2}$

Using this relation inductively and taking, we have$| \Delta_{T,0}|=\max_{1\leq J\leq J+1}t_{j}arrow 0$$e^{\pi^{(x-xo)\cdot\xi_{0}}}U(T,0,x, \xi_{0})=i\int e^{\pi^{i_{\phi[q,p]}}}D[q,p]$$= \lim_{|\Delta_{T,0}|arrow 0}(\frac{1}{2\pi\hslash})^{J}\int_{R^{2J}}e^{\hslash^{i_{\Sigma_{j=1}^{J+1}\phi_{T_{j},T_{j-1}}(x_{j},\xi_{j-1},x_{j-1})}}}\prod_{j=1}^{J}dx_{j}d\xi_{j}$$=| \Delta_{T,0}|arrow 0^{e^{i}}\lim\hslash^{\phi_{T,0}(x,\xi_{0^{X}0})}(\frac{\prod_{j=1}^{J+1}\cosh t_{j}}{\cosh T})^{1/2}$$= \frac{1}{(\cosh T)^{1/2}}\exp\frac{i}{\hslash}(-x_{0}\cdot\xi_{0}+\frac{2x\cdot\xi_{0}-(x^{2}-\xi_{0}^{2})\sinh T}{2\cosh T})$.

The operator $U(T,0)$ of (1.2) satisfies the equation$(i\hslash\theta_{T}-\hslash^{2}\triangle/2-\mathscr{S}/2)U(T,O)=0,$$U(O,O)=I$ .

\S 3.5. Exception: $d=1,$ $H(t,x,\xi)=-iF/2-i\xi^{2}/2,$ $F[q,p]=1$
This complex-valued function $H(t,x,\xi)$ does not satisfy Assumption 1. However we can get
the function $U(T,0,x,\xi_{0})$ of the fundamental solution in a similar way:
and $(\partial_{x}H)=-ix$. By the canonical equationNote$(\partial_{\xi}H)=-i\xi$$\partial_{t}\overline{q}_{T_{j},T_{j-1}}(t)=-i\overline{p}_{T_{j},T_{j-1}}(t),$with$\overline{q}_{T_{j},T_{j-1}}(T_{j})=x_{j}$and$\partial_{t}\overline{p}_{T_{j},T_{j-1}}(t)=i\overline{q}_{T_{j},T_{j-1}}(t),$$\overline{p}_{T_{j},T_{j-1}}(T_{j-1})=\xi_{j-1}$$T_{j-1}\leq t\leq T_{j}$, we have the bicharacteristic paths$\overline{q}_{T_{j},T_{j-1}}(t)=\frac{x_{j}\cosh(t-T_{j-1})+i\xi_{j-1}\sinh(T_{j}-t)}{\cosh(T_{j}-T_{j-1})}$$\overline{p}_{T_{j},T_{j-1}}(t)=\frac{ix_{/}\cdot\sinh(t-T_{j-1})+\xi_{j-1}\cosh(T_{j}-t)}{\cosh(T_{j}-T_{j-1})}$

,

.

181
$Q$

$eo$

$\bullet 0$$0\bullet$$(0,x_{0})\ovalbox{\tt\small REJECT}\cdots\cdots\cdot\bullet\circ$.
.

$:^{(T,x)}(0,\xi_{0})\ovalbox{\tt\small REJECT}\cdot\cdot\cdot\cdot\cdot Q\bullet\cdot\cdot\cdot\cdot(T_{j-1},\xi_{j1})\bullet 0_{\bullet\cdot;}\bullet\cdot\cdot.Q\ldots.....\ddot{\Phi}\bullet...$$\bullet.\cdot\cdot\cdot\cdot\cdot 0\ddot{o\prime}..(.T..’x..)\ldots.\bullet 0’$.

$\overline{0T_{j-1}T_{j}T}$The complex-valued path$\overline{0T_{j-1}T_{j}T}$The complex-valued path$q_{\Delta_{T,0}}$$p_{\Delta_{T,0}}$Figure 10.

Let

be the piecewise bicharacteristic paths of (2.4) (Figure 10).
Then the functional
becomes the function$q_{\Delta_{T,0}},$$p_{\Delta_{T,0}}$$\phi[q_{\Delta_{T,0}},p_{\Delta_{T,0}}]$$\phi_{\Delta_{T,0}}=\phi[q_{\Delta_{T,0}},p_{\Delta_{T,0}}]=\sum_{j=1}^{J+1}\phi_{T_{j},T_{j-1}}(x_{j},\xi_{j-1},x_{j-1})$

,

where

$\phi_{T_{j},T_{j-1}}(x_{j},\xi_{j-1},x_{j-1})$$=-x_{j-1} \cdot\xi_{j-1}+\frac{2x_{j}\cdot\xi_{j-1}+i(_{j}+\xi_{j-1}^{2})\sinh(T_{j}-T_{j-1})}{2\cosh(T_{j}-T_{j-1})}$Let$(\xi_{1}^{*},x_{1}^{*})$

be the solution of$\partial_{(\xi_{1},x_{1})}(\phi_{T_{2},T_{1}}+\phi)(x,\xi_{1}^{*},x_{1}^{*},\xi_{0})=0$.

(Figure 11).

$\bullet Q\cdots$$0\prime i!.\cdot$$x_{1^{t.\bullet}}^{*}\bullet(T\ldots’ x)..\bullet.0\Phi..\ldots....\ldots\ldots\bullet\dot{}$$(0,x_{0})\ovalbox{\tt\small REJECT}$$\bullet(T,x)(0,\xi_{0})\ovalbox{\tt\small REJECT}\ldots.....\zeta_{1}^{*}\bullet\bullet(T_{2_{Q\prime}},.\cdot\cdot\xi_{2})\bullet 0’\ldots\ldots\cdot\cdot 0’.\ddot{o}’\bullet\cdots\cdot\cdot\cdot\cdot\cdot\cdot\bullet.$.

$\overline{0T_{1}T_{2}T}$
The path$q_{(\Delta_{T,T_{2}},0)}$and$\overline{0T_{1}T_{2}T}$
The path$x_{1}^{*}$

Figure 11.

$p_{(\Delta_{T,T_{2}},0)}$and$\xi_{1}^{*}$

182
Then we have$\phi 2,1$$= \phi_{T_{2},0}(x2,\xi_{0,0}x)+\frac{1}{2}\partial_{(\xi_{1},x_{I})}^{2}(\phi\tau_{2},r_{1}+\phi_{T_{1},0})\{\begin{array}{l}\xi_{l}-\xi_{l}^{*}x_{l}-x_{1}^{*}\end{array}\}\cdot\{\begin{array}{l}\xi_{1}-\xi_{l}^{*}x_{l}-x_{l}^{*}\end{array}\}$,$(-1)\det\partial_{(\xi_{1},x_{1})}^{2}(\phi r_{2},\tau_{1}+\phi_{T_{1},0})$$=(-1) \det[^{i\frac{\sinh(T_{2}-T_{1})}{\cosh(T_{2}-T_{1})-1}}i\frac{\sin^{-1}h(T_{1}-0)}{\cosh(T_{1}-0)}]=\frac{\cosh T_{2}}{\cosh t_{2}\cosh t_{1}}$Performing the integration with respect to$(\xi_{1},x_{1})$.

, we have$( \frac{1}{2\pi\hslash})\int_{R^{2}}e^{ii}\hslash\hslash,dx_{1}d\xi_{1}$$=e^{\hslash^{\phi_{T_{2},0}(x_{2},\xi_{0},\triangleleft)}}i( \frac{\cosh t_{2}\cosh t_{1}}{\cosh T_{2}})^{1/2}$

Using this relation inductively and taking$| \Delta_{T,0}|=\max\iota\leq J\leq J+1^{f}jarrow 0$, we have$e^{i} \hslash^{(x-x_{0})\cdot\xi_{0}}U(T,0,x,\xi_{0})=\int\hslash^{\phi[q,p]}$$= \lim_{|\Delta_{T,0}|arrow 0}(\frac{1}{2\pi\hslash})^{J}\int_{R^{y}}^{i}e^{\hslash^{\Sigma_{j=1}^{J+1_{\phi_{T_{j},T_{j-1}}}}(x_{j},\xi_{1-1},x_{j-1})}}\prod_{j=1}^{J}dx_{j}d\xi_{j}$$= \lim_{|\Delta_{T,0}|arrow 0}e^{\frac{i}{\hslash}\phi_{T.0}(x,\xi_{0^{X}0})}(\frac{\prod_{j=1}^{J+1}\cosh t_{j}}{\cosh T})^{1/2}$$= \frac{1}{(\cosh T)^{1/2}}\exp\frac{i}{\hslash}(-x_{0}\cdot\xi_{0}+\frac{2x\cdot\xi_{0}+i(F+\xi_{0}^{2})\sinh T}{2\cosh T})$.

The operator $U(T,0)$ of (1.2) satisfies the heat equation$(\hslash\partial_{T}-\hslash^{2}\triangle/2+x^{2}/2)U(T,0)=0,$$U(O,O)=I$ .

\S 4. Properties of Phase Space Path Integrals
We sketch some properties of (1.4) with general functional $F[q,p]$ .

\S 4.1. Fubini-type theorem
Theorem 3 (Fubini-type). Let
Assume that for any multi-index$T$$\alpha,$be sufficiently small. Let $m\geq 0$ and $0\leq T’\leq T’’\leq T$ .
is continuous and satisfies$\partial_{x}^{\alpha}B(t,x)$$|\partial_{x}^{\alpha}B(t,x)|\leq C_{a}(1+$183$|x|)^{m}$with a positive constant1$C_{\alpha}$. Then, for any$F[q,p]\in \mathcal{F}$including $F[q,p]=1$ , we have$e^{\frac{i}{\hslash}\phi[q,p]}( \int_{rT’,T’’)}B(t,q(t))dt)F[q,p]\mathcal{D}[q,p]$$= \int_{[T,T)}(\int e^{\frac{i}{\hslash}\phi[q,p]}B(t,q(t))F[q,p]\mathcal{D}[q,p])dt$.

Remark We do not treat $B(t,q(t),p(t))$ at the time .
$t$Remark (Perturbation expansion formula). If$|\partial_{x}^{\alpha}B(t,x)|\leq C_{\alpha}$, we have$\int e^{\frac{i}{\hslash}\phi[q,p]+\frac{i}{\hslash}\int_{l0,T)}B(\tau,q(\tau))d\tau}\mathcal{D}[q,p]$$= \sum_{n=0}^{\infty}(\frac{i}{\hslash})^{n}\int_{I0,T)}d\tau_{n}\int_{[0,r_{n})}d\tau_{n-\iota\cdots\int_{l0,\tau_{2})}d\tau l}$$\cross./\cdot e^{\frac{i}{h}\phi[q,p]}B(\tau_{n},q(\tau_{n}))B(\tau_{n-1},q(\tau_{n-1}))\cdots B(\tau_{1},q(\tau_{1}))\mathcal{D}[q,p]$.

\S 4.2. Semiclassical approximation of Hamiltonian type
Let $4_{K_{2}}dT<1/2$ . Let $q_{T,0}=q_{T,0}(t,x,\xi_{0},x_{0})$ and $p_{T,0}=p_{T,0}(t,x,\xi_{0})$ be the piecewise bicharacteristic paths for the simplest division $0<T$ (Figure 12).
For any, there exists the stationary pointof the phasefunction
given by$(x_{J+1},\xi_{0})\in R^{d}\cross R^{d}$$(x_{J}^{*},\xi_{J}^{*}, \ldots,x_{1}^{*},\xi_{1}^{*})$$\phi_{\Delta_{T,0}}=\phi[q_{\Delta_{T,0}},p_{\Delta_{T,0}}]$$(\partial_{(\xi_{J},x_{J},\ldots,\xi_{1},x_{1})}\phi_{\Delta_{T,0}})(x_{J+1},\xi_{J}^{*},x_{J}^{*}, \ldots,\xi_{1}^{*},x_{1}^{*},\xi_{0})=0$

Pushing the stationary pointby$(x_{J}^{*},\xi_{J}^{*}, \ldots,x_{1}^{*},\xi_{1}^{*})$.

into the Hessian matrix of$\phi_{\Delta_{T,0}}$, we define$D(T,x_{J+1},\xi_{0})$$D(T,x_{J+1},\xi_{0})$$= \lim_{|\Delta_{T,0}|arrow 0}(-1)^{dJ}\det(\partial_{(\xi_{J},x_{J},\ldots,\xi_{1},x_{1})}^{2}\phi_{\Delta_{T,0}})(x_{J+1},x_{J}^{*},\xi_{J}^{*}, \ldots,x_{1}^{*},\xi_{1}^{*},\xi_{0})$The path$q_{T,0}$

for the division $T>0$

The path

Figure 12.

$p_{T,0}$.

for the division $T>0$

184
Then the remainder estimate for the semiclassical approximation of Hamiltonian type as$\hslasharrow 0$

are the following.
Theorem 4 (Semiclassical approximation of Hamiltonian type as
small. Then, for any $F[q,p]\in \mathcal{F}$, we have$\hslasharrow 0$). Let $T$ be sufficiently$\int e\pi^{i_{\phi[q,p]}}F[q,p]\mathcal{D}[q,p]$$=e^{i}\hslash^{\phi[q_{T.0},p_{T,0}]}(D(T,x,\xi_{0})^{-1/2}F[q_{T,0},p_{T,0}]+\hslash T(T,\hslash,x,\xi_{0},x_{0}))$Here for any multi-indices, the remainder term $Y(T,\hslash,x,\xi_{0},x_{0})$ satisfies$\beta$$\alpha,$$|\partial_{X}^{a}\partial_{\xi_{0}}^{\beta}Y(T,\hslash,x,\xi_{0},x\circ)|\leq C_{a\beta}(1+|x|+|\xi_{0}|+|x0|)^{m}$with a positive constant.

,

independent of $0<\hslash<1$ .

$C_{\alpha,\beta}$\S 5. Proof for Theorems 1, 2 and 4
We sketch the process of the proof for Theorems 1, 2 and 4. In order to prove the convergence of the multiple integral
(5.1)

as

$|\Delta_{T,0}|arrow 0$$( \frac{1}{2\pi\hslash})^{dJ}\int_{R^{2dJ}}^{i}e^{\hslash^{\phi[q_{\Delta}}r,0^{p_{\Delta}}r,0^{]}}F[q_{\Delta_{T,0}},p_{\Delta_{T,0}}]\prod_{j=1}^{J}dx_{j}d\xi_{j}$, we have only to add many assumptions for$F_{\Delta_{T,0}}(x_{J+1},\xi_{J},x_{J}, \ldots,x_{1},\xi_{0},x\circ)=F[q_{\Delta_{T,0}},p_{\Delta_{T,0}}]$.

The assumptions should be closed under addition and multiplication. Then will be an algebra.
Do not consider other things. Then will become larger as a set. If lucky, will contain atleast one example $F[q,p]=1$ as the fundamental solution for the Schr\"odinger equation. Our
proof consists of 3 steps. As the first step, we use an estimate of H. Kumano-go-Taniguchi’saswith a positive constant . As thetype [18, (6.94), p.360] to control (5.1) bysecond step, we use a stationary phase method of Fujiwara‘s type [9] to control (5.1) bywith a positive constant . At the last step, we add assumptions so thatindependent of$\mathcal{F}$$\mathcal{F}$$\mathcal{F}$$Jarrow\infty$$C^{J}$

$C$

$C$

$C$

$Jarrow\infty$(5.1) converges as$|\Delta_{T,0}|arrow 0$.
\S 6. Definition of ClassThe definition of the class$\mathcal{F}$$\mathcal{F}$of functionals $F[q,p]$ is the following.

185
Definition 1 (Class

of functionals $F[q,p]$ ). Let $F[q,p]$ be a functional whose domain contains all the piecewise bicharacteristic pathsof (2.4). We say that
if
Assumption
2.
satisfies$\mathcal{F}$$q_{\Delta_{T,0}},$$F[q,p]\in \mathcal{F}$$p_{\Delta_{T,0}}$$F_{\Delta_{T,0}}=F[q_{\Delta_{T,0}},p_{\Delta_{T,0}}]$Assumption 2. Let $m\geq 0$ . Let$j=1,2,$ $\ldots,J,J+1$ are non-negative pammeters$\sum_{j=1}^{J+1}u_{j}=U<\infty$
depending on the divisionsuch that. For any integer $M\geq 0$, there
exist positive constants, any multi-indices,such that for any
with$1\leqk\leq
J$$j=1,2,$ $\ldots,J,J+1$ and any,$u_{j}\geq 0,$$\Delta_{T,0}$$A_{M},$$X_{M}$$\Delta_{T,0}$$\alpha_{j},$$\beta_{j-1}$$|\alpha_{j}|$$|\beta_{j-1}|\leq M,$(6.1)$|( \prod_{j=1}^{J+1}\partial_{x_{j}}^{\alpha_{/}}\partial_{\xi_{j-1}}^{\beta_{j-1}})F_{\Delta_{T,0}}(x_{J+1},\xi_{J},x_{J}, \ldots,\xi_{1},x_{1},\xi_{0^{X}0})|$$\leq A_{M}(X_{M})^{J+1}(\prod_{j=1}^{J+1}(t_{j})^{\min(|\beta_{j-1}|,1)})(1+\sum_{j=1}^{J+1}(|x_{j}|+|\xi_{j-1}|)+|x_{0}|)^{m}$

(6.2)

,

$|( \prod_{j=1}^{J+1}\partial_{x_{j}}^{\alpha_{/}}\partial_{\xi_{j-1}}^{\beta_{j-1}})\partial_{x_{k}}F_{\Delta_{T,0}}(x_{J+1},\xi_{J},x_{J}, \ldots,\xi_{1},x_{1},\xi_{0},x_{0})|$$\leq A_{M}(X_{M})^{J+1}u_{k}(\prod_{j\neq k}(t_{j})^{\min(|\beta_{j-1}|,1)})(1+\sum_{j=1}^{J+1}(|x_{j}|+|\xi_{j-1}|)+|x_{0}|)^{m}$.

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