On the Feynman path integral for the magnetic Schroedinger equation with a polynomially growing electromagnetic potential

TL;DR

This paper rigorously defines Feynman path integrals for magnetic Schrödinger equations with polynomially growing electromagnetic potentials, expanding the mathematical framework to include more physically relevant unbounded potentials.

Contribution

It establishes a rigorous mathematical definition of Feynman path integrals for magnetic Schrödinger equations with polynomially growing potentials, including unbounded electromagnetic potentials.

Findings

Path integrals are well-defined as $L^2$-valued continuous functions in time.

Includes potentials with polynomial growth in spatial variables.

Extends previous definitions to unbounded electromagnetic potentials.

Abstract

The Feynman path integrals for the magnetic Schroedinger equations are defined mathematically, in particular, with polynomially growing potentials in the spatial direction. For example, we can handle electromagnetic potentials $(V,A_{1},A_{2},...,A_{d})$ such that $V(t,x) = |x|^{2(l+1)} + $`` a polynomial of degree $(2l + 1)$ in $x$ " ($l = 0,1,2,...$) and $A_{j}(t,x)$ are polynomials of degree $l$ in $x$. The Feynman path integrals are defined as $L^2$-valued continuous functions with respect to the time variable.

Equations

$$E=-\frac{\partial A}{\partial t}-\frac{\partial V}{\partial x},$$

$$B_{jk}=\frac{\partial A_k}{\partial x_j}-\frac{\partial A_j}{\partial x_k}(1\le j<k\le d),$$

$$\mathcal{L}(t,x,\dot{x})=\frac{m}{2}|\dot{x}{|}^2+e\dot{x}\cdot A(t,x)-eV(t,x),\dot{x}\in {\mathbb{R}}^d$$

$$i\hslash \frac{\partial u}{\partial t}(t)=H(t)u(t)$$

$$:=\left[\frac{1}{2m}\sum _{j=1}^d{\left(\frac{\hslash }{i}\frac{\partial }{\partial x_j}-e{A}_j(t,x)\right)}^2+eV(t,x)\right]u(t),$$

$$S(t,s;q)={\int }_s^t\mathcal{L}(\theta ,q(\theta ),\dot{q}(\theta ))d\theta$$

$$K(t,0)f=\int e^{iS(t,0;q)}f(q(0))\mathcal{D}q$$

$$V(t,x)=|x{|}^{2(l+1)}+\sum _{|\alpha |\le 2l+1}{a}_{\alpha }(t)x^{\alpha },$$

$$A_j(t,x)=\sum _{|\alpha |\le l}{b}_{j\alpha }(t)x^{\alpha }(j=1,2,\dots ,d)$$

$$\underset{x,\xi ,x^{\prime }}{\sup }|{\partial }_{\xi }^{\alpha }{\partial }_x^{\beta }{\partial }_{x^{\prime }}^{\gamma }p(x,\xi ,x^{\prime })|\le C_{\alpha ,\beta ,\gamma }<\infty$$

$$\int e^{ix\cdot \xi }\text{mathchar}22d\xi \int e^{-i{x}^{\prime }\cdot \xi }p(x,\hslash \xi ,x^{\prime })f(x^{\prime })d{x}^{\prime },\text{mathchar}22d\xi =(2\pi {)}^{-d}d\xi$$

$$\Vert P(X,\hslash D_x,X^{\prime }){\Vert }_{L^2\to L^2}=\underset{x,\xi ,x^{\prime }}{\sup }|p(x,\xi ,x^{\prime })|+O(\hslash ),$$

$$K_{\Delta }(t,0)f=\underset{ϵ\to 0+}{\lim }$$

$$\times f(x^{(0)})\prod _{j=1}^{\nu -1}\chi (ϵx^{(j)})d{x}^{(0)}d{x}^{(1)}\cdots d{x}^{(\nu -1)}.$$

$$\prod _{j=0}^{\nu -1}{\sqrt{\frac{m}{2\pi i({\tau }_{j+1}-{\tau }_j)}}}^d\text{Os}-\int \cdots {\int }_{{\mathbb{R}}^d}{e}^{iS(t,0;q_{\Delta })}f(x^{(0)})d{x}^{(0)}d{x}^{(1)}\cdots d{x}^{(\nu -1)}$$

$$C_0<x{>}^{2(M_{\ast }+1)}-C_1\le V(t,x)\le C_2<x{>}^{2(M_{\ast }+1)}$$

$$|{\partial }_x^{\alpha }V(t,x)|\le C_{\alpha }<x{>}^{2(M_{\ast }+1)},|\alpha |\ge 1,$$

$$|{\partial }_x^{\alpha }{\partial }_{t}V(t,x)|\le C_{\alpha }<x{>}^{2(M_{\ast }+1)}$$

$$|A_j(t,x)|\le C<x{>}^{M_{\ast }+1-\delta }$$

$$|{\partial }_x^{\alpha }{\partial }_t{A}_j(t,x)|\le C_{\alpha }<x{>}^{M_{\ast }+1}$$

$$C_{\ast }|x{|}^{2M_{\ast }}-C_1\le -\frac{1}{2m}\left(\frac{\partial E}{\partial x}(t,x)+\frac{{}^t\partial E}{\partial x}(t,x)\right)$$

$$|{\partial }_x^{\alpha }{E}_j(t,x)|\le C_{\alpha }<x{>}^{2M_{\ast }},|\alpha |\ge 1,$$

$$|{\partial }_x^{\alpha }{A}_j(t,x)|\le C_{\alpha }<x{>}^{M_{\ast }},|\alpha |\ge 1.$$

$$|{\partial }_x^{\alpha }{B}_{jk}(t,x)|\le C_{\alpha }<x{>}^{-(1+{\delta }_{\alpha })},|\alpha |\ge 1$$

$$|{\partial }_x^{\alpha }{\partial }_t{B}_{jk}(t,x)|\le C_{\alpha }<x{>}^{M_{\ast }}$$

$$|{\partial }_x^{\alpha }{\partial }_t{B}_{jk}(t,x)|\le C_{\alpha }<x{>}^{-(1+{\delta }_{\alpha })},|\alpha |\ge 1$$

$$|{\partial }_x^{\alpha }{\partial }_t{B}_{jk}(t,x)|\le C_{\alpha }<x{>}^{M_{\ast }-1},|\alpha |\ge 1$$

$$\Vert K_{\Delta }(t,0)f\Vert \le e^{Kt}\Vert f\Vert$$

$$V^{\prime }=V-\frac{\partial \psi }{\partial t},A_j^{\prime }=A_j+\frac{\partial \psi }{\partial x_j}(j=1,2,\dots ,d).$$

$$K_{\Delta }^{\prime }(t,0)f=e^{i\psi (t,\cdot )}{K}_{\Delta }(t,0)\left(e^{-i\psi (0,\cdot )}f\right)$$

$${\mathcal{L}}_s(t,x,\dot{x})=\mathcal{L}(t,x,\dot{x})-H_1(t,x),$$

$$i\frac{\partial u}{\partial t}(t)=\bigl{[}H(t)I+H_{1}(t)\bigr{]}u(t),$$

Full text

“ a polynomial of degree $(2l+1)$ in $x$ ” ($l=0,1,2,...$) and $A_{j}(t,x)$ are polynomials of degree $l$ in $x$. The Feynman path integrals are defined as $L^{2}$-valued continuous functions with respect to the time variable.

Department of Mathematics, Shinshu University, Matsumoto 390-8621, Japan.

E-mail: ichinose@math.shinshu-u.ac.jp

1 Introduction

Let $T>0$ be an arbitrary constant, $0\leq t\leq T$ and $x=(x_{1},\dotsc,x_{d})\in\mathbb{R}^{d}$. Let $E(t,x)=(E_{1},\dots,E_{d})\in\mathbb{R}^{d}$ and $B(t,x)=(B_{jk}(t,x))_{1\leq j<k\leq d}\in\mathbb{R}^{d(d-1)/2}$ denote the electric strength and the magnetic strength tensor, respectively and $(V(t,x),A(t,x))=(V,A_{1},\dots,A_{d})\in\mathbb{R}^{d+1}$ an electromagnetic potential, i.e.

[TABLE]

where $\partial V/\partial x=(\partial V/\partial x_{1},\dots,\partial V/\partial x_{d})$. Then the Lagrangian function is given by

[TABLE]

with mass $m>0$ and charge $e\in\mathbb{R}$. Then the corresponding Schrödinger equation is given by

[TABLE]

where $\hbar$ is the Planck constant. Throughout this paper we always consider solutions to the Schrödinger equations in the sense of distribution. Hereafter we suppose $\hbar=1$ and $e=1$ for simplicity.

Let $L^{2}=L^{2}(\mathbb{R}^{d})$ denote the space of all square integrable functions on $\mathbb{R}^{d}$ with inner product $(f,g):=\int f(x)g(x)^{*}dx$ and norm $\|f\|$, where $g(x)^{*}$ denotes the complex conjugate of $g(x)$. Let $S(t,s;q)$ be the classical action

[TABLE]

for a path $q(\theta)\in\mathbb{R}^{d}\ (s\leq\theta\leq t)$, where $\dot{q}(\theta)=dq(\theta)/d\theta$. Our aim in the present paper is to prove that for any $f\in L^{2}$ we can determine the Feynman path integral

[TABLE]

in $L^{2}(\mathbb{R}^{d})$ for the system (1.2) with a potential $(V,A)$ growing polynomially in the spatial direction $x$. As shown in Example 3.1, a typical example of potentials that we can handle is

[TABLE]

[TABLE]

with an integer $l\geq 0$ and functions $a_{\alpha}(t)\in\mathbb{R},b_{j\alpha}(t)\in\mathbb{R}$ in $C^{1}([0,T])$, i.e. continuously differentiable functions, where $|x|^{2}=\sum_{j=1}^{d}x_{j}^{2}$ and for a multi-index $\alpha=(\alpha_{1},\dots,\alpha_{d})$ we write $|\alpha|=\sum_{j=1}^{d}\alpha_{j}$, $x^{\alpha}=x_{1}^{\alpha_{1}}\cdots x_{d}^{\alpha_{d}},\partial_{x_{j}}=\partial/\partial x_{j}$ and $\partial_{x}^{\alpha}=\partial_{x_{1}}^{\alpha_{1}}\cdots\partial_{x_{d}}^{\alpha_{d}}$.

In the present paper the Feynman path integral (1.5) is defined by the time-slicing method in terms of piecewise free moving paths or piecewise straight lines. The time-slicing approach in terms of piecewise free moving paths to path integrals is actually the classical approach in the physics literature (cf. p.32 in [8] and p.278 in [20]).

The Feynman path integral for the system (1.2) with potentials $A=0$ and $V$ satisfying $|V(t,x)|\leq C(1+|x|^{2})$ has been studied mathematically by many authors for a long time since Feynman had published his famous paper [7] in 1948 (cf. §10 in [1] and [4]). See [9], [10], [11], [12], [19] and their references for the recent study. We note that in [10], [11] and [12] we studied the Feynman path integrals defined by the time-slicing method in terms of piecewise free moving paths.

On the other hand, if $|V(t,x)|\geq C(1+|x|^{2})^{1+\delta}$ holds with positive constants $C$ and $\delta$, it may be not simple to construct the Feynman path integral for (1.2) mathematically as stated in §10.2 of [1] and in §3.5 of [16]. In fact, there seems to be only a few papers on it, which will be referred below, as far as the author knows. In addition, we note that as well known, the uniqueness of solutions to (1) with $u(0)=f$ in the space $C^{0}_{t}([0,T];L^{2})$ doesn’t hold in general if $V(t,x)$ satisfies $V(t,x)\leq-C(1+|x|^{2})^{1+\delta}$ with positive constants $C$ and $\delta$ (cf. pp. 157-159 in [5], Theorem VIII.7 in [21], Theorems X.2 and X.3 in [22]), where $C^{0}_{t}([0,T];L^{2})$ denotes the space of all $L^{2}$-valued continuous functions in $t\in[0,T]$.

Nelson in [18] has constructed the Feynman path integral (1.5) for (1.2) in $L^{2}$ for $f\in L^{2}$ with $A=0$ and a continuous function $V(x)$ outside a set of capacity [math] in $\mathbb{R}^{d}$, independent of $t\in[0,T]$, by using the Trotter product formula. It is to be noted that in [18] the classical action $S(t,0;q)$ is replaced with a certain approximation. See (9) on p. 333 of [18].

Daubechies and Klauder in [6] have showed the following. Take $(V,A)=(V(x),A(x))$, independent of $t$, satisfying $|V(x)|\leq C(1+|x|^{2})^{M}$ and $|A(x)|\leq C(1+|x|^{2})^{M}$ with constants $C\geq 0$ and $M\geq 0$. Let $H$ be the operator defined by (1) with a core consisting of finite linear spans generated by eigenvalues of $(-\Delta+|x|^{2})/2$, where $\Delta=\sum_{j=1}^{d}\partial_{x_{j}}^{2}$. Let $F_{0}(x)$ be the ground state of $(-\Delta+|x|^{2})/2$ and define the canonical coherent states $|p,q>=e^{ip\cdot x}F_{0}(x-q)$ for all $(q,p)\in\mathbb{R}^{2d}$, where $p\cdot x=\sum_{j=1}^{d}p_{j}x_{j}$. Let $H^{\prime}$ be a maximal extension of $H$ on $L^{2}$ and denote the deficiency indices of $H^{\prime}$ by $n_{+}(H^{\prime})$ and $n_{-}(H^{\prime})$. Daubechies and Klauder have constructed the phase space Feynman path integral in the form of weak topology of $L^{2}$, i.e. giving $(|p^{\prime\prime},q^{\prime\prime}>,e^{-itH^{\prime}}|p^{\prime},q^{\prime}>)$ if $n_{+}(H^{\prime})=0$ and $(|p^{\prime\prime},q^{\prime\prime}>,e^{-itH^{\prime{\dagger}}}|p^{\prime},q^{\prime}>)$ if $n_{-}(H^{\prime})=0$ in terms of the Winer measure pinned at $(p^{\prime},q^{\prime})$ at $t=0$ and at $(p^{\prime\prime},q^{\prime\prime})$ at $t$, where $H^{\prime{\dagger}}$ denotes the adjoint operator of $H^{\prime}$.

Albeverio and Mazzucchi in [2] and [3] have studied the Feynman path integrals for the systems (1.2) with $A=0$, $V(x)=|\Omega x|^{2}/2+\lambda C(x,x,x,x)\ (\lambda\in\mathbb{R})$ and with $A=0$, a positive homogeneous polynomial $V(x)$ of $2M$-order $(M=1,2,\dots)$, respectively in terms of infinite dimensional oscillatory integrals and the Wiener measure, where $\Omega$ is a $d\times d$ regular matrix and $C(x,y,w,z)$ is a completely symmetric positive fourth-order covariant tensor on $\mathbb{R}^{d}$. It is noted that all Feynman path integrals in [2, 3] are defined in the form of weak topology of $L^{2}$. See §10.2 in [1] and §3.5 in [16] for topics relating to [2, 3].

The present paper is having four points to be emphasized: (1) In our system (1.2) there exists a magnetic field $B(t,x)$. (2) Our magnetic field $B(t,x)$ and electric field $E(t,x)$ can vary on time $t$. (3) Our Feynman path integral can be defined as an $L^{2}$- valued function on $[0,T]$ as in [18], not in the form of weak topology of $L^{2}$ as in [2, 3, 6]. (4) Our method of constructing the Feynman path integral can not be applied to systems with potentials satisfying $V(t,x)\leq-C(1+|x|^{2})^{1+\delta}\ (C>0,\delta>0)$, though in [2, 6, 18] the Feynman path integrals for such systems are constructed.

In the present paper the Feynman path integrals will be constructed not only for the one-particle systems (1.2), but also the multi-particle systems with spin. In addition, we will construct the Feynman path integrals for bosons and fermions, i.e. quantum systems consisting of many identical particles with spin.

We will prove the results in the present paper, following the proofs in [10, 11, 12, 13]. That is, we introduce the fundamental operator $\mathcal{C}(t,s)$ in §5, and prove its stability and consistency. Combining these results and the existence theorem proved in [14] to the Schrödinger equations (1) in both of $L^{2}$ and the Schwartz space ${\cal S}(\mathbb{R}^{d})$ of all rapidly decreasing functions on $\mathbb{R}^{d}$, we can prove our main results. In particular, in the present paper we will use the delicate result below concerning the $L^{2}$-boundedness of pseudo-differential operators, which is stated as Theorem 13.13 on p. 322 in [24].

Theorem 1.A. Suppose $p(x,\xi,x^{\prime})\in S^{0}(\mathbb{R}^{3d})$, i.e.

[TABLE]

for all $\alpha,\beta$ and $\gamma$. Let $P(X,\hbar D_{x},X^{\prime})$ be the pseudo-differential operator defined by

[TABLE]

for $f\in{\cal S}(\mathbb{R}^{d})$. Then we have

[TABLE]

where $\|P\|_{L^{2}\to L^{2}}$ denotes the operator norm from $L^{2}$ into $L^{2}$.

The plan of the present paper is as follows. In §2 our main results are stated. In §3 we will state examples to which our results can be applied. In §4 we will construct the Feynman path integrals for bosons and fermions. In §5 and §6 the stability and the consistency of $\mathcal{C}(t,s)$ will be proved, respectively. In §7 Theorems 2.1 - 2.2 and in §8 Theorems 2.3 - 2.4 will be proved.

2 Main theorems

Let $t$ in $[0,T]$. For an arbitrary integer $\nu\geq 1$ we take $\tau_{j}\in[0,T]\ (j=1,2,\dots,\nu-1)$ satisfying $0=\tau_{0}<\tau_{1}<\dots<\tau_{\nu-1}<\tau_{\nu}=t$, set $\Delta:=\{\tau_{j}\}_{j=1}^{\nu-1}$ and write $|\Delta|:=\max\{\tau_{j+1}-\tau_{j};j=0,1,\dots,\nu-1\}$. Let $x\in\mathbb{R}^{d}$ be fixed. We take arbitrary points $x^{(j)}\in\mathbb{R}^{d}\ (j=0,1,\dotsc,\nu-1)$ and determine the piecewise free moving path or the piecewise straight line $q_{\Delta}(\theta;x^{(0)},\dotsc,x^{(\nu-1)},x)\in\mathbb{R}^{d}\ (0\leq\theta\leq t)$ by joining $x^{(j)}$ at $\tau_{j}\ (j=0,1,\dotsc,\nu,x^{(\nu)}=x)$ in order. Let $\mathcal{L}(t,x,\dot{x})$ be the Lagrangian function defined by (1.2) and $S(t,s;q)$ the classical action defined by (1.4). Take $\chi\in C^{\infty}_{0}(\mathbb{R}^{d})$, i.e. an infinitely differentiable function on $\mathbb{R}^{d}$ with compact support, such that $\chi(0)=1$ and determine the approximation of the Feynman path integral (1.5) for $f\in C^{\infty}_{0}(\mathbb{R}^{d})$ by

[TABLE]

From now on we always suppose that $\chi$ is a real-valued function belonging to $C^{\infty}_{0}(\mathbb{R}^{d})$ such that $\chi(0)=1$. The RHS is an oscillatory integral and will be denoted by

[TABLE]

(cf. p. 45 of [15]).

In the present paper we often use symbols $C,C_{\alpha},C_{\alpha,\beta}$, $C_{a}$ and $\delta$ to write down constants, though these values are different in general.

**Assumption 2.1. ** We assume that $\partial_{x}^{\alpha}\partial_{t}^{k}V(t,x)$ and $\partial_{x}^{\alpha}\partial_{t}^{k}A_{j}(t,x)\ (j=1,2,\dots,d)$ are continuous in $[0,T]\times\mathbb{R}^{d}$ for all $\alpha$ and $k=0,1$. Moreover, we assume the existence of constants $M_{*}\geq 0,C_{0}>0,C_{1}\geq 0,C_{2}\geq 0$ with

[TABLE]

in $[0,T]\times\mathbb{R}^{d}$, where $<x>=\sqrt{1+|x|^{2}}$. We also assume

[TABLE]

[TABLE]

for all $\alpha$,

[TABLE]

with a constant $\delta>0$ and

[TABLE]

for all $\alpha$.

**Assumption 2.2. ** Let $M_{*}$ be the constant in Assumption 2.1. We assume

[TABLE]

with constants $C_{*}>0$ and $C_{1}\geq 0$, and

[TABLE]

where $\partial E/\partial x=(\partial E_{i}/\partial x_{j};i\downarrow j\rightarrow 1,2,\dots,d)$ is a $d\times d$ matrix and ${}^{t}\partial E/\partial x$ its transposed matrix. We assume

[TABLE]

In addition, we assume either

[TABLE]

with constants $\delta_{\alpha}>0$ and

[TABLE]

for all $\alpha$, or

[TABLE]

with constants $\delta_{\alpha}>0$ if $0\leq M_{*}<1$ and

[TABLE]

if $M_{*}\geq 1$.

**      Theorem 2.1****.**

Suppose that Assumptions 2.1 and 2.2 are satisfied. Then there exist constants $\rho^{*}>0$ and $K\geq 0$ such that the following statements hold for all $\Delta$ satisfying $|\Delta|\leq\rho^{*}$ and all $t\in[0,T]$:

(1) $K_{\Delta}(t,0)f$ defined on $f\in C^{\infty}_{0}(\mathbb{R}^{d})$ by (2) is determined independently of the choice of $\chi$ and $K_{\Delta}(t,0)f$ can be uniquely extended to a bounded operator on $L^{2}$ with

[TABLE]

for all $f\in L^{2}$.

(2) For all $f\in L^{2}$, as $|\Delta|\to 0$, $K_{\Delta}(t,0)f$ converges in $L^{2}$ uniformly in $t\in[0,T]$ to an element $K(t,0)f\in L^{2}$, which we call the the Feynman path integral of $f$.

(3) For all $f\in L^{2}$, $K(t,0)f$ belongs to $C_{t}^{0}([0,T];L^{2})$. In addition, $K(t,0)f$ is the unique solution in $C^{0}_{t}([0,T];L^{2})$ to (1) with $u(0)=f$.

(4) Let $\psi(t,x)$ be a real-valued function such that $\partial_{x_{j}}\partial_{x_{k}}\psi(t,x)$ and $\partial_{t}\partial_{x_{j}}\psi(t,x)$ $(j,k=1,2,\dotsc,d)$ are continuous in $[0,T]\times\mathbb{R}^{d}$ and consider the gauge transformation

[TABLE]

We write (2) for this $(V^{\prime},A^{\prime})$ as $K^{\prime}_{\Delta}(t,0)f$. Then we have the formula

[TABLE]

for all $f\in L^{2}$, and we have the analogous relation between the limits $K^{\prime}(t,0)f$ and $K(t,0)f$ as in **[11]**.

Next we consider the Lagrangian function for the spin system

[TABLE]

where $H_{1}(t,x)=(h_{1jk}(t,x);j\downarrow k\rightarrow 1,2,\dots,l)$ is a Hermitian matrix of degree $l$ and $\mathcal{L}(t,x,\dot{x})$ the Lagrangian function defined by (1.2). Then the corresponding quantized equation is given by

[TABLE]

where $u(t)={}^{t}(u_{1}(t),\dots,u_{l}(t))\in\mathbb{C}^{l}$, $H(t)$ is the operator defined by (1) and $I$ the identity matrix of degree $l$.

For a continuous path $q(\theta)\in\mathbb{R}^{d}\ (s\leq\theta\leq t)$ let us define an $l\times l$ matrix $\mathcal{F}(\theta,s;q)\ (s\leq\theta\leq t)$ by the solution to

[TABLE]

Then, for the piecewise free moving path $q_{\Delta}(\theta;x^{(0)},x^{(1)},\dots,x^{(\nu-1)},x)$ we define the probability amplitude by

[TABLE]

using $S(t,s;q)$ defined by (1.4). Let $f={}^{t}(f_{1},f_{2},\dots,f_{l})\in C^{\infty}_{0}(\mathbb{R}^{d})^{l}$. Then we define the approximation $K_{s\Delta}(t,0)f$ of the Feynman path integral $K_{s}(t,0)f$ for the system (2.17) by replacing $e^{iS(t,0;q_{\Delta})}$ in (2) with $e^{*iS_{s}(t,0;q_{\Delta})}$ as in [12].

**      Theorem 2.2****.**

Besides Assumptions 2.1 and 2.2 we assume

[TABLE]

for all $\alpha$. Let $\rho^{*}>0$ be the constant in Theorem 2.1. Then we get the same assertions for $K_{s\Delta}(t,0)f$ as for $K_{\Delta}(t,0)f$ in Theorem 2.1 with another constant $K\geq 0$, where $K_{s}(t,0)f=\lim_{|\Delta|\to 0}K_{s\Delta}(t,0)f\in C^{0}_{t}([0,T];(L^{2})^{l})$ for $f\in(L^{2})^{l}$ is the unique solution in $C^{0}_{t}([0,T];(L^{2})^{l})$ to (2.18) with $u(0)=f$.

*      Remark 2.1**.*

Since we see from (2.19) that $e^{iS(t,0;q)}\mathcal{F}(t,0;q)$ is the solution to

[TABLE]

we can write $\exp*iS_{s}(t,0;q_{\Delta})$ formally as $\exp i\displaystyle{\int_{0}^{t}}\mathcal{L}_{s}(\theta,q_{\Delta}(\theta),\dot{q}_{\Delta}(\theta)d\theta.$ This is the reason why we express the right-hand side of (2.20) as $\exp*iS_{s}(t,0;q_{\Delta})$.

*      Remark 2.2**.*

We write

[TABLE]

for $x$ and $y$ in $\mathbb{R}^{d}$ when $s\not=t$. Then from Lemma 2.1 of [12] we have

[TABLE]

*      Remark 2.3**.*

Letting $M_{*}=0$, we assume (2.8), (2.10) and (2.21). Let $(V(t,x),A(t,x))$ be an arbitrary potential such that $V,\partial V/\partial x_{j},\partial A_{j}/\partial t$ and $\partial A_{j}/\partial x_{k}\ (j,k=1,2,\dots,d)$ are continuous in $[0,T]\times\mathbb{R}^{d}$. Then we have proved in [11] and [12] the same assertions as in Theorems 2.1 and 2.2 . Aside from this, letting $M_{*}=0$, we assume (2.8), (2.9), (2.12), (2.21),

[TABLE]

and

[TABLE]

for a constant $M\geq 0$. Using (1), from (2.8) and (2.23) we have

[TABLE]

for all $\alpha$. We note (3.3) in [13] or (5.2) in the present paper. Then, under the assumptions above we can prove the same assertions as in Theorems 2.1 and 2.2 as in the proofs of the theorems stated in [11] and [12].

In the end we will consider the multi-particle system. For simplicity we will consider the 4-particle system

[TABLE]

where $x=(x(1),x(2),x(3),x(4))\in\mathbb{R}^{4d}$. The corresponding Schrödinger equation is given by

[TABLE]

Assumption 2.3. (1) Each \bigl{(}V_{l}(t,x(l)),A^{(l)}(t,x(l))\bigr{)}\ (l=1,2) satisfies Assumption 2.1 with $M_{*}=M_{l*}>0$. (2) Each $V_{l}(t,x(l))\ (l=3,4)$ satisfies (2.23) and (2.24).

We define $E^{(l)}(t,x(l))$ and $B^{(l)}(t,x(l))\ (l=1,2,3,4)$ by (1) where $A=A^{(l)}$ and $V=V_{l}$.

Assumption 2.4. Let $M_{l*}\ (l=1,2)$ be the constants in Assumption 2.3 and $M_{l*}=0\ (l=3,4)$. (1) $A^{(l)}$ and $(E^{(l)},B^{(l)})\ (l=1,2,3,4)$ satisfy Assumption 2.2 with $M_{*}=M_{l*}$. (2) $V_{jk}(t,z)\ (z\in\mathbb{R}^{d})$ satisfies

[TABLE]

for all $1\leq j<k\leq 4.$

We define the approximation $K_{\Delta}^{\sharp}(t,0)f$ of the Feynman path integral $K^{\sharp}(t,0)f$ for the 4-particle system (2) in the same way as (2).

**      Theorem 2.3****.**

Suppose Assumptions 2.3 and 2.4. Then we have the same assertions for $K_{\Delta}^{\sharp}(t,0)f$ as for $K_{\Delta}(t,0)f$ in Theorem 2.1 with other constants $\rho^{*}>0$ and $K\geq 0$, where the Feynman path integral $K^{\sharp}(t,0)f\in C^{0}_{t}([0,T];L^{2})$ for $f\in L^{2}(\mathbb{R}^{4d})$ is the unique solution in $C^{0}_{t}([0,T];L^{2})$ to (2) with $u(0)=f$.

Let us consider the spin system. Taking a Hermitian matrix $H_{1}(t,x)=(h_{1jk}(t,x);j\downarrow k\rightarrow 1,2,\dots,l_{0})\ (x\in\mathbb{R}^{4d})$ of degree $l_{0}$ and using $\mathcal{L}^{\sharp}(t,x,\dot{x})$ defined by (2), we determine

[TABLE]

For a path $q(\theta)\in\mathbb{R}^{4d}\ (s\leq\theta\leq t)$ we define $\mathcal{F}^{\sharp}(\theta,s;q)$ by the solution to (2.19). Then we define $\exp*iS^{\sharp}_{s}(t,0;q_{\Delta})$ by (2.20) and $K^{\sharp}_{s\Delta}(t,0)f$ for $f\in C^{\infty}_{0}(\mathbb{R}^{4d})^{l_{0}}$ in the same way as we did $K_{s\Delta}(t,0)f$.

**      Theorem 2.4****.**

Besides Assumptions 2.3 and 2.4 we assume (2.21). Let $\rho^{*}>0$ be the constant in Theorem 2.3. Then we have the same assertions for $K^{\sharp}_{s\Delta}(t,0)f$ as for $K_{s\Delta}(t,0)f$ in Theorem 2.2 with another constant $K\geq 0.$

3 Examples

In this section we will give some examples satisfying Assumptions 2.1 and 2.2 in §2.

**      Lemma 3.1****.**

Let $f\in C^{2}([0,\infty))$ and set

[TABLE]

Let $x\not=0$ be an arbitrary point in $\mathbb{R}^{d}$. Then there exists an orthogonal matrix $\mathfrak{R}$ such that

[TABLE]

Proof.

Let $\mathfrak{R}=(R_{ij};i\downarrow j\rightarrow 1,2,\dots,d)$ be an orthogonal matrix. Then we have

[TABLE]

which shows

[TABLE]

and so

[TABLE]

Hence

[TABLE]

On the other hand, from (3.1) we see

[TABLE]

and so

[TABLE]

Consequently, letting $\overrightarrow{e_{1}}=(1,0,\dots,0)\in\mathbb{R}^{d}$, we have

[TABLE]

Let $x\not=0$ be an arbitrary point in $\mathbb{R}^{d}$. Then we can take an orthogonal matrix $\mathfrak{R}$ such that $\mathfrak{R}x=|x|\overrightarrow{e_{1}}$. Then from (3.8) we have

[TABLE]

and hence have (3.1) from (3.9). ∎

From Lemma 3.1 we can easily get the following.

**      Corollary 3.2****.**

Let $f\in C^{2}([0,\infty))$ such that

[TABLE]

We define $V(x)$ by (3.1). Then we have

[TABLE]

**      Proposition 3.3****.**

Let $f\in C^{2}([0,\infty))$ satisfying (3.10) and $f^{\prime}(0)\geq 0$. Let $\mathfrak{A}$ be a regular real matrix. We denote the smallest eigenvalue of ${}^{t}\mathfrak{A}\mathfrak{A}$ by $\beta>0$. We set

[TABLE]

Then we have

[TABLE]

Proof.

Setting $W(x)=f(|x|^{2})$, we have $V(x)=W(\mathfrak{A}x),$ which shows

[TABLE]

as in the proof of (3.8). Hence

[TABLE]

for $u\in\mathbb{R}^{d}$, where $(\cdot,\cdot)$ is the inner product in $\mathbb{R}^{d}$. Since we have $\partial^{2}W(x)/\partial x^{2}\geq 2f^{\prime}(|x|^{2})I$ from (3.11), we have

[TABLE]

which leads to

[TABLE]

because of $f^{\prime}(\theta)\geq 0\ (0\leq\theta<\infty)$. Hence we obtain the first inequality of (3.13). The second inequality follows from the fact that $f^{\prime}(\theta)$ is an increasing function and $|\mathfrak{A}x|^{2}=({}^{t}\mathfrak{A}\mathfrak{A}x,x)\geq\beta|x|^{2}$. ∎

*      Example 3.1**.*

Let $(V,A)$ be the potential defined by (1.6) and (1.7) with an integer $M\geq 0$, real-valued $a_{\alpha}(t)$ and $b_{j\alpha}(t)$ in $C^{1}([0,T])$. I will prove that this $(V,A)$ satisfies Assumptions 2.1 and 2.2 with the integer $M_{*}=M$.

Noting (1), we can easily see that we have only to prove (2.7). Letting $f(\theta)=\theta^{M+1}$ in Corollary 3.2, we have

[TABLE]

Hence we can prove (2.7), because we have

[TABLE]

from (1), (1.6) and (1.7).

*      Example 3.2**.*

Let $\mathfrak{A}(t)$ be a regular real matrix whose components are continuously differentiable on $[0,T]$. We set

[TABLE]

with an integer $M\geq 0$, where we assume

[TABLE]

for all $\alpha$ with constants $\delta_{\alpha}>0$ and

[TABLE]

for all $\alpha$. In addition, we assume

[TABLE]

for all $\alpha$ and

[TABLE]

for all $\alpha$. Then this potential $(V,A)$ satisfies Assumptions 2.1 and 2.2 with $M_{*}=M$.

In fact we have only to prove (2.7) as in the arguments in Example 3.1. Letting $f(\theta)=\theta^{M+1}$ in Proposition 3.3, we have

[TABLE]

with $\beta>0$. Hence we can prove (2.7) as in the proof of Example 3.1.

*      Example 3.3**.*

Let $\mathfrak{A}(t)$ be the matrix in Example 3.2. We set

[TABLE]

with a constant $M\geq 0$, where $V_{1}(t,x)$ is assumed to satisfy (3.16) and (3.17). Suppose that $A(t,x)$ satisfies (3.18) and (3.19). In addition, when $M$ in (3.21) is in $(0,1)$, we assume (3.18) with $M=0$. Then this potential $(V,A)$ satisfies Assumptions 2.1 and 2.2. In fact we have only to prove (2.7). Letting $f(\theta)=(1+\theta)^{M+1}$ in Proposition 3.3, we have

[TABLE]

Hence we can prove (2.7) as in the proof of Example 3.2.

4 The Feynman path integrals for bosons and fermions

In this section we consider the quantum spin system consisting of $N$ particles. We write $x=(\mathbf{x}_{1},\mathbf{x}_{2},\dots,\mathbf{x}_{N})\in\mathbb{R}^{3N}$. The Lagrangian function is given by

[TABLE]

in terms of the tensor product, where $\mathbf{A}_{i}\in\mathbb{R}^{3},\mathbf{B}_{i}\in\mathbb{R}^{3},V_{i}\in\mathbb{R},V_{jk}\in\mathbb{R},$ $\widehat{\mathbf{s}}_{i}=(\hat{s}_{1},\hat{s}_{2},\hat{s}_{3})$ are spin matrices with three components and $I_{j}$ the identity matrix for the j-th particle. In particular we suppose that all particles are identical. Hence we suppose $m_{i}=m,e_{i}=e,\mathbf{A}_{i}=\mathbf{A},\mathbf{B}_{i}=\mathbf{B},V_{i}=V,V_{jk}=W$ and $\widehat{\mathbf{s}}_{i}=\widehat{\mathbf{s}}$. Let $L$ be the magnitude of spin of particles. We note that the N-fold tensor product $L^{2}(\mathbb{R}^{3})^{2L+1}\otimes\cdots\otimes L^{2}(\mathbb{R}^{3})^{2L+1}$ is isomorphic to $L^{2}(\mathbb{R}^{3N})^{l}$ with $l=(2L+1)^{N}$ (cf. Theorem II.10 on p. 52 in [21]), which we write as $\mathfrak{H}$. The Schrödinger equation for the Lagrangian (4) is given by

[TABLE]

We note that if $W=0$ and $u(t,x)=u_{1}(t,\mathbf{x}_{1})\otimes\cdots\otimes u_{N}(t,\mathbf{x}_{N})$, (4) is written as

[TABLE]

where $H_{j}(t)=|i^{-1}\partial_{\mathbf{x}_{j}}-e\mathbf{A}(t,\mathbf{x}_{j})|^{2}/(2m)+eV(t,\mathbf{x}_{j})-e\mathbf{B}(t,\mathbf{x}_{j})\cdot\widehat{\mathbf{s}}/m$.

Let $S^{\sharp}_{i}(t,s;q)$ be the classical action for $\mathcal{L}^{\sharp}_{i}(t,x,\dot{x})$ defined by (4). We define the approximation $K^{\sharp}_{is\Delta}(t,0)f$ of the Feynman path integral $K^{\sharp}_{is}(t,0)f$ for (4) in the same way as we did $K^{\sharp}_{s\Delta}(t,0)f$ before Theorem 2.4, where f=\bigl{\{}f(\mathbf{x}_{1},s_{1},\mathbf{x}_{2},s_{2},\dots,\mathbf{x}_{N},s_{N});s_{j}=-L,-L+1,\dots,L\ (j=1,2,\dots,N)\bigr{\}}\in C^{\infty}_{0}(\mathbb{R}^{3N})^{l}. That is, we define $\mathcal{F}^{\sharp}_{i}(\theta,s;q)$ for a path $q(\theta)\in\mathbb{R}^{3N}\ (s\leq\theta\leq t)$ by the solution

[TABLE]

where $H_{1}(t,x)=-\sum_{j=1}^{N}I_{1}\otimes\cdots\otimes I_{j-1}\otimes e\mathbf{B}(t,\mathbf{x}_{j})\cdot\widehat{\mathbf{s}}/m\otimes I_{j+1}\otimes\cdots\otimes I_{N}$. Next we define the probability amplitude by (2.20) and eventually define $K^{\sharp}_{is\Delta}(t,0)f$ by (2).

We define $\mathcal{F}(\theta,s;\mathbf{q})\ (s\leq\theta\leq t)$ for a continuous path $\mathbf{q}(\theta)\in\mathbb{R}^{3}\ (s\leq\theta\leq t)$ by the solution

[TABLE]

Then we can easily have

[TABLE]

Hence we have

[TABLE]

because of the uniqueness of solutions to the ordinary differential equation, where $\mathbf{B}(t,\mathbf{x})$ are assumed to be continuous in $[0,T]\times\mathbb{R}^{3}$.

Let $\widehat{P}_{ij}\ (i,j=1,2,\dots,N)$ be the operator exchanging the $i$-th particle and the $j$-th one. That is, we define

[TABLE]

for $f_{j}(\mathbf{x}_{j})\in L^{2}(\mathbb{R}^{3})^{2L+1}\ (j=1,2,\dots,N)$ and extend $\widehat{P}_{ij}$ for $f=\sum_{n=1}^{\infty}f_{1}^{(n)}(\mathbf{x}_{1})\otimes\cdots\otimes f_{N}^{(n)}(\mathbf{x}_{N})\in\mathfrak{H}$.

The following theorem shows that the Feynman path integrals $K^{\sharp}_{is}(t,0)f$ are expressing bosons and fermions.

**      Theorem 4.1****.**

Assume that $(V(t,\mathbf{x}),\mathbf{A}(t,\mathbf{x}))\ (\mathbf{x}\in\mathbb{R}^{3}),\mathbf{B}(t,\mathbf{x})$ and $W(t,\mathbf{x})$ satisfy Assumptions 2.1-2.2, (2.21) and (2.28), respectively. Then we have: (1) The same assertions for $K^{\sharp}_{is\Delta}(t,0)f$ as for $K_{\Delta}(t,0)f$ in Theorem 2.1 hold, where $K^{\sharp}_{is}(t,0)f\in C^{0}_{t}([0,T];\mathfrak{H})\ (f\in\mathfrak{H})$ is the unique solution in $C^{0}_{t}([0,T];\mathfrak{H})$ to (4) with $u(0)=f$. (2) If $f\in\mathfrak{H}$ is symmetric, i.e. $\hat{P}_{ij}f=f$ for all $i$ and $j$, so is $K^{\sharp}_{is}(t,0)f$. (3) If $f\in\mathfrak{H}$ is antisymmetric , i.e. $\hat{P}_{ij}f=-f$ for all $i\not=j$, so is $K^{\sharp}_{is\Delta}(t,0)f$.

Proof.

The first assertion (1) follows from Theorem 2.4. Let’s prove the second assertion. For simplicity suppose $N=2$. Let $c(\mathbf{x}_{1},\mathbf{x}_{2})\in\mathbb{C}$ be a bounded measurable function. Then we can prove

[TABLE]

from the definition of $\hat{P}_{12}$, approximating $c(\mathbf{x}_{1},\mathbf{x}_{2})$ by $\sum_{j=1}^{n}c_{1}^{(j)}(\mathbf{x}_{1})c_{2}^{(j)}(\mathbf{x}_{2})$.

Let $0\leq t-s\leq\rho^{*}$. Setting $\mathbf{q}_{j}(\theta):=\mathbf{q}^{t,s}_{\mathbf{x}_{j},\mathbf{y}_{j}}(\theta)\ (s\leq\theta\leq t,j=1,2)$, we write

[TABLE]

for $f_{j}(\mathbf{x}_{j})\in C^{\infty}_{0}(\mathbb{R}^{3})^{2L+1}\ (j=1,2)$, which belongs to $L^{2}(\mathbb{R}^{6})^{l}$ from (1) of Theorem 2.4. From (4.4) we can write

[TABLE]

Making the same arguments as in the proof of (4.6), by the exchange of $(\mathbf{x}_{1},\mathbf{y}_{1})$ and $(\mathbf{x}_{2},\mathbf{y}_{2})$ in the above equation we can prove

[TABLE]

Letting $\epsilon\to 0$, we have

[TABLE]

Using $S^{\sharp}_{i}(t,s;\mathbf{q}_{1},\mathbf{q}_{2})=S^{\sharp}_{i}(t,s;\mathbf{q}_{2},\mathbf{q}_{1})$, we have

[TABLE]

We have proved in (1) of Theorem 2.4 that $\mathcal{C}^{\sharp}_{is}(t,s)$ is a bounded operator on $\mathfrak{H}$. Hence from (4) we see

[TABLE]

for $f\in\mathfrak{H}$.

Noting Remark 2.2, from (2) and (2.20) we can write

[TABLE]

for $f\in C^{\infty}_{0}(\mathbb{R}^{6})^{l}$. Since we have

[TABLE]

by (1) of Theorem 2.4 we obtain

[TABLE]

for $f\in C^{\infty}_{0}(\mathbb{R}^{6})^{l}$ and so for $f\in\mathfrak{H}$. Therefore, by (4.10) we have

[TABLE]

Since $\hat{P}_{12}f=f$ holds from the assumption, we obtain

[TABLE]

This shows that $K^{\sharp}_{is\Delta}(t,0)f$ is symmetric, which completes the proof of the second assertion. In the same way the third assertion is proved from (4.12). ∎

*      Remark 4.1**.*

We have supposed Assumptions 2.1 and 2.2 for $(V(t,\mathbf{x}),\mathbf{A}(t,x))$ in Theorem 4.1. In place of these assumptions we suppose the assumptions stated in Remark 2.3 for $(V,\mathbf{A})$. Then we can prove the same assertions as in Theorem 4.1 as in the proof of Theorem 4.1.

5 Stability of $\mathcal{C}(t,s)$

Let $\mathcal{L}(t,x,\dot{x})$ and $S(t,s;q)$ be the Lagrangian function and the classical action defined by (1.2) and(1.4), respectively. Let $q^{t,s}_{x,y}$ be the path defined by (2.22) and write

[TABLE]

Then we have

[TABLE]

Let $M\geq 0$ and suppose that $p(x,w)\in C^{\infty}(\mathbb{R}^{2d})$ satisfies

[TABLE]

for all $\alpha$ and $\beta$, where $<x;w>=\sqrt{1+|x|^{2}+|w|^{2}}$. We write the semi-norms of ${\cal S}={\cal S}(\mathbb{R}^{d})$ as |f|_{l}=\sum_{|\alpha+\beta|\leq l}\sup\bigl{\{}|x^{\alpha}\partial_{x}^{\beta}f(x)|;x\in\mathbb{R}^{d}\bigr{\}}\ (l=0,1,2,\dots). For $f\in{\cal S}$ we define

[TABLE]

Then the formal adjoint operator $P(t,s)^{{\dagger}}$ of $P(t,s)$ on ${\cal S}$ is given by

[TABLE]

We have the following from Lemma 2.1 of [11].

**      Lemma 5.1****.**

We define $P(t,s)f$ by (5.4) for $f\in{\cal S}$. Assume (2.3) and (2.9). Then, $\partial_{x}^{\alpha}(P(t,s)f)$ are continuous in $0\leq s\leq t\leq T$ and $x\in\mathbb{R}^{d}$ for all $\alpha$.

Taking $1$ as $p(x,w)$ in (5.4), for $f\in{\cal S}$ we define

[TABLE]

Using Lemma 5.1, we can write $K_{\Delta}(t,0)f$ defined by (2) as

[TABLE]

for $f\in{\cal S}$ under the assumptions of Lemma 5.1.

**      Lemma 5.2****.**

We assume that $\partial_{x}^{\alpha}V(t,x),\partial_{x}^{\alpha}A_{j}(t,x)$ and $\partial_{x}^{\alpha}\partial_{t}A_{j}(t,x)$ are continuous in $[0,T]\times\mathbb{R}^{d}$ for $|\alpha|\leq 1$. Let $p(x,w)$ be a function satisfying (5.3). Then for any $0<\epsilon\leq 1$ and $0\leq s<t\leq T$ we have

[TABLE]

[TABLE]

[TABLE]

or

[TABLE]

where

[TABLE]

Proof.

We have proved (5.2), (5.9) and (5.2) in Proposition 3.3 of [10] and Lemma 2.2 of [11]. So we will prove (5.2), (5.9) and (5.2), though these have been proved in Lemma 3.1 of [13] in essentials. Let $\mathbf{\Delta}$ be the 2-dimensional plane with oriented boundary consisting of $-\{(s,y+\theta(x-y));0\leq\theta\leq 1\},\{(s+\theta\rho,y+\theta(z-y));0\leq\theta\leq 1\}$ and $-\{(s+\theta\rho,x+\theta(z-x));0\leq\theta\leq 1\}$. Then we have

[TABLE]

Hence from the proof of Lemma 3.2 in [10] we have

[TABLE]

for all $x,y$ and $z$ in $\mathbb{R}^{d}$. Multiplying (5.13) by $(t-s)/m$ and adding this to $(x-y)\cdot\Phi$ where $\Phi$ is defined by (5.2), we have (5.2), (5.9) and (5.2). ∎

**      Lemma 5.3****.**

Assume (2.7) and let $C_{1}\geq 0$ be the constant in (2.7). Then, there exists a constant $C^{\prime}_{*}>0$ such that for all $X=(x,y,z)\in\mathbb{R}^{3d}$ we have

[TABLE]

Proof.

For a while we write

[TABLE]

and $X^{\prime}=(z,x-z,y-x)=|X^{\prime}|(\omega^{\prime}_{1},\omega^{\prime}_{2},\omega^{\prime}_{3})\in\mathbb{R}^{3d}$. From (2.7) we have

[TABLE]

for $X^{\prime}\not=0$. If the right-hand side of (5) is equal to zero for a point $(\omega^{\prime}_{1},\omega^{\prime}_{2},\omega^{\prime}_{3})$ such that $|\omega^{\prime}_{1}|^{2}+|\omega^{\prime}_{2}|^{2}+|\omega^{\prime}_{3}|^{2}=1$, we have

[TABLE]

for all $0<\sigma_{1}<1$ and $0\leq\sigma_{2}\leq 1$, which means $\omega^{\prime}_{1}=\omega^{\prime}_{2}=\omega^{\prime}_{3}=0$. This is contradiction. Hence there exists a constant $C^{\prime}_{*}>0$ such that

[TABLE]

which shows (5.3) with another constant $C^{\prime}_{*}>0$ because of $|X^{\prime}|\geq C|X|$ with a constant $C>0$. ∎

For a while we write the constant $C_{1}$ in (2.7) and (5.3) as $a$. Let us write

[TABLE]

[TABLE]

We can now easily prove the following auxiliary result.

**      Lemma 5.4****.**

(1) Let us define $\Phi(t,s;x,y,z)$ by (5.2). Then we have

[TABLE]

[TABLE]

(2) Let us define $\Phi(t,s;x,y,z)$ by (5.2). Then we have (5.4) where $B^{\prime}(t,s;x,y,z)$ is given by

[TABLE]

**      Lemma 5.5****.**

(1) Let us write the fifth term on the right-hand side of (5.2) as $\rho\widetilde{B}(t,s;x,y,z)$. Assume (2.10). Then we have

[TABLE]

(2) Let us write the fifth term on the right-hand side of (5.2) as $\rho\widetilde{B}(t,s;x,y,z)$. Assume (2.12). Then we have

[TABLE]

Proof.

Both of (1) and (2) are proved by Lemma 3.5 in [10]. ∎

The following lemma is crucial in the present paper.

**      Lemma 5.6****.**

We assume (2.7) and (2.8). (1) Let us define $\Phi$ by (5.2). Assume (2.12) if $0\leq M_{*}<1$ and (2.13) if $M_{*}\geq 1$. Then there exist constants $\rho^{*}>0,\delta>0$ and $C\geq 0$ such that for $0\leq t-s\leq\rho^{*}$ and $X=(x,y,z)\in\mathbb{R}^{3d}$ we have the estimates

[TABLE]

[TABLE]

where $|\Omega|$ denotes the Hilbert-Schmidt norm \bigl{(}\sum_{i,j=1}^{d}|\Omega_{ij}|^{2}\bigr{)}^{1/2} of a matrix $\Omega=(\Omega_{ij};i\downarrow j\rightarrow 1,2,\dots,d)$ . Furthermore, for all fixed $0\leq t-s\leq\rho^{*}$ and $(x,y)\in\mathbb{R}^{2d}$, the map: $\mathbb{R}^{d}\ni z\to\xi=\Phi(t,s;x,y,z)\in\mathbb{R}^{d}$ is a homeomorphism, whose inverse will be denoted by the map: $\mathbb{R}^{d}\ni\xi\to z=z(t,s;x,\xi,y)\in\mathbb{R}^{d}$. (2) Let us define $\Phi$ by (5.2). Assume (2.10) and (2.11). Then we have the same assertions as in (1).

Proof.

We will first prove (1). Lemma 5.3 and (5) show

[TABLE]

for all $X=(x,y,z)\in\mathbb{R}^{3d}$. Hence we have

[TABLE]

together with (2.8). We note Faraday’s law

[TABLE]

which follows from (1). Hence, using the assumption (2.12) if $0\leq M_{*}<1$ and (2.13) if $1\leq M_{*}$, from (5) we have

[TABLE]

for all $\alpha,\beta$ and $\gamma$. Here we used that if (2.12) holds, $\partial_{x}^{\alpha}\partial_{t}B(t,x)$ are bounded on $\mathbb{R}^{d}$ for all $\alpha$. This follows from Lemma 3.5 in [10]. From (5.4) we also get

[TABLE]

for all $\alpha,\beta$ and $\gamma$ together with (2) of Lemma 5.5.

Noting (5.23), we can rewrite (5.4) as

[TABLE]

Noting $\theta/(1+\theta^{2})\leq 1$ for all $\theta\geq 0$, from (2.8), (5.24) and (5.26) we have

[TABLE]

for all $\alpha,\beta$ and $\gamma$. In the same way from (5.27) we also have

[TABLE]

for all $\alpha,\beta$ and $\gamma$. Therefore, from (5.23) and (5) we have

[TABLE]

with a constant $C\geq 0$. Thereby we can see together with (5.24) and (5)-(5.30) that there exists a constant $\rho^{*}>0$ satisfying (5.21) and (5.22). Hence we can complete the proof of the assertion (1) by using Theorem 1.22 on p. 16 in [23].

We will prove (2). As in the proof of (1) we can prove (5.23) - (5.26), and so prove (5.29). Now, $B^{\prime}$ is given by (5.19). Then from (1) of Lemma 5.5 we have

[TABLE]

for all $\alpha,\beta$ and $\gamma$. Consequently we can prove (5.30). Hence we can complete the proof of (2) as in the proof of (1). ∎

The constant $\rho^{*}>0$ defined in Lemma 5.6 is fixed from now on throughout sections 5, 6 and 7.

**      Proposition 5.7****.**

We assume (2.7) - (2.9). (1) Let us define $\Phi$ by (5.2). Assume (2.12) if $0\leq M_{*}<1$ and (2.13) if $M_{*}\geq 1$. Let $0\leq t-s\leq\rho^{*}$ and $z(t,s;x,\xi,y)$ the function defined in Lemma 5.6. Then we have

[TABLE]

for $(x,\xi,y)\in\mathbb{R}^{3d}.$ (2) Let us define $\Phi$ by (5.2). Assume (2.10) and (2.11). Then we have the same assertions as in (1).

Proof.

Let $0\leq t-s\leq\rho^{*}$. We will first prove (1). Let $w=x,\xi$ or $y$. It follows from $\xi=\Phi(t,s;x,y,z(t,s;x,\xi,y))$ that we have

[TABLE]

and so from (1) of Lemma 5.6

[TABLE]

Using (2.8)-(2.9) and (2.12)-(2.13), from (5.2) and (2) of Lemma 5.5 we get

[TABLE]

with non-negative constants $C$ and $C^{\prime}$. Hence, using (5.22), we can prove

[TABLE]

with a constant $C^{\prime\prime}\geq 0$. Next from (5.33) we have

[TABLE]

Hence, using (5.32) with $|\alpha+\beta+\gamma|=1,$ we can prove (5.32) with $|\alpha+\beta+\gamma|=2$ as in the proof of the case of $|\alpha+\beta+\gamma|=1$. In the same way we can complete the proof of (5.32) by induction.

We consider the assertion (2). $\Phi$ is given by (5.2). Then we see from (1) of Lemma 5.5 that the corresponding inequalities to (5) are given by

[TABLE]

Hence we can prove (5.32) as in the proof of (1). ∎

**      Theorem 5.8****.**

Suppose (2.3) and Assumption 2.2. Let $\mathcal{C}(t,s)$ be the operator on ${\cal S}(\mathbb{R}^{d})$ defined by (5.6) and $0\leq t-s\leq\rho^{*}$. Then $\mathcal{C}(t,s)$ can be extended to a bounded operator on $L^{2}(\mathbb{R}^{d})$ and satisfies

[TABLE]

for all $f\in L^{2}$ with a constant $K\geq 0$.

Proof.

Since we can prove from (2.3) and (2.8) as in the proof of (2.25) that $\partial_{x}^{\alpha}\partial_{t}A_{j}(t,x)$ are continuous in $[0,T]\times\mathbb{R}^{d}$, Lemma 5.2 holds. We will first prove the case that (2.12) and (2.13) are assumed. Let us define $\Phi$ by (5.2). Then from Lemma 5.2 we have

[TABLE]

and so, changing variables from $z$ to $\xi=\Phi(t,s;x,y,z)$ by Lemma 5.6,

[TABLE]

From (2.8), (5.24) and (5) - (5.30) we have

[TABLE]

with $p_{j}(t,s;x,\xi,y)\in S^{0}(\mathbb{R}^{3d})\ (j=0,1)$. In particular, from (5.23) we have

[TABLE]

Noting (5.32), from (5) and (5) we can prove

[TABLE]

in ${\cal S}$ for $f\in{\cal S}$. Therefore, applying Theorem 1.A to the above, we have

[TABLE]

with a constant $K\geq 0$, which shows (5.35).

Next we consider the case that (2.10) and (2.11) are assumed. Let us define $\Phi$ by (5.2). Then we can prove Theorem 5.8 as in the proof of the first case. ∎

**      Corollary 5.9****.**

Suppose the assumptions of Theorem 5.8. Let $K_{\Delta}(t,0)f$ for $f\in C^{\infty}_{0}(\mathbb{R}^{d})$ be the approximation defined by (2) of the Feynman path integral. Let $|\Delta|\leq\rho^{*}$. Then $K_{\Delta}(t,0)f$ can be uniquely extended to a bounded operator on $L^{2}(\mathbb{R}^{d})$, which can be written as

[TABLE]

for $f\in L^{2}$, and one has (2.14) with the same constant $K$ as in Theorem 5.8.

Proof.

As in the proof of (4.11) from Theorem 5.8 we can prove (5.39), which shows (2.14) by (5.35). ∎

**      Theorem 5.10****.**

Suppose (2.3) and Assumption 2.2. Let $q(x,w)$ be a function satisfying (5.3) with $M=0$. We set $p(t,s;x,w)=q(x,\sqrt{t-s}w)$ and define $P(t,s)$ by (5.4). Then we have

[TABLE]

for $f\in L^{2}$ with a constant $C\geq 0$.

Proof.

Let $f\in{\cal S}$. As in the proof of (5) we have

[TABLE]

Noting (5.32), we have

[TABLE]

in ${\cal S}$ with $z=z(t,s;x,(t-s)\eta/m,y)$ and hence we can prove (5.40) by Theorem 1.A as in the proof of (5.35). ∎

6 Consistency of $\mathcal{C}(t,s)$

**      Lemma 6.1****.**

Suppose the assumptions of Proposition 5.7. Let $0\leq t-s\leq\rho^{*}$ and $z(t,s;x,\xi,y)$ the function defined in Lemma 5.6. Then we have

[TABLE]

Proof.

We first consider the case that $\Phi$ is given by (5.2), which we write as

[TABLE]

Then, using (2) of Lemma 5.5, from the assumptions (2.8) - (2.9) and (2.12) - (2.13) we have

[TABLE]

[TABLE]

[TABLE]

From (5.32) we have

[TABLE]

which shows

[TABLE]

We take $z=z(t,s;x,\rho\eta/m+\sqrt{\rho}\zeta/m,x+\sqrt{\rho}y)$ in (6). Then we have

[TABLE]

Hence we get

[TABLE]

Applying (6.3) - (6) to (6), we have (6.1).

We consider the case that $\Phi$ is given by (5.2). We write $\Phi$ as (6). Then from (1) of Lemma 5.5 we have

[TABLE]

correspondingly to (6.5). Hence we can also prove (6.1). ∎

From (5.32) we have the following.

**      Lemma 6.2****.**

Suppose the assumptions of Proposition 5.7. Let $0\leq t-s\leq\rho^{*}$. Then we have

[TABLE]

**      Lemma 6.3****.**

Suppose the assumptions of Proposition 5.7. Let $0\leq t-s\leq\rho^{*}.$ Take a $p(x,w)$ satisfying (5.3) and set

[TABLE]

for $0<\epsilon\leq 1$, where $z=z(t,s;x,\rho\eta/m+\sqrt{\rho}\zeta/m,x+\sqrt{\rho}y)$. Then we have

[TABLE]

for all $\alpha$ with constants $C_{\alpha}$ independent of $\epsilon$.

Proof.

We write $<D_{y}>^{2}\,=1-\sum_{j=1}^{d}\partial_{y_{j}}^{2}$. Let $l_{j}\geq 0\ (j=0,1)$ be integers. Using (6.1), (6) and (6.2), from (6.3) we have

[TABLE]

Hence, taking $l_{0}$ and $l_{1}$ so that $2l_{j}-2M(2M_{*}+1)>d,$ we get

[TABLE]

In the same way we can prove (6.11), using (6.2). ∎

**      Proposition 6.4****.**

Suppose (2.3) and Assumption 2.2. Let $p(x,w)$ be a function satisfying (5.3) and define $P(t,s)$ by (5.4). Let $0\leq t-s\leq\rho^{*}$. Then there exists an integer $l\geq 0$ such that we have

[TABLE]

for $f\in{\cal S}(\mathbb{R}^{d})$.

Proof.

If $t=s$, the inequality (6.12) follows from (5.4). Let $0<t-s\leq\rho^{*}$. Let us define $q_{\epsilon}(t,s;x,\eta)$ by (6.3) for $p(x,w)$. Then, using Lemma 5.2 and (5.32), we can prove

[TABLE]

for $f\in{\cal S}$, which has been proved in (4.12) of [11]. Hence we see

[TABLE]

Consequently, using (6.11), we have

[TABLE]

with an integer $l$. Taking $l_{0}$ so that $2l_{0}-2M(2M_{*}+1)>d$, we obtain (6.12). ∎

**      Theorem 6.5****.**

Suppose (2.3) and Assumption 2.2. Let $p(x,w)$ be a function satisfying (5.3) and define $P(t,s)$ by (5.4). Let $0\leq t-s\leq\rho^{*}$. Then, for any $\alpha$ there exists an integer $l(\alpha)\geq 0$ such that we have

[TABLE]

for $f\in{\cal S}$.

Proof.

Setting $p^{\prime}(x,s)=x^{\alpha}\,p(x,w)$, we have $P^{\prime}(t,s)f=x^{\alpha}P(t,s)f$ from (5.4). Hence from Proposition 6.4 we can prove

[TABLE]

for $f\in{\cal S}$ with an $l(\alpha)\geq 0$, which shows the first inequality of (6.15).

Next we can write $P(t,s)f$ as

[TABLE]

as in §2 of [11]. Then we have

[TABLE]

where $\beta\leq\alpha$ indicates $\beta_{j}\leq\alpha_{j}$ for all $j=1,2,\dots,d$. Using the assumptions (2.3) and (2.9), from (6) we have

[TABLE]

for all $\alpha^{\prime}$ and $\beta^{\prime}$. Hence, applying Proposition 6.4 to $P_{\beta}(t,s)$, from (6.17) we obtain the second inequality of (6.15). ∎

**      Proposition 6.6****.**

We assume (2.3), (2.8) and (2.9). Let $H(t)$ and $\mathcal{C}(t,s)$ be the operators defined by (1) and (5.6), respectively. Then there exists a continuous function $r(t,s;x,w)$ in $0\leq s\leq t\leq T$ and $(x,w)\in\mathbb{R}^{2d}$ satisfying (5.3) for an $M\geq 0$ such that

[TABLE]

for $f\in{\cal S}(\mathbb{R}^{d})$.

Proof.

From (2.3) and (2.8) we have

[TABLE]

for all $\alpha$ as in the proof of (2.25). Consequently we get Proposition 6.6 from Proposition 3.5 in [12] or Proposition 2.3 in [10]. ∎

7 Proofs of Theorems 2.1 and 2.2

We suppose Assumption 2.1 and let $M_{*}\geq 0$ be the constant in Assumption 2.1. Let us introduce the weighted Sobolev spaces

[TABLE]

We denote the dual space of $B^{a}$ by $B^{-a}$ and the $L^{2}$ space by $B^{0}$.

We have proved the following in Theorem 2.1 of [14] and its proof.

Theorem 7.A. Suppose Assumption 2.1 and (2.9). Then for any $f\in B^{a}\ (a=0,\pm 1,\pm 2,\dots)$ there exists a solution $u(t)=U(t,0)f\in C^{0}_{t}([0,T];B^{a})\cap C^{1}_{t}([0,T];B^{a-1})$ with $u(0)=f$ to the equation (1). This solution $u(t)$ is uniquely determined in the space $\bigcup_{a^{\prime}=1}^{\infty}C^{0}_{t}([0,T];B^{-a^{\prime}})\cap C^{1}_{t}([0,T];B^{-a^{\prime}-1})$. We also have

[TABLE]

and in particular

[TABLE]

Corollary 7.B. Suppose Assumption 2.1 and (2.9). Then for any integer $l\geq 0$ there exists an integer $l^{\prime}\geq 0$ such that

[TABLE]

*for all $f\in{\cal S}$. *

Proof.

The Sobolev lemma indicates

[TABLE]

where $[\cdot]$ denotes the Gauss symbol (cf. (2.24) on p. 78 in [17]). Hence, for any integer $l\geq 0$ there exist integers $l_{1}\geq 0$ and $l_{2}\geq 0$ such that

[TABLE]

for $f\in{\cal S}$. Therefore from (7.2) we have

[TABLE]

with an integer $l^{\prime}$. ∎

**      Lemma 7.1****.**

Suppose (2.3) - (2.4) and Assumption 2.2. Let $H(t)$ and $\mathcal{C}(t,s)$ be the operators defined by (1) and (5.6), respectively. Then there exists an integer $l\geq 0$ such that

[TABLE]

for all $f\in{\cal S}$.

Proof.

Using (6.19), we can write

[TABLE]

and so

[TABLE]

From (2.3) and (2.9) we can see that for $a=0,1,2,\dots$ there exist integers $l(a)\geq 0$ satisfying

[TABLE]

Consequently we see that the $L^{2}$ norm of the second term on the right-hand side of (7) is bounded by

[TABLE]

Applying (7) to this term, and applying (2.4), (2.9) and (6.20) to the third term on the right-hand side of (7), we have

[TABLE]

with an integer $l^{\prime}(0)\geq 0$. Hence, applying Theorem 6.5 to $\mathcal{C}(t,s)f$ and $R(t,s)f$, and using (7.5), we can prove (7.6). ∎

**      Lemma 7.2****.**

Suppose Assumptions 2.1 and 2.2. Then there exists an integer $l\geq 0$ such that we have

[TABLE]

for $f\in{\cal S}$.

Proof.

Correspondingly to (7) - (7) we have

[TABLE]

[TABLE]

[TABLE]

Hence from Theorem 7.A and (7.5) we can see

[TABLE]

for all $f\in{\cal S}$ with an integer $l^{\prime}\geq 0$. Writing

[TABLE]

we can prove (7.10) from (7.6) and (7.11). ∎

Now we will prove Theorem 2.1. Hereafter we assume $|\Delta|\leq\rho^{*}$. We have proved (2.14) in Corollary 5.9. First we assume $f\in{\cal S}$. From (5.39) we can write

[TABLE]

Using (5.35), we have

[TABLE]

which leads to

[TABLE]

from (7.4) and (7.10). Hence we see that as $|\Delta|\to 0$, $K_{\Delta}(t,0)f$ for $f\in{\cal S}$ converges to $U(t,0)f$ in $L^{2}$ uniformly in $t\in[0,T]$.

Let $f\in L^{2}$ be arbitrary. For any $\epsilon>0$ we take a $g\ \in{\cal S}$ such that $\|g-f\|<\epsilon$. Then from (2.14) and (7.3) we see

[TABLE]

which shows

[TABLE]

because of $g\in{\cal S}$. This indicates

[TABLE]

In the end, to complete the proof of Theorem 2.1 we have only to prove (2.16). From (2.15) and (5) we have

[TABLE]

which shows (2.16) from (5.6) and (5.7).

Next we consider the Lagrangian function defined by (2.17). Let $\mathcal{F}(\theta,s;q^{t,s}_{x,y})\\ (s\leq\theta\leq t)$ be the $l\times l$ matrix defined as the solution to (2.19). The following has been proved in Lemma 3.1 of [12].

**      Lemma 7.3****.**

We assume

[TABLE]

in $[0,T]\times\mathbb{R}^{d}$ for all $j,k=1,2,\dots,l$. Then we have

[TABLE]

in $0\leq s\leq t^{\prime}\leq t\leq T$ for all $\alpha$ and $\beta$.

Using $S(t,s;q^{t,s}_{x,y})$, we define

[TABLE]

if $0\leq s<t\leq T$ and $\mathcal{C}_{s}(t,t)f=f$ for $f={}^{t}(f_{1},\dots,f_{l})\in{\cal S}(\mathbb{R}^{d})^{l}$. Then we have the following correspondingly to Theorem 5.8.

**      Proposition 7.4****.**

We assume (2.3), Assumption 2.2 and (2.21). Then $\mathcal{C}_{s}(t,s)$ on ${\cal S}^{l}$ can be extended to a bounded operator on $(L^{2})^{l}$ and satisfies

[TABLE]

for $f={}^{t}(f_{1},\dots,f_{l})\in(L^{2})^{l}$ with a constant $K^{\prime}\geq 0$, where $\|f\|^{2}=\sum_{j=1}^{l}\|f_{j}\|^{2}.$

Proof.

From (2.19) we have

[TABLE]

Then from the assumption (2.21) and Lemma 7.3 we have

[TABLE]

for all $\alpha$ and $\beta$. Using $\mathcal{C}(t,s)$ defined by (5.6), we can write

[TABLE]

Then, noting (7.21), from Theorems 5.8 and 5.10 we obtain

[TABLE]

with constants $C_{0}\geq 0$ and $K^{\prime}\geq 0$, which shows (7.19). ∎

We have the following correspondingly to Proposition 6.6.

**      Lemma 7.5****.**

We consider the equation (2.18). Assume (2.3), (2.8), (2.9) and (7.16). Then there exist $r_{jk}(t,s;x,w)\ (j,k=1,2,\dots,l)$ satisfying (5.3) with an $M\geq 0$ such that

[TABLE]

for $f\in{\cal S}(\mathbb{R}^{d})^{l}$.

Proof.

From (2.3) and (2.8) we had (6.20). Hence we can prove (7.5) from Proposition 3.5 of [12]. ∎

Now we will prove Theorem 2.2. Let $|\Delta|\leq\rho^{*}.$ Using Proposition 7.4, from (2.20), Remark 2.2 and (7.18) we can write

[TABLE]

for $f\in(L^{2})^{l}$ and get the estimates (2.14) by (7.19). Next consider the equation (2.18). Then since we are assuming (2.21), we get the same assertions as in Theorem 7.A and so get (7.4). Hence we can complete the proof of Theorem 2.2 as in the proof of Theorem 2.1, using Theorem 6.5, Proposition 7.4 and Lemma 7.5.

8 Proofs of Theorems 2.3 and 2.4

In this section we always suppose Assumptions 2.3 and 2.4. Let $\mathcal{L}^{\sharp}(t,x,\dot{x})$ be the Lagrangian function defined by (2). We write

[TABLE]

Let’s define $q^{t,s}_{x,y}$ by (2.22) and write $\gamma^{t,s}_{x,y}$ as (5.1). Then the classical action for $q^{t,s}_{x,y}$ is written as

[TABLE]

correspondingly to (5).

Let $p(x,w)$ be a function satisfying (5.3) and define $P(t,s)$ by (5.4). Then we have the same assertions as in Lemma 5.1. We define $\tau(\sigma)$ and $\zeta^{(l)}(\sigma)$ by (5.12) for $x=x(l)$, $y=y(l)$ and $z=z(l)$. Hereafter for simplicity we suppose $m=m_{l}\ (l=1,2,3,4)$

**      Lemma 8.1****.**

Let $p(x,w)$ be a function satisfying (5.3). Then for any $0<\epsilon\leq 1$ and $0\leq s<t\leq T$ we have (5.2), where

[TABLE]

[TABLE]

or

[TABLE]

Proof.

We note

[TABLE]

Then we can prove Lemma 8.1 as in the proof of Lemma 5.2. ∎

We have the following consequence from Lemma 5.3.

**      Lemma 8.2****.**

We have (5.3), where $E=E^{(l)}(t,x(l))\ (l=1,2,3,4),C_{1}=C_{1}(l)\geq 0,x=x(l),\zeta(\sigma)=\zeta^{(l)}(\sigma),X=X(l)=(x(l),y(l),z(l))\in\mathbb{R}^{3d}$ and $M_{*}=M_{l*}$.

We set $a=\max\{C_{1}(l);l=1,2,3,4\}\geq 0$. Let us write (5) and (5) for $E=E^{(l)}(t,x(l))\ (l=1,2,3,4)$ as $E^{\prime(l)}_{0}(t,s;x(l),y(l),z(l))$ and $E^{\prime(l)}_{1}(t,s;x(l),y(l)\\ ,z(l))$, respectively. In the same way we write (5.19) and (5.4) for $B_{jk}=B_{jk}^{(l)}(t,x(l))$ as $B^{\prime(l)}(t,s;x(l),y(l),z(l))$. We define

[TABLE]

In the same way we define $E^{\prime}_{1s}(t,s;x,y,z)$ and $B^{\prime}_{s}(t,s;x,y,z)$. Then from (8.1) and (8.1) we have

[TABLE]

which is correspondent to (5.4).

**      Lemma 8.3****.**

There exist constants $\rho^{*}>0$ and $\delta>0$ such that for $0\leq t-s\leq\rho^{*}$ and $(x,y,z)\in\mathbb{R}^{12d}$ we have (5.21) and

[TABLE]

where $1+\rho^{2}|X|^{2M_{*}}$ in (5.21) is replaced with $\prod_{l=1}^{4}(1+\rho^{2}|X(l)|^{2M_{l*}})$.

Proof.

Noting the assumption (2.28), we can easily prove Lemma 8.3 from Lemma 8.2 and (8) as in the proof of Lemma 5.6. ∎

We see from Lemma 8.3 that the mapping:$\mathbb{R}^{4d}\ni z\to\xi=\Phi(t,s;x,y,z)\in\mathbb{R}^{4d}$ is homeomorphic if $0\leq t-s\leq\rho^{*}.$ We write its inverse mapping as $\mathbb{R}^{4d}\ni\xi\to z=z(t,s;x,\xi,y)\in\mathbb{R}^{4d}$. Then, noting the assumption (2.28), we can prove (5.32), (5.35) and (5.40) as in the proofs of Proposition 5.7, Theorems 5.8 and 5.10. In the same way we can prove (6.15) and (6.19) as in the proofs of Theorem 6.5 and Proposition 6.6.

We introduce the weighted Sobolev spaces

[TABLE]

for $a=1,2,\dots$, where $\omega_{a}(x)=\sum_{l=1}^{4}<x(l)>^{2a(M_{l*}+1)}$. We denote the dual space of $B^{\prime a}$ by $B^{\prime-a}$ and the $L^{2}$ space by $B^{\prime 0}$. Then from Theorem 2.4 in [14] we get the same assertions as in Theorem 7.A. Joining the results above, we can prove Theorems 2.3 and 2.4 as in the proofs of Theorems 2.1 and 2.2.