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semiclassical, Van-Vleck, Aharonov-Bohm, 
 time dependent Schrodinger equation.
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\begin{document} 
\title
{The Semi-classical Van-Vleck Formula. \\ Application to the  Aharonov-Bohm Effect} 
\author{J.M Bily  and  D. Robert\\ D\'epartement de Math\'ematiques, CNRS-UMR $N^o$ 6629,\\ Universit\'e de Nantes,  44072
Nantes-Cedex 03, FRANCE.\\
email robert@math.univ-nantes.fr}
 \date{} \maketitle



{\bf Abstract} {\footnotesize  At the very beginning of the quantum theory, Van-Vleck (1928) proposed 
 a nice approximation formula  for the integral kernel of the time dependent propagator for the Schr\"odinger equation.
 This formula can be deduced from the Feynman path integral by a formal stationary phase argument. After the fondamental 
works by  H\"ormander  and Maslov  on Fourier-integral operators,  it became  possible 
 to give a rigorous mathematical proof of the Van Vleck formula.
 We present here a more direct and elementary proof, using propagation of coherent states.
 We apply this  result to give a  mathematical proof of the Aharonov-Bohm effect observed on  the time dependent propagator. 
 This effect concerns a phase factor depending on the flux of a magnetic field, which can be non trivial, even
 if the particle never meets the magnetic field.}

\section{Introduction}
Let us consider the time dependent  Schr\"odinger equation
\beq\label{sch}
i\hbar\frac{\partial \psi(t)}{\partial t} = \hat{H}(t)\psi(t),\;\;  
  \psi(t_0) = f 
\edq
$t_0$  is the initial time, $f$ an initial state,
  $\hat{H}(t)$ 
a quantum Hamiltonian  defined as a smooth family  
 of self-adjoint operators in the Hilbert space $L^2(\R^n)$. In this paper, 
 $\hat{H}(t)$  is obtained as the  $\hbar$-Weyl-quantization  (cf \cite{ro}),  of
 a classical observable 
  $H(t, q,p)$, $q, p \in \R^n$. So we write, 
 $\hat{H}(t)= Op^w_{\hbar}H(t)$.\\ 
For an electron moving in an electro-magnetic field, we have the classical Hamiltonian
 $H(t,q,p) = \vert p - A(t,q)\vert^2 + V(t,q))$, 
$V(t,q)\in\R$ is  the electric potential ,   $ A(t,q)\in \R^n$ the 
magnetic potential, depending smoothly on position  $q\in\R^n$ 
and time $t\in \R$. (the detailed assumptions will be given later).


The solution $\psi(t)$ of equation (\ref{sch}) defines a quantum motion $f\mapsto \psi(t)$, 
 denoted by  $ \psi(t) = U_\hbar(t,t_0)f$.  Under suitable assumptions, 
the quantum evolution
 $U_\hbar(t,t_0)$ is a unitary operator in 
$L^2(\R^n)$. Anyway, 
$U_\hbar(t,t_0)$  has a distribution integral kernel, 
 $K_\hbar(t,t_0;x,y)$  ( it is the Green  kernel of the evolution problem (\ref{sch}))
 and will be called here the quantum propagator.
 In the semi-classical r\'egime ($\hbar \searrow 0$), Van-Vleck \cite{vv}, in a particular case   and later Gutzwiller \cite{gu},  have computed
 an approximation of 
 $K_\hbar(t,t_0;x,y)$  as $\hbar \searrow 0$  and obtained the following formula 
\beq\label{vv1}
K_\hbar(t,t_0;x,y) \approx (2\pi i\hbar)^{-n/2}{\rm e}^{-i\frac{\pi}{2}\sigma}
 \vert{\rm det}\partial^2_{x,y}S(t,t_0; y, \bar p)\vert^{1/2}{\rm e}^{\frac{i}{\hbar}S(t,t_0; y, \bar{p})}.
\edq
This approximation is  valid 
if  there exists only one classical trajectory, in the configuration space, connecting
 the starting point $(t_0, y)$ to the arrival point $(t, x)$.
 The momentum  $\bar p$ of this trajectory 
is  determined 
 by the following equation 
\beq\label{mt}
 q(t,t_0; y,\bar p) = x, \;\; {\rm where}\;\;
  q(t,t_0;  q,\bar p)=q(t),\;\;  p(t,t_0; q,\bar{p})=p(t),  
\edq
 are defined as the solution of the Hamilton equations in the phase space,  
\beqn\label{ham}
\frac{\partial q(t)}{\partial t}  =  \frac{\partial H}{\partial p}(t;q(t),p(t)),\;\;
\frac{\partial p(t)}{\partial t} &=&   -\frac{\partial H}{\partial q}(t;q(t),p(t)),\\
q(t_0) & = & q,\;\; p(t_0) = \bar{p}. \nonumber
\edqn
In (\ref{vv1}), $S$  is the action along the trajectory,
 $\partial^2_{x,y}S(t,t_0; y, \bar p)$  is  the  Van-Vleck matrix, 
$$
\partial^2_{x,y}S(t,t_0; y, \bar p) = \{\partial^2_{x_j,y_k}S(t,t_0; y, \bar p)\}_{1\leq k,j\leq n}, 
$$
  and 
 $\sigma\in\Z$  is an index,  the well known 
 H\"ormander-Maslov-Morse index. Van-Vleck (1928) did  not really discovered $\sigma$.
 This index was identified much later,  after several   works
 (Pauli, Van Hove, Morette, Keller, Gutzwiller  and  many others).
Formula (\ref{vv1}) is well known in quantum mechanics and is usually derived from the Feynman path
 integral with  a stationary  phase argument. 
Let us recall here  the Feynman path integral expression for the quantum
 propagator.
$$
 K_\hbar(t,t_0;x,y) = \int_{\Omega_{y,x}^{t_0, t}}{\rm e}^{\frac{i}{\hbar}S}\vert d\gamma\vert, 
$$
 where  $\Omega_{y,x}^{t_0, t}$ is the set of path  $\gamma$ in $\R^{n}$ such that $\gamma(t_0) =y$, $\gamma(t) = x$, 
  $S$ is the classical action along $\gamma$ and  $\vert d\gamma\vert$ is a ``pseudomeasure" on $\Omega_{y,x}^{t_0, t}$.
A recent tentative to give a rigorous meaning to Feynman path integral is the paper
 by Cartier-Morette \cite{camo}. But it is not clear how to justify the stationary phase theorem, 
 in infinite dimensional space, for general Hamiltonians with electric and magnetic fields. \\
 Another approach for a proof of  the formula (\ref{vv1}) is the B.K.W method. This approach
 works easily when  $\vert t-t_0\vert$ is small (this is the case considered in 1928 by Van-Vleck)  but is
 much more difficult for arbitrary large time.  In this case it is is necessary to use the geometrical Fourier integral
 operator theory of Duistermaat-H\"ormander-Maslov. To our knowledge,
 this approach was checked to prove Van-Vleck formula 
 for the quantum propagator  in  particular cases  only (Feydoriuk-Maslov, \cite{fm}).
 In \cite{lasi} the authors consider also approximations of the time dependent Schr\"odinger equation  
 for  every time using Fourier-integral operators with complex phases. They compute Maslov index and 
 the Aharonov-Bohm effect for  a line of flux.\\
The main goal of this paper  is to give a rigorous proof of (\ref{vv1})  using a 
coherent states analysis of the propagation  of solutions of (\ref{sch}) as it is  explained in \cite{coro}.
We shall see that this approach is rather elementary and gives directly accurate estimates, including Maslov index.
Our result is applied to give a mathematical proof for the time dependent Aharonov-Bohm effect, 
 for  time dependent electric potentials and  for stationary  magnetic potentials. 



\section{Coherent states and quantum propagator}
Coherent states are used, to construct solutions for  partial differential equations,
  from  a long time (see \cite{coraro} for
 an  application to  a trace formula  and  historical comments). 
Here we shall follow the presentation given in  \cite{coro,coraro}.
For  $z = (q,p)\in \R^{2n}$ in the phase space,  $\varphi_z$  denotes the coherent state 
  concentrated at the the point $z$. It is well known  that   $\{\varphi_z\}_{z\in \R^{2n}}$  is an ``overcomplete basis"
 in $L^2(\R^n)$. This property means that for every $u\in L^2(\R^n)$ we have the Plancherel formula
\beq
\int_{\R^n}\vert u(x)\vert^2dx = (2\pi\hbar)^{-n}\int_{\R^{2n}}\vert <u, \varphi_z>\vert^2dz, 
\edq
where  $<\cdot,\cdot>$ is the scalar product in $L^2(\R^n)$. Let us remark  that we could
 introduce here the  Bargmann transform $\tilde{u}(z) \stackrel{def}{=} <u, \varphi_z>$. Using
 this, we can reformulate the Schwartz kernel theorem in the following way, where 
  ${\cal S}^\prime(\R^n)$ is the dual  Schwartz space  (tempered distributions).
\begin{lemma} 
Every continuous and linear operator  $R$  from  ${\cal S}(\R^n)$ into
  ${\cal S}^\prime(\R^n)$ has an integral-distribution kernel,  $K^R$  given by the formula
\beq
K^R(x,y) = (2\pi\hbar)^{-n}\int_{\R^{2n}}\left(R\varphi_z\right)(x)\overline{\varphi_z(y)} dz.
\edq
\end{lemma}
Applying this lemma to the quantum  evolution operator  $U(t,t_0)$,  we get
 for the quantum propagator $K_\hbar(t,t_0)$, 
\beq\label{reprog}
K_\hbar(t,t_0;x,y) = (2\pi\hbar)^{-n}\int_{\R^{2n}}[U(t,t_0)\varphi_z](x)\overline{\varphi_z(y)} dz.
\edq

Let us now recall  some definitions concerning coherent states.
  Let us introduce
\beq\label{bs}
\psi_0(x) = (\hbar \pi )^{-n/4} \ \exp \left(-\frac{ \vert x\vert
^2}{2\hbar}\right),  
\edq
 which is the ground state of the $n$-dimensional harmonic oscillator, and for
$z= (q, p) \in \R^{2n}$, 
\beq
{\cal T}(z) = \exp \left\{ {i \over \hbar} (p \cdot x - q \cdot
\hbar D_x) \right \}, 
\edq
 which is the Weyl-Heisenberg operator of translation by $z$ in phase space, 
where $D_x = \frac{\partial}{i\partial x}$. 
  We also have 
$\varphi_z = {\cal T}(z) \psi_0 $. 
It is useful to rewrite (\ref{bs}) as
\beq
\psi_0 = \Lambda_{\hbar} \widetilde{\psi}_0, \;
 {\rm where}\; 
\Lambda_{\hbar}\psi(x) = \hbar^{-n/4}\psi\left((x\hbar^{-1/2}\right)  \;\;
{\rm  and}\;\;
\widetilde{\psi}_0 (x) = \pi^{-n/4} \exp \left ( - \vert x\vert^2/2 \right).
\edq
A more explicite expression is
\beq
\varphi_z(x) = {\rm e}^{\frac{i}{\hbar}(p\cdot x - \frac{q\cdot p}{2})}\psi_0(x-q)
\edq
Under rather  general assumptions,   an asymptotic expansion
in $\hbar$ for  the quantum evolution of coherent states,  
$U(t,t_0)\varphi_z$,  was obtained in \cite{coro}. 
This result will be used  here to get  an expression of the propagator as a Fourier integral~:
\beq 
K_\hbar(t,t_0;x,y) = (2\pi\hbar)^{-n}\int_{\R^{2n}}a(\hbar;t,t_0;x;z){\rm e}^{\frac{i}{\hbar}
\Phi(t,t_0;x,y;z)}dz,
\edq
 where $\Phi$  is a complex value global phase,  with non negative imaginary part, 
  $\Im \Phi \geq 0$,   and 
$a(\hbar)$  is an amplitude term which can be expanded in  $\hbar^{1/2}$,
$$
a(\hbar;t,t_0;x;z) \approx \sum_{j\geq 0}\hbar^{j/2}a_j(t,t_0;x;z), \;
\; x\in\R^n,\; z\in\R^{2n}.
$$
 Let us now make explicite our assumptions on the Hamiltonian $H$. In what follows, 
 $H(t,z)$ is supposed to satisfy the following properties.
\begin{description}
\item[A.0] $H(t,z) \in C^{\infty}(\R\times\R^{2n})$. Moreover there exist
 $m\in \N$, $M\in\R$,   $T>0$,  such that
 for all $\gamma\in\N^{2n}$, $\vert \gamma\vert \geq m$,   there exists  $C_{H,T,\gamma}$ such that
\beq
<z>^{-M/2}\vert\partial_z^\gamma H(t,z)\vert \leq C_{H,T,\gamma}, \;\;{\rm for}\;\;
  z\in \R^{2n},\;\;\vert t-t_0 \vert \leq T,.
\edq
where $<z> = (1+\vert z\vert^2)^{1/2}$.
\item[A.1] 
For all $z\in\R^{2n}$  
the Hamilton equation  (\ref{ham}) has a  unique solution starting from $z$ at $t=t_0$, $z(t)=(q(t), p(t))$,  defined
 for $\vert t - t_0\vert \leq T$.
\item[A.2] The quantum evolution for the Schr\"odinger equation (\ref{sch}),  $U(t,s)$, 
 exists, as a unitary operator in  
$L^2(\R^n)$, for $\vert t - t_0 \vert\leq T, \vert s - t_0\vert\leq T$. (see \cite{coro,ya2} for more details).
\end{description}
\begin{remark}
In  \cite{coro}  assumption  {\bf A.0} was  formulated in a too restrictive sense. 
The  constant $C_{H,T,\gamma}$ may depend on  $\gamma$. This is clear from the proof given in \cite{coro}. 
\end{remark}
Let us recall that  $z_t=(q_t,p_t)$  is the solution of (\ref{ham}) ($z_{t_0}=z$). Let us introduce: 
\beq\label{act}
 S(t, z) = \int_{t_0}^t (p_s\cdot\dot{q_s}-H(s, z_s))ds\;\; {\rm (classical\;action)}, \;{\rm  and \;define}\;
 \delta(t, z) = S(t, z)-\frac{q_t\cdot p_t-q\cdot p}{2}.
\edq

We need to introduce the Jacobi stability matrix,  $F(t, z)$, which is the
  linearized flow  associated with  (\ref{ham}) at the point  $z_t$ of
 the classical trajectory.
 This means that  $F(t,z)$ is the flow defined by the quadratic Hamiltonian
 $H_{qe}(t, \zeta) = \frac{1}{2}\partial^2_{z,z}H(t,z_t)\zeta\cdot\zeta$, $\zeta\in\R^{2n}$,
 with  $F(t_0, z)={\bf 1}$.  $F(t,z)$  is a symplectic   $2n\times 2n$ matrix 
  which can be written as four $n\times n$ blocks~:
 \beq 
F_t = F(t, z) = \pmatrix{A_t &\quad B_t \cr
C_t &\quad D_t \cr} 
\edq
We also need to introduce the metaplectic representation,  Met($F$),
 (see \cite{fo}) and \\
 ${\rm Met}_{\hbar}(F) = \Lambda_{\hbar}^{-1}{\rm Met}(F)\Lambda_{\hbar}$
  where $F$  is a symplectic matrix,  with four $n\times n$ blocks, $A, B, C, D$ as above.
 It is well known that $ A + iB$  is invertible, so we can define 
\beq
\Gamma = (C + iD)(A + iB)^{-1}.
\edq
The action of  Met($F$) on a Gaussian is given by  the following useful formula \cite{fo}.
\beq
 {\rm Met}(F)\widetilde{\Psi}_0 (x) = \pi^{-n/4}\left({\rm det}(A+iB)\right)^{-1/2}{\rm e}^{i\Gamma x\cdot x/2}.
\edq
In what follows,  $\left({\rm det}(A_t+iB_t)\right)^{-1/2}$  will be  defined by continuity starting from  $t = t_0$.\\
Let us remark that  ${\rm Met}_{\hbar}(F_t)$ is also the quantum propagator defined by the quadratic
 Hamiltonian $Op_\hbar^wH_0(t, .)$ where 
$$
H_0(t, \zeta) = \frac{1}{2}\partial^2_zH(t, z_t)\zeta\cdot\zeta.
$$
From Theorem (3.5)
 of  \cite{coro} (see also \cite{coraro}),  we get the following asymptotic expansion, in  $L^2$-norm sense.
 \begin{theorem}[propagation of Gaussians] Under assumptions {\bf A.0} to {\bf A.2},  for all $N\in\N$   
 there  exists a real number  $E_{N, T, z}>0$  such that for $\vert t - t_0\vert \leq T$, we have
\beq
\left\Vert U_\hbar(t,t_0)\varphi_z - 
 \exp\left(\frac{i\delta(z, t)}{\hbar}\right){\cal T} (z_t){\rm Met}_\hbar(F_t)
 \left(\Lambda_\hbar Q_N(t,\hbar;x)\widetilde{\psi_0}\right)\right\Vert_{L^2} \leq 
E_{N, T, z}\hbar^{N/2}
 \edq
with
\beqn
 Q_N(t,\hbar;x) = 1+ \sum_{(k,j)\in I_N}\hbar^{k/2 - j} Q_{kj}(t,x) \nonumber \\
{ \rm with} \; I_N = \{(k,j)\in \N\times\N  ,\;1\leq j\leq N-1,\;k\geq 3j,\;
 1 \leq k-2j < N \}
\edqn
 and  $Q_{kj}$  is  polynomial in  $x$ of degree $k$, odd or even with $k$,  with coefficients depending on $t$ and $z$.\\
 More explicitely , we have, modulo an error term in $L^2(\R^n)$, 
\beqn
U_\hbar(t,t_0)\varphi_z(x) &  =  & 
(\pi\hbar)^{-n/4}\left({\rm det}(A_t+iB_t)\right)^{-1/2} \nonumber\\
&  &\cdot\exp[\frac{i}{\hbar}\left(\delta(z, t) + p_tx-\frac{q_t\cdot p_t}{2} + \frac{\Gamma}{2}(x-q_t)\cdot(x-q_t)\right)] \nonumber\\
 &  & \cdot 
 \left(1 +  \sum_{(k,j)\in I_N}\hbar^{k/2 - j} b_{kj}(t,x-q_t)\right) + O(\hbar^{N/2}) 
\edqn
 where $b_{kj}(t,y)$ are polynomials of degree $\leq k$ in $y\in\R^n$  with time dependent coefficients and   with the same parity as $k$. 
\end{theorem}


\begin{remark}
It would be interesting to give explicit bounds on the  remainder term $E_{N, T, z}$ according
 to the three parameters $N, T, z$. Only partial results are known. 
\begin{enumerate}
\item Large $T$  estimates are given in \cite{coro}, related with the Ehrenfest time $T_\hbar$ of order $\log(\hbar^{-1})$. 
\item For  large $N$ Hagedorn-Joye \cite{hajo} recently proved analytic estimates (with bounds in $N!$)
 for a class of analytic electric potentials.
\item When  {\bf A.0} holds with $M = 0$ and $m =2$ it is proved in \cite{coro}
 that \\$\sup_{z\in\R^{2n}}E_{N,T,z} < +\infty$.
\end{enumerate}

\end{remark}


\section{Semi-classical approximation for the  propagator}
The  general Van-Vleck formula, for arbitrary Hamiltonians, 
  can be understood starting from  the particular case of quadratic Hamiltonians. So, let us begin with  a quadratic, 
 time dependent Hamiltonian $H(t, z)$, with real coefficients. Using the same notations as above,
 $F_t$ is the classical flow ($F_t={\bf 1}$) defined by $H(t, z)$. It is  well known that
 $F_t$ is a time dependent symplectic matrix.
  Let us introduce  
the quantum propagator  $U(t,t_0)$  defined by the Weyl quantization, $\hat{H}(t)$ of $H(t)$.
 The following  explicit formula for the integral kernel $K_\hbar(t,t_0;x,y)$ of $U(t,t_0)$
 is well known   (\cite{le, rosa2}).
\begin{proposition} If at time  $t$  we have {\rm det}$(B_t) \neq 0$
   then  $K_\hbar(t,t_0)$  is $C^{\infty}$-smooth for  $(x,y)\in\R^{2n}$.  Moreover we have the exact formula
 \beq\label{vvq}
K_\hbar(t,t_0;x,y) = (2\pi i\hbar)^{-n/2}{\rm e}^{-i\frac{\pi}{2}\sigma}
 \vert{\rm det}B_t\vert^{-1/2}{\rm e}^{\frac{i}{\hbar}S(t,t_0; x,y)}
\edq
where $ S(t,t_0; x,y)$  is the generating function of the flow $F_t$  and $\sigma$ is
 a  Maslov index.
\end{proposition}
\begin{remark} In \cite{rosa2} the definition of the  Maslov index is slightly different.
 Here we have used the usual convention in physics. 
\end{remark}

For general Hamiltonians, the integral kernel of the propagator may be 
not smooth for $t\neq t_0$, as it can be seen  from the above proposition (in particular for the harmonic oscillator). 
This subject was intensively studied by several people, in particular  
 Yajima \cite{ya2}. It is well known that smoothness  depends  on  large momentum behaviour.
 To get pointwise asymptotics for the kernel of the propagator we explore here two possibilities.
 In a first part  we shall consider the propagator restricted to bounded momentum states. This will be used
 to study the Aharonov-Bohm effect in the next section of this paper. In a second part 
  we shall follow Yajima ideas \cite{ya1}  by considering subquadratic Hamiltonians (one  difference
 between  our work and Yajima's work is that we consider here the semiclassical {\em r\'egime}).

Let us introduce   $\Theta \in C_0^{\infty}(\R^{n})$ , $U_\Theta(t,t_0) := U(t,t_0)\Theta(\hbar D_x)$ and 
 the integral kernel 
 $K_\hbar^\Theta(t,t_0;x,y)$  of  $U_\Theta(t,t_0)$. 
\begin{proposition}
For every  $N\geq 1$  we have
 $$
U^\Theta(t,t_0) = U^\Theta_N(t,t_0) + R^\Theta_N(t,t_0),
$$
such that  $\Vert R^\Theta_N\Vert = O(\hbar^{N/2})$ and the integral kernel  $K^{\Theta,N}(t,t_0)$ of  $U^\Theta_N$
is a Fourier integral,  
\beq\label{approg}
K^{\Theta, N}_\hbar(t,t_0;x,y) = 
(2\pi\hbar)^{-n}\int_{\R^{2n}}a^{\Theta,N}(\hbar;t,t_0;x;z){\rm e}^{\frac{i}{\hbar}
\Phi(t,t_0;x,y;z)}dz, 
\edq
 where $a^{\Theta,N}(\hbar, x)$ is a polynomial in  $x, \sqrt{\hbar}$, with the following property
\beq
a^{\Theta,N}(\hbar,t, z, x) = a^{\Theta}_0(t,z) + \sum_{k\in I_N}\hbar^{k/2-j}a^\Theta_{k,j}(t,z,x-q_t), 
\edq
where $a^\Theta_{k,j}(t,z,u)$  is a polynomial in $u\in \R^n$ of degree $k$ odd or even with $k$,
 which  vanishes if $p$ is outside Supp($\Theta$) ( $z = (q, p)$).  
 In particular, 
\beq
a^{\Theta}_0(t,z) = (\pi\hbar)^{-n/2}\Theta(p){\rm det}(A_t+iB_t)^{-1/2}. 
\edq 
  
The phase $\Phi$  has the following expression
\beqn\label{phase}
\Phi(t,t_0;x,y;z) = \int_{t_0}^t (\dot{q_s}\cdot p_s -H(s,q_s,p_s))ds  + p\cdot(q-y) +p_t\cdot(x-q_t)\\ \nonumber
 +\frac{i}{2}\vert y-q\vert^2 +\frac{1}{2} \Gamma_t(x-q_t)\cdot(x-q_t).
\edqn
\end{proposition}
The phase $\Phi$  comes from the Gaussians propagation theorem and from 
the action on Gaussians of metaplectic operators. The proof of the proposition
 follows easily from the following lemma.
 \begin{lemma}
\beq
\Theta(\hbar D)\varphi_z(x) = \sum_{\vert \gamma\vert \leq N}
\frac{\hbar^{\vert \gamma\vert /2}}{\gamma!}\partial^\gamma_p\Theta(p)\psi_{\gamma,z}(x)
 + \hbar^{(N+1)/2}R_{N,\hbar,z}(x), 
\edq
where  $\psi_{\gamma,z} = {\cal T}(z)\Lambda_\hbar D^\gamma)\tilde{\psi_0}$ and
$$
\forall N^\prime,\; \exists C_{N, N^{\prime}}, {\rm tel\;que}\;
 \Vert R_{N,\hbar,q,p} \Vert_{L^2} \leq  C_{N, N^{\prime}}<p>^{- N^{\prime}},\; \forall (q,p)\in\R^{2n},\;
 \forall \hbar \in ]0, 1].
$$
\end{lemma}
The proof is omitted because it  easily  follows from the Taylor formula expansion. 

 Let us  now introduce  the main  assumption  for the validity of the Van-Vleck formula.\\ \noindent
{\bf (V-V-I)}{\em There  exists a finite number $J$  of classical trajectories connecting
 in the configuration space the initial point
 $(t_0,y)$ to the final  point  $(t,x)$,  with initial momenta in supp($\Theta$),   such that
 $(t_0,y)$ and  $(t,x)$ are not conjugate. More precisely,
there exist 
 $ p^{(j)}\in supp(\Theta)$, $1\leq j\leq J$,   such that }
\beqn
q(t,t_0; y, p^{(j)}) & = & x,\;\; {\rm and} \\
{\rm det}\left(\frac{\partial q}{\partial p}(t,t_0;y, p^{(j)})\right) & \neq & 0,\;\; 1\leq j \leq J.
\edqn
\begin{theorem}[Van-Vleck formula]\label{vv2} a) If $q(t,t_0; y, p) \neq  x$ for all $p\in supp(\Theta)$ then 
$$
K^{\Theta}_\hbar(t,t_0;x,y)  = O(\hbar^{\infty}).
$$
b) Under assumptions {\bf (V-V-I)}, the Van-Vleck formula holds true in the following sense
\beqn
K^{\Theta}_\hbar(t,t_0;x,y) =
(2\pi i\hbar)^{-n/2}\sum_{1\leq j\leq J} { \rm e}^{-i\frac{\pi}{2}\sigma_j}
 \vert{\rm det}[\partial^2_{x,y}S(t,t_0; y,p^{(j)})]\vert^{1/2}{\rm e}^{\frac{i}{\hbar}S(t,t_0; y, p^{(j)})} \nonumber\\ 
\cdot\bigg(\Theta(p^{(j)}) + \hbar c_j^\Theta(t,t_0;x,y) + 
\sum_{m\geq 1}\hbar^{1 +m/2}c_{j,m}^\Theta(t,t_0;x,y) \bigg) + O(\hbar^{\infty})
 \edqn
where 
\begin{itemize}
\item[(i)]  $c_{j,m}^\Theta(t,t_0;x,y)$ are smooth coefficients depending on the classical trajectory $j$, 
\item[(ii)] $\sigma_j$ is the Maslov index of the path number $j$, 
\item[(iii)] $S$ is the classical action along the trajectory starting at $(t_0, y, p^{(j)})$
 and arriving at $(t, x, p(t,t_0,y,p^{(j)})$, 
\item[(vi)]   $\partial^2_{x,y}S(t,t_0; y,p^{(j)})$ is the Van-Vleck matrix,
 \item[v)] $O(\hbar^{\infty})$ holds in the $L^2$-norm operator sense.
\end{itemize}
\noindent
Moreover, if the Hamiltonian $H(t)$ is elliptic (like the usual Shr\"odinger one) then the remainder
 estimates $O(\hbar^{\infty})$ holds in $L^\infty$-norm in $(x,y)$.
\end{theorem}
{\bf Proof}\\
The proof  uses the stationary phase theorem with complex phase \cite{ho},
 applied to the Fourier integral (\ref{approg}).
First of all let us remark that the imaginary part of the matrix  $\Gamma_t$ is positive,  
 non degenerate, because $F_t$ is symplectic. This follows from a  direct computation giving
 $\Im(\Gamma_t) = (A_tA_t^{^\dagger}+B_tB_t^{^\dagger})^{-1}$, 
where  $X{^\dagger}$  is the the transposed matrix of  the matrix $X$.   So we have
$$
\{\Im \Phi = 0\} \Longleftrightarrow \{ q = y,\; q_t = x \}.
$$
Then we compute the gradient $\partial_z\Phi$ and the Hessian matrix $\partial^2_z\Phi$, 
 in the variables  $z = (q,p)$. A straightforward computation gives
\beq \label{action}
\partial_p S(t;z)  =  \left(\frac{\partial q_t}{\partial p}\right){^\dagger}p_t,\;\;
\partial_q S(t;z)  = \left(\frac{\partial q_t}{\partial q}\right){^\dagger}p_t - p.
\edq
where  $S$ the action defined in (\ref{act}). 
\beqn
\partial_q \Phi & = &  C_t{^\dagger}(x - q_t) - A_t{^\dagger}\Gamma(x - q_t) - i(y - q)\\
\partial_p \Phi & = & D_t{^\dagger}(x - q_t) + (q - y) - B_t{^\dagger}\Gamma_t(x- q_t).
\edqn
Let us recall that the four $n\times n$ blocks of the linearized flow $F_t$ are given by
$$
 A_t = \frac{\partial q_t}{\partial q},\; B_t = \frac{\partial q_t}{\partial p},\;
 C_t = \frac{\partial p_t}{\partial q},\; D_t = \frac{\partial p_t}{\partial p}
$$
So,  on the critical set of $\Phi$ given by $\{ q = y,\; q_t = x \}$, 
we have 
\beqn
\partial^2_z\Phi= 
\left(\begin{array}{cc}
2iI + (A_t +iB_t)^{-1}B_t &i(A_t +iB_t)^{-1}B_t\\
i(A_t +iB_t)^{-1}B_t &-(A_t +iB_t)^{-1}B_t\\
\end{array}\right).
\edqn 
The determinant can be easily computed, 
$$
{\rm det}(\partial^2_z\Phi) = {\rm det}[(-2i)(A_t +iB_t)^{-1}B_t].
$$
From assumption {\bf(V-V-I)} we know that  ${\rm det}B_t \neq 0$. Hence
 the critical points of $\Phi$ are non degenerate and the stationary phase theorem applies.
The only non trivial point here is to recognize the Maslov index.
 For that we have to clarify the argument of the complex number
 $({\rm det}(\partial^2_{z,z}\Phi)^{-1/2}$  appearing in the stationary phase Theorem, which is now recalled
 for the reader convenience.
\begin{theorem}[stationary phase expansion, \cite{ho}]  Let ${\cal O} \subset {I\hskip-1truemm R}^d$ be an
open set, and let $a, f \in C^{\infty}({\cal O})$ with $\Im f \geq 0$ in ${\cal O}$
and supp $a \subset {\cal O}$. We assume 
$$
 \{x \in {\cal O}, \Im f(x) = 0, \partial_x f(x) = 0 \} \Longleftrightarrow \{ x=x_0\}, 
$$
and the Hessian, $\partial^2_x f(x)$  of $f$ is
nondegenerate $x_0$.
 Under this  conditions, the integral  $J(\omega ) = \int_{\R^d} e^{i \omega
f(x)} a(x)dx$ has the following asymptotic expansion as $\omega \to + \infty$,
modulo $O(\omega^{- \infty})$,
$$ 
J(\omega ) = \left ( {2 \pi \over \omega} \right )^{{d \over 2}} \sum_{j
\geq 0} c_j\omega^{-j} \quad  + O(\omega^{-\infty}). 
$$
 The main coefficient $c_0$ is given by
$$
c_0 = e^{i\omega f(x_0)}{\det}_+^{-1/2}\left(\frac{\partial^2_x f(x_0)}{i}\right) a(x_0) , 
$$
$\det_+^{1/2}$ is defined as follows. Let us denote  by ${\cal M}_+$ the linear space of complex, symmetric,
 $d\times d$ matrices $V$ such that $\Re(V)$ is positive-definite. For $V$ real ($\Im(V) = 0$)  we define
 $\arg[\det_+(V)]=0$. Then for $V\in{\cal M}_+$, $\arg[\det_+(V)]$ 
is defined by analytic continuation of $\log$ (Liouville formula) and by continuity  to the closure of ${\cal M}_+$. 
\end{theorem}
In our case  we  have
\begin{lemma}
$$
{\rm det}_+^{-1/2}\left(i^{-1}\partial^2_{z,z}\Phi\right) =
 2^{-n/2}{\rm det}_+^{-1/2}\left(i(A+iB)^{-1}B\right).
$$
\end{lemma}
{\bf Proof}\\ It is well known that  for $V\in {\cal M}_+$  we have
$$
{\rm det}_+^{-1/2}(V) = (\pi)^{-d/2}\int_{\R^d}{\rm e}^{-V u\cdot u}du.
$$
Let us compute the r.h.s with  $V = i^{-1}\partial^2_z\Phi$.  We have
$$
\int_{\R^{2n}} {\rm e}^{-\Gamma(x,y)\cdot(x,y)}dxdy =
 \int_{\R^{2n}} {\rm e}^{-2x^2-iE(y-ix)\cdot(y-ix)}dxdy
$$
where $E = (A+iB)^{-1}B$. \\
By analytic deformation we get 
 $$
 \int_{\R^{2n}} {\rm e}^{-\Gamma(x,y)\cdot(x,y)}dxdy  =
 \left(\int_{\R^n}{\rm e}^{-2x^2}dx\right)\left(\int_{\R^n}{\rm e}^{-iEy\cdot y}dy\right).
$$
Let us remark that 
$$
iE = B{^\dagger}(AA{^\dagger} + BB{^\dagger})^{-1} + iA{^\dagger}(AA{^\dagger} + BB{^\dagger})^{-1}B
$$
hence $iE$ has a positive-definite real part.
The lemma follows. \QED
With the next lemma  we recover the usual Van-Vleck matrix.
\begin{lemma}
Under assumption {\bf(V-V-I)} we have
\beq
\partial^2_{x,y}S = -B^{-1}
\edq
\end{lemma}
{\bf Proof}\\ 
Let us recall that  $S = S(t,t_0; q, p^{(j)})$,  where   $q = y$ and  $p^{(j)}$  is
 determined by the implicit function theorem, as a function  
 of  $x$ and $y$. 
So computing  the derivatives of  $S$ we get:
 \beq
\partial_pS = B{^\dagger}p(t),\;\; \partial^qS = A{^\dagger}p(t) - p, 
\edq
and 
$$\partial_yS = -p(t,t_0, x;y),\;\;
\partial^2_{x,y}S  = - \partial_xp(t,t_0, x;y) = -B^{-1}.
$$
\QED

\begin{remark}
The index  $\sigma$ comes obviously from computation of 
 {\rm arg}$\det_+(iE)$. We can easily see that $\sigma \in \Z$, 
 using the elementary identity
 $$
 { \rm arg det}_c(A+iB)^{-1} - { \rm arg det }_+(iE) \in \pi(\Z + \frac{1}{2})
$$
where ${ \rm arg det }_c(A+iB)^{-1}$ denotes the determination defined by continuity from $t=t_0$.
Several equivalent definitions and topological interpretations of $\sigma$ are given in \cite{rosa2}.
 In particular it is proved that {\rm sgn}($\det B) = (-1)^{n/2+\mu}$ where $\mu\in\Z/(2\Z)$ 
 is a geometrical Maslov factor. 
\end{remark}

\begin{remark} If the Hamiltonian $H$ is such that $\partial^2_p H$ is invertible,
 (like $H(q,p) = \vert p\vert^2 + V(q)$)  then for $0<\vert t-t_0\vert <\delta$,
 $\delta$ small enough, the condition (V-V-1) is fulfilled with $J=1$. In this case $\sigma = 0$.
\end{remark}
\begin{remark}
It is also possible to get other Van-Vleck formulas by considering the propagator
 in different representations, like momentum representation. This case corresponds
 to the integral kernel for the operator $\tilde{U}(t,t_0) = {\cal F}_\hbar^{-1}U(t,t_0){\cal F}_\hbar$
 where ${\cal F}_\hbar$ is the $\hbar$-Fourier transform,
$$
{\cal F}_\hbar f(\xi) = (2\pi\hbar)^{-n/2}\int_{\R^n}{\rm e}^{-ix\xi/\hbar}f(x)dx.
$$
More generaly we could  get as well  semi-classical asymptotics for the propagation of oscillating
 states, $U(t;t_0)[a(x){\rm e}^{i\theta(x)/\hbar}]$,  outside caustics,   where $a$ is a  smooth function
 with  compact support and $\theta$ is a smooth real function. This gives 
 a different proof of results usually obtained using H\"ormander-Maslov theory \cite{fm} (see also \cite{lasi}). \\
Moreover it seems    possible to get   with our  method  explicit results when (V-V-1) is not fulfilled, 
 with suitable assumptions on the rank of  the matrix  $\frac{\partial q_t}{\partial p}$.
This could be a  different approach  of  caustics phenemena. (compare with \cite{du}). 
\end{remark}
 

Let us now consider some  specific Hamiltonians having  smooth propagators for $t\neq t_0$,
 modulo a small error in $\hbar$.
Our models are the following  Hamiltonians 
 \beq
H(t;q,p) = \frac{1}{2}\vert p\vert^2 + V(t,q)
\edq
where $V(t,q)$ is
 an electric potential, satisfying the following assumptions. 
There exists $\varepsilon>0$ such that for all multiindex $\alpha$ we have
\beqn
 \vert\partial_q^\alpha V(t,q)\vert & \leq &  C_\alpha<q>^{(2-\vert\alpha\vert)_+}\\
\vert \partial_qV(t,q)\vert & \leq &  C_1<q>^{1-\varepsilon}  
\edqn
The following lemma is inspired from Yajima \cite{ya1}.

\begin{lemma}
For  every $t, T$ such that $0<\vert t -t_0�\vert\leq T$, and  $q\in \R^n$,  
  there exists $ C > 0$ depending only on $\vert t -t_0�\vert$, $T$, $q$ and 
 $C_1$  such that we have
\beq
\vert q(t,t_0, q,p)\vert \geq \frac{\vert t -t_0�\vert}{2}\vert p\vert
\edq
 for all  $p\in\R^n$ satisfying  $\vert p\vert \geq C$.  
\end{lemma}

{\bf Proof}.  We use the notation  $q_t =  q(t,t_0, q,p)$,
 and  $p_t =  p(t,t_0, q,p)$. 
 From Hamilton equation we get 
\beq\label{newton}
q_t -(q + (t - t_0)p) =  - \int_{t_0}^t\int_{t_0}^s\partial_qV(\tau,q(\tau))d\tau ds
\edq
Let us introduce
$$
\Vert q\Vert_T = 1 + \sup_{\vert s-t_0\vert \leq T}\vert q(s)\vert.
$$
We get  from (\ref{newton}),
 \beq\label{CI1}
\vert q_t -(q + (t - t_0)p)\vert \leq C_1T^2\vert q\vert_T^{1-\varepsilon} 
\edq
\beq
\Vert q\Vert_T \leq   C_1T^2\Vert q\Vert_T^{1-\varepsilon} + \vert q\vert + T\vert p\vert +1
\edq
 Hence
\beq
\{\vert p\vert \geq  \frac{(2C_1T^2)^{1/\varepsilon}}{2T} \}\; \Longrightarrow \;
\{\Vert q\Vert_T \leq 2(1 + \vert q\vert + T\vert p\vert ) \}
\edq
But from (\ref{CI1}) we have
\beq 
\vert q_t\vert \geq \vert t - t_0\vert\vert p\vert -\vert q\vert  -
 C_1T^2(1 + \vert q\vert + T\vert p\vert)^{1-\varepsilon}
\edq
The lemma follows by choosing
$$
C = \frac{(C_1T^2)^{1/\varepsilon} + 8C_1T^{3-\varepsilon} + 1 + 5\vert q\vert}{\vert t -t_0\vert}.
$$ \QED
Using this lemma, by revisiting the proof given for the modified  propagator $K_\hbar^\Theta $, 
 we see  by applying the non stationary
 phase Theorem (the phase $\Phi$ is  non critical outside
 a compact set)  that the  same argument works for the full propagator.\\ 
 For the remainder term,  $R_N^1 = R_N$,   we get 
 an estimate, using \cite{coro} (see Remark 2.4, 3) and  the equality, 
\beq
<R_Nf, g> = (2\pi\hbar)^{-n}\int\,\int R_N\varphi_z(x)\bar g(x)\tilde f(z)dxdz
\edq
where  $\tilde f(z) = <f, \varphi_z>$. ($f\mapsto \tilde f$  is the Bargmann transform).\\
We get easily the estimate 
 \beq\label{RE} 
 \vert<R_Nf, g>\vert \leq C\hbar^{N-n}\Vert g\Vert_\infty\Vert\tilde f \Vert_1,
\edq
 where $\vert \cdot\Vert_r$  is the usual norm in the Lebesgue space $L^r$, $1\leq r\leq +\infty$.
To state the final result,  let us introduce  the following  assumption.\\ 
{\bf (V-V-II)} 
{\em There  exist a finite number of classical trajectories connecting
 $y$ at time $t_0$ to $x$ at time $t$,  with initial momenta in $\R^n$,   such that
 $y$ and  $x$ are not conjugate. More precisely,
there exist 
 $(y, p^{(j)})\in \R^n$, $1\leq j\leq J$  such that }
\beqn
q(t_0,t; y, p^{(j)}) & = & x,\;\; {\rm and} \\
{\rm det}\left(\frac{\partial q}{\partial p}(t_0,t;y, p^{(j)})\right) & \neq & 0,\;\; 1\leq j \leq J.
\edqn
We can state a result for the full propagator, i.e without momentum cutoff.
\begin{theorem}
Under assumption  {\bf (V-V-II)}, we have 
\beq
U(t) = U_N(t) + R_N(t), 
\edq
where $R_N(t) = O(\hbar^N)$  with the meaning given in (\ref{RE})
and   $ U_N(t)$ has a smooth integral kernel in the variables
 $(x,y)$ , $K_\hbar(t,t_0; x,y)$,  
having  the following  asymptotic expansion
\beqn
K^N_\hbar(t,t_0;x,y) =
(2\pi i\hbar)^{-n/2}\sum_{1\leq j\leq J} { \rm e}^{-i\frac{\pi}{2}\sigma_j}
 \vert{\rm det}\partial^2_{x,y}S(t,t_0; y,p^{(j)})\vert^{1/2}{\rm e}^{\frac{i}{\hbar}S(t,t_0; y, p^{(j)})}\nonumber\\ 
\bigg( 1  + \hbar c_j(t,t_0;x,y) + 
\sum_{m\geq 1}\hbar^{1 +m/2}c_{j,m}(t,t_0;x,y) \bigg) + O(\hbar^{\infty})
 \edqn
where  $\sigma_j$  Maslov-Morse index along the path $\gamma_j$.
\end{theorem}
\begin{remark}
The coefficients $c_{j,m}$ probably vanish for $m\geq 1$ odd  but we are not able to prove this.
 We  could compute explicitely the coefficient $c_j$. 
\end{remark}
 
\section{The time-dependent Aharonov-Bohm Effect}
 The Aharonov-Bohm effect,   discovered in 1959  by theoretical computations
   and  verified  afterwards  in  experiments in 1978 \cite{mp},
 is a purely quantum phenomena. It asserts that an electromagnetic potential
 can act on a quantum particle even if the field vanishes in a neighborhood
 of the particle trajectory.  
More recently this effect was   considered in the mathematical litterature 
 for bounded states in \cite{he} and for scattering states  in \cite{ni}. Here
 we consider the time dependent case. From a physicist  point of view this 
 phenomena is clearly explained in the book \cite{ba} (p.69-79), using Feynman path integrals.
 We want to present now  a rigorous mathematical setting for the
 Aharonov-Bohm effect using the results
 already proved for the time dependendent propagator in {\bf section 3} of this paper. \\
Let us begin here with the two dimensional Schr\"odinger Hamiltonian,
\beq
\hat{H}_{a,V}(t) = \frac{1}{2}\left(\hbar D_x -  a(x)\right)^2 + V(t,x),
\edq
where $a$ is a time independent  magnetic potential. The magnetic field is given  by 
\beq
b = d_xa = (\partial_{x_2}a_1 - \partial_{x_1}a_2)dx_1\wedge dx_2,\; x =(x_1,x_2).
\edq
We assume that $a, V$ are smooth in $(t,x)$, supp($b$) is  a  compact set  and there exists $m\in\R$ such that
$$
\vert\partial^\alpha_x a(x)\vert + \vert\partial^\alpha_xV(t,x)\vert \leq C_\alpha<x>^m
$$
Moreover the quantum Hamiltonians,   $\hat{H}_{a,V}$ and $\hat{H}_{0,V}(t)$, are such that they define
 propagators  $U_{a,V}(t,t_0)$ as  unitary operators in  $L^2(\R^2)$ (see \cite{ya1} for sufficient conditions).\\ \noindent
 {\em Let us assume that condition {\bf(V-V-I)} holds for $J =2$. So there exist 
two classical trajectories $\gamma_1, \gamma_2$ for $H_{0,V}$ such that
 $q_j(t_0) =  y$, $q_j(t) = x$, for $j = 1, 2$ and supp($b$) is inside the loop 
$\gamma_1\cup(-\gamma_2)$}
So we have the following picture.

%\input{AB-fig1}

\unitlength=2cm
\begin{picture}(6,3)
\bezier{1000}(1,1)(2.5,4)(5.5,1.5)
\put(0.5,1){$(t_0,y)$}
\put(3,1.5){\circle*{1}}
\bezier{1000}(1,1)(3,0.1)(5.5,1.5)
\put(5.6,1.5){$(t,x)$}
\put(2.7, 2.7){$\gamma_1$}
\put(3.3,.5){$\gamma_2$}
\put(3.5,0.6){\vector(4,1){0.4}}
\put(3,2,7){\vector(4,0){0,4}}
\put(2.6,1.2){supp($b$)}
\put(2.9,1.4){\circle{2}}
\put(2.9,1.8){$\partial\Omega$}
\put(3.8,.2){fig.1}
\end{picture}



Moreover  we need to put a screen (or a filter) to prevent classical particles starting from $y$
 to penetrate in supp($b$). This can be carried out  by  a filter $\Theta$ as follows.
 Let be $p^{(j)}$, $j = 1, 2$, the momentum of the path $\gamma_j$ at origine time $t_0$.
 For $\varepsilon > 0$ small enough, let us assume that $\Theta$ is smooth  and 
  satisfies the support condition
 \beq
 {\rm supp}(\Theta) \subset \{\vert p -p^{(1)}\vert < \varepsilon\} \cup \{\vert p -p^{(2)}\vert < \varepsilon\}. 
\edq
We choose $\varepsilon$ such that for $p\in {\rm supp}(\Theta)$ and 
 $t_0\leq s\leq t$, $q(s, y, p)$ stays outside supp($b$).
 To measure the Aharonov-Bohm effect we propose to compare  the propagators 
  $K_{\Theta,\hbar}^{\kappa a,V}(t,t_0;x,y)$ generated by
  $U_{\kappa a,V}(t,t_0)\Theta(\hbar D_x)$,  for $\kappa = 0$ and $\kappa = 1$, 
  at the meeting  point $(t,x)$. 
\begin{theorem}\label{AB}
Under the above assumptions,  we have for every  $\kappa \in[0, 1]$, 
\beqn\label{abf}
\left\vert 2\pi\hbar K_{\Theta,\hbar}^{\kappa a,V}(t,t_0;x,y)\right\vert^2 & = &  \noindent\\
\left\vert f_1(t,t_0;x,y){\rm e}^{\frac{i}{\hbar}S_{0,V}(\gamma_1)} +  
f_2(t,t_0;x,y){\rm e}^{\frac{i}{\hbar}[S_{0,V}(\gamma_2) - \kappa\varphi(b))]}\right\vert^2  & + & 
 O(\hbar), 
\edqn
 where $S_{a,V}(\gamma)$ denotes  the classical action for $H_{a,V}$ along the path $\gamma$,
 $f_1, f_2$ are real amplitudes not depending on $\kappa$, $a$ nor $\hbar$. 
More precisely, with the  notations
 of Theorem (\ref{vv1}) we have 
\beq\label{ampl}
f_j(t,t_0;x,y) =  \Theta(p^{(j)})\vert{\rm det}\partial^2_{x,y}S(t,t_0; y,p^{(j)})\vert^{1/2}{ \rm e}^{-i\frac{\pi}{2}\sigma_j}.
\edq
\end{theorem}  
{\bf Proof}\\This a direct consequence of Theorem \ref{vv2} plus an elementary action computation.\\
First of all, our assumptions imply that, for all $\kappa$,
  the only classical path for $H_{\kappa a,V}$, connecting
 $(t_0,y)$ to  $(t,x)$ with momentum in supp($\Theta$),    are $\gamma_1$
 and $\gamma_2$. This explains the formula giving  the amplitudes $f_j$
 by applying Theorem \ref{vv2}.  
For the phase computation, let us 
 consider the Lagrangian
  ${\cal L}_{\kappa a,V}$  for the potential $(\kappa a, V)$. We have
\beq
 {\cal L}_{\kappa a,V} = {\cal L}_{0,V} + \kappa a(q)\cdot\dot{q}\;.
\edq
Let us recall that  for  a  classical particle in an electromagnetic field
 with potential $(\kappa a, V)$  the equation of motion is given by 
$$
\ddot{q} = \partial_q V + \kappa b(q)\cdot\dot{q}
$$
where the magnetic field $b$ is given by the anti-symmetric matrix~: 
 $ b(q) =  \partial_q a - (\partial_q a)^\dagger$ (as usual the dots express  derivatives in time).
So we have the following equality for the actions
\beq
S_{\kappa a,V}(\gamma) = S_{0,V}(\gamma) + \kappa\int_\gamma adq.
\edq
Then we can conclude with Stokes formula giving the flux of $b$, 
\beq
 \int_{\gamma_2}adq = \int_{\gamma_1}adq - \varphi(b).
\edq
\QED
\begin{remark} The (A.B) effect seems difficult to observe if the magnetic field $b$ is time dependent. In that case
 the variations of the field induces an electric  field so that the equation  of motion is
$$ 
\ddot{q} = \partial_q V + \kappa\frac{\partial a}{\partial t} + \kappa b(q)\cdot\dot{q}.
$$
The support of  $\frac{\partial a}{\partial t}$ can meet $\gamma_1$ or  $\gamma_2$ and modify
 the dynamics in a neighborhood of $\gamma_1\cup\gamma_2$  .
\end{remark}

\begin{remark}
We can get   the same result as in Theorem \ref{AB} if we put an impenetrable barrier, $\partial \Omega$,  around supp($b$).
 More precisely the same asymptotic result holds for the propagator defined with Dirichlet
 boundary conditions on the complement of a neighborhood  of supp($b$). This can be
 easily proved  by using the Duhamel principle.
\end{remark}
\begin{remark}
If we choose $\Theta(p^{(j)}) \neq 0$, $j =1, 2$, and $\frac{\kappa \varphi(b)}{2\pi\hbar}\notin \Z$,
 then we can compute the Aharonov-Bohm effect. More precisely let us compute the perturbation coming form
 the magnetic potential
 on the transition amplitudes. We have 
\beqn
\left\vert 2\pi\hbar K_{\Theta,\hbar}^{\kappa a,V}(t,t_0;x,y)\right\vert^2 = \nonumber\\
\rho_1^2 + \rho_2^2 + 2\rho_1\rho_2\Im[\exp i\left(\frac{\pi}{2}(\sigma_2-\sigma_1) + 
\frac{S_1-S_2-\kappa\varphi(b)}{\hbar}\right)],
\edqn
 where $\rho_j=\vert f_j\vert$.
So we get the following  measure of  the Aharonov-Bohm Effect, modulo $O(\hbar)$, 
\beqn
\left\vert 2\pi\hbar K_{\Theta,\hbar}^{\kappa a,V}(t,t_0;x,y)\right\vert^2
 - \left\vert 2\pi\hbar K_{\Theta,\hbar}^{0,V}(t,t_0;x,y)\right\vert^2 & = & \nonumber\\
 2\rho_1\rho_2\left[\sin\left(\frac{\pi}{2}(\sigma_2-\sigma_1) + 
\frac{S_1-S_2-\kappa\varphi(b)}{\hbar}\right) -\sin\left(\frac{\pi}{2}(\sigma_2-\sigma_1) + 
\frac{S_1-S_2}{\hbar}\right)\right] & + &  O(\hbar).
\edqn
In particular if $\sigma_1 = \sigma_2$,  $S_1 = S_2$ and  $\kappa \varphi(b) = \frac{\pi\hbar}{2}$
 we get a  non null effect. With more work it is possible to get the above result modulo
 a remainder term with order $O(\hbar^{\infty})$ (see \cite{bi}).
\end{remark}

\begin{remark} With the same method, we can get  an analogue result in $\R^3$ if the magnetic field is supported inside
 a solid torus $\Pi$   and if $\gamma_1\cup(-\gamma_2)$ is a non trivial loop in $\R^3\backslash\Pi$.
 (see fig.2 below)
\end{remark}

%\input{AB-fig2}

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\put(4,4.6){$(t_0,y)$}
\put(1.7,2.4){supp($b$)}
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\put(1.9,4){\vector(1,-2){.1}}
\end{picture}



Now we want to explain, from a mathematical point of view, the Aharonov-Bohm effect  coming from
 a time dependent electric potential (for a physicist point of view see \cite{ab, pt}).\\
We construct a perturbation $W(s,q)$ of the electric potential  $V(s,q)$ 
\beq
W(s,q) = \chi_1(s)Y_1(q) + \chi_2(s)Y_2(q),
\edq
in the following way. 

%\input{AB-fig3}

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\put(.5,1){$(t_0,y)$}

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\put(2.9,2.4){${\cal T}_1$}
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\end{picture}


\noindent
Let us consider a time interval, $[t_1, t_2]\subset]t_0, t[$ and small tubes ${\cal T}_j$
 around the pieces of path $ P_j =:\{q^{(j)}(s),\;t_1\leq s\leq t_2\}$.
Let us introduce two  smooth functions
 on configuration space, $F_j$, $j =1, 2$,   such that supp$F_j \subset {\cal T}^\prime_j$, where
${\cal T}^\prime_j$ is a small neighborhood of ${\cal T}_j$ such that
 ${\cal T}^\prime_1\cap{\cal T}^\prime_2 = \emptyset$ and $F_j = 1$ on ${\cal T}_j$,
and two functions $\chi_j$
 of time  $s$, supp$(\chi_j\subset ]t_1, t_2[$ (j=1, 2). \\
 We want to compare  the propagators $ K_\hbar^V(t,t_0)$, $ K_\hbar^{V+\kappa W}(t,t_0)$,
defined  respectively by $\hat{H}_V$ and  $\hat{H}_{V+W}$.  We get a statement
 for the electric A.B effect. Let uis assume $n=2$ for simplicity. 
\begin{theorem}
Under the above assumptions, choosing $\varepsilon$ small enough for the support of  $\Theta$,
 then  for $\kappa\in[-1, 1]$,  we have
\beqn
\left\vert 2\pi\hbar K_{\Theta,\hbar}^{V+\kappa W}(t,t_0;x,y)\right\vert^2 & = & \noindent\\
\left\vert f_1{\rm e}^{\frac{i}{\hbar}S_{0,V}(\gamma_1)} +  
f_2{\rm e}^{\frac{i}{\hbar}(S_{0,V}(\gamma_2) - \delta(\kappa))})\right\vert^2 & + & O(\hbar)
\edqn
where $f_j$ are given by the same formula as in (\ref{ampl}) and  the phase shift $\delta$ is
 given by  the integral 
 $\delta(\kappa) =  \kappa\int_{t_0}^t(\chi_2(u) - \chi_1(u))du$.
\end{theorem}
{\bf Proof}\\
It is the same proof as for Theorem \ref{AB} after seeing that for $p\in {\rm supp}(\Theta)$
 we have 
\\ \noindent$\partial_q W(s,q(s,t_0;y,p)) = 0$ for $t_0 \leq s\leq t$. That means that this
 particle never meets the electric field. Nevertheless we get a phase shift in the
 transition amplitudes from $(t_0,y)$ to $(t,x)$ which is generically non trivial.
\QED
\begin{remark} The  magnetic and  electric Aharonov-Bohm effect could be formulated in an unified way using
 the integral $\oint_\gamma(Vdt + adq)$ along  the curve $\gamma$ in  space-time.
 \end{remark}
 
 


 
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\end{document}

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