The Anisotropic Gaussian Semi-Classical Schrödinger Propagator

Abstract

We present a construction of the anisotropic Gaussian semi-classical Schrödinger propagator, emblematic of a class of Fourier integral operators of quadratic phase kernels related to the Schrödinger equation. We deduce a set of algebraic relations of the variational matrices, solutions of the variational system pertaining to single Gaussian wave packet semi-classical time evolution, some already known in the literature, representing the symplectic and other invariances of the dynamics, which are subsequently utilized in order to derive the Van Vleck formula from the semi-classical Schrödinger propagator.

1. Introduction

The anisotropic Gaussian semi-classical approximation has been studied in the physics literature, in relation to initial value representations [1], in the context of semi-classical time evolution problems in atomic physics and theoretical chemistry. The initial focus of the methodology was single semi-classical Gaussian wave packet dynamics, on the basis of the nearby orbit approximation, which is related to the variational system and the matrix Riccati equation [2,3,4,5,6], while the algebraic properties of the solutions of the variational system, as constituent elements of anisotropic wave packet dynamics, have been studied [4]. Further advances have been made for semi-classical time evolution problems of more general initial states, on the basis of the continuous frame structure constituted by Gaussian wave packets and there have been detailed analyses of the their accuracy [7,8,9]. The semi-classical Schrödinger propagator, on the premises of the anisotropic Gaussian semi-classical approximation, has also been constructed [6]. Similar ideas have been implemented in the construction of asymptotic solutions for the Cauchy problem of the wave equation, referred to as Gaussian beams [10,11,12,13].

Insofar as the underlying algebraic structure, the underlying matrix Riccati dynamics is related to the nearby orbit method, in essence, a relation between the anisotropy matrix controlling the direction and spread of the propagated Gaussian wave packet, a solution of the matrix Riccati equation, and the variational matrices, solutions of the corresponding variational system [4,14].

More recently, Fourier integral operators of quadratic phase kernels related to the Schrödinger equation have been studied in the mathematics literature, with complex-valued quadratic phases of their kernels and common algebraic and geometric structure underlying their dynamics [14,15,16,17,18].

Following previous joint work, where the anisotropic Gaussian semi-classical Schrödinger propagator was used in order to construct a semi-classical phase space Schrödinger propagator and its relation to the Van Vleck formula hinted [19], in this paper we construct the anisotropic Gaussian semi-classical Schrödinger propagator based on the Weyl shift, for the Weyl quantization of the Hamiltonian, and prove the semi-classical inference to the Van Vleck kernel.

In Section 2 we define the semi-classical Gaussian wave packets and establish the tight continuous frame structure which they constitute; subsequently, we define the Cauchy problem for the Schrödinger propagator, describing time evolution for generic initial data; we consider the problem of semi-classical time evolution of single semi-classical Gaussian wave packets and construct the relevant asymptotic solution along a generic Hamiltonian orbit, parametrized by the variational matrices, solutions of the related variational system, and the anisotropy matrix, solution of the matrix Riccati equation; we derive a set of identities involving the variational matrices, following Hagedorn [4] highlighting them as representations of the symplectic invariance of the underlying Hamiltonian flow and certain symmetries of the Riccati dynamics, of particular practical use in the relevant semi-classical calculus.

In Section 3 we provide a definition of the anisotropic Gaussian semi-classical Schrödinger propagator.

In Section 4 we derive the Van Vleck formula by semi-classical asymptotics of the kernel of the anisotropic Gaussian semi-classical Schrödinger propagator, utilizing the algebraic relations of the anisotropy matrices derived in Section 2.

Finally, in Appendix A we provide a proof of the algebraic identities satisfied by the variational matrices, and in Appendix B a reference table of basic concepts notation.

2. The Anisotropic Gaussian Semi-Classical Approximation

2.1. The Semi-Classical Gaussian Wave Packet

We begin with a constructive definition of the semi-classical Gaussian wave packet.

Definition 1.

The semi-classical Gaussian wave packet in $L^2\left({\mathbb{R}}^d\right)$, centered at the phase space point $(q,p)\in {\mathbb{R}}^{2d}$, is defined by the action of the Weyl shift [6,8,20]

$${\mathcal{T}}_{(q,p)}=\exp {\displaystyle \frac{i}{ħ}}\left(p·x-q·\left(-iħ{\displaystyle \frac{\partial }{\partial x}}\right)\right)$$

(1)

on the Gaussian vacuum state, $G_0(x;ħ)={\left(\pi ħ\right)}^{-d/4}{e}^{-{\left|x\right|}^2/2ħ}$, i.e.,

$$G_{(q,p)}(x;ħ):={\mathcal{T}}_{(q,p)}{G}_0(x;ħ),$$

(2)

for some $ħ>0$.

Remark 1.

The parameter $ħ$ appears as a free positive parameter in representations of the Heisenberg group ${\mathbb{H}}_{2d+1}$ into $L^2\left({\mathbb{R}}^d\right)$ [20]. In the frame-work of semi-classical analysis, it is a small parameter, the semi-classical parameter. Physically, it is interpreted as the inverse of an action characteristic to the underlying classical mechanical motion in units of the reduced Planck constant, in the case when the motion is approximated ever more accurately by the latter, so that $ħ\to 0^{+}$. This constitutes the semi-classical limit of quantum mechanics.

The following property of the wave packets allows for the continuous phase space representation of the Schrödinger propagator.

Proposition 1.

According to the above definition, we have

$$G_{(q,p)}(x;ħ)={\left(\pi ħ\right)}^{-d/4}\exp {\displaystyle \frac{i}{ħ}}\left({\displaystyle \frac{p·q}{2}}+p·(x-q)+{\displaystyle \frac{i}{2}}{|x-q|}^2\right).$$

(3)

The set of semi-classical Gaussian wave packets, ${\left\{G_{(q,p)}\right\}}_{(q,p)\in {\mathbb{R}}^{2d}}$, constitutes a tight continuous frame in $L^2\left({\mathbb{R}}^d\right)$.

Proof. 

The action of the Weyl shift is understood by means of the Baker–Campbell–Hausdorff formula for the Heisenberg algebra ${\mathfrak{h}}_{2d+1}$. Elements $h,h^{\prime }\in {\mathfrak{h}}_{2d+1}$ are uniquely expressed as $h=a·X+b·Y+cE$ and $h^{\prime }=a^{\prime }·X+b^{\prime }·Y+c^{\prime }E$, for $(a,b,c),(a^{\prime },b^{\prime },c^{\prime })\in {\mathbb{R}}^{2d+1}$, where $X=(X_1,\dots ,X_d)$, $Y=(Y_1,\dots ,Y_d)$ and E are generators of the algebra satisfying the commutation relations $[X_j,Y_k]={\delta }_{jk}E$, $[X_j,E]=0$ and $[Y_j,E]=0$, for $j,k=1,\dots ,d$ [6,20]. For the exponential map onto the Heisenberg group ${\mathbb{H}}_{2d+1}$, we have

$$e^h{e}^{h^{\prime }}=\exp \left(h+h^{\prime }+{\displaystyle \frac{1}{2}}[h,h^{\prime }]\right),$$

(4)

so that unambiguous meaning is given to the two non-commuting argument function

$$e^{h+h^{\prime }}=\exp \left(-{\displaystyle \frac{1}{2}}(a^{\prime }·b-a·b^{\prime })E\right)e^h{e}^{h^{\prime }}.$$

(5)

The action of the Weyl shift follows, as it realizes the Schrödinger representation of ${\mathbb{H}}_{2d+1}$ on $L^2\left({\mathbb{R}}^d\right)$,

$$\begin{array}{c}{\mathcal{T}}_{(q,p)}{G}_0(x;ħ)\hfill \\ =\exp (-{\displaystyle \frac{1}{2}}\left[{\displaystyle \frac{i}{ħ}}p·x,-{\displaystyle \frac{i}{ħ}}q·\left(-iħ{\displaystyle \frac{\partial }{\partial x}}\right)\right])e^{{\displaystyle \frac{i}{ħ}}p·x}\exp (-{\displaystyle \frac{i}{ħ}}q·\left(-iħ{\displaystyle \frac{\partial }{\partial x}}\right))G_0(x;ħ)\\ =\exp {\displaystyle \frac{i}{ħ}}\left(p·x-{\displaystyle \frac{p·q}{2}}\right)\exp \left(-q·{\displaystyle \frac{\partial }{\partial x}}\right)G_0(x;ħ)\\ \hfill ={\left(\pi ħ\right)}^{-d/4}\exp {\displaystyle \frac{i}{ħ}}\left({\displaystyle \frac{p·q}{2}}+p·(x-q)+{\displaystyle \frac{i}{2}}{|x-q|}^2\right),\end{array}$$

(6)

noting that $[p·x,q·{\displaystyle \frac{\partial }{\partial x}}]=-p·q\mathrm{Id}$.

As for the second point, the set of semi-classical Gaussian wave packets defines a map from the phase space, as a measure space endowed with the Liouville measure, to $L^2\left({\mathbb{R}}^d\right)$, $(q,p)\mapsto G_{(q,p)}$, such that [21,22]:

1. It is weakly measurable, in the sense that for all $\psi \in L^2\left({\mathbb{R}}^d\right)$, the function $\langle G_{(q,p)},\psi \rangle$ is measurable with respect to the Liouville measure; the later is proportional to the phase space wave function, which possesses strong smoothness properties [19].

2. For all $\psi \in L^2\left({\mathbb{R}}^d\right)$ the inequalities $c_1{\parallel \psi \parallel }_{L^2}^2\le \int |\langle G_{(q,p)},\psi \rangle {|}^{2}dqdp\le c_2{\parallel \psi \parallel }_{L^2}^2$ are saturated for $c_1=c_2={\left(2\pi ħ\right)}^d$ [19]. □

We consider a Hamiltonian function $H(q,p,t)$ of appropriate smoothness and growth in $(q,p)\in {\mathbb{R}}^{2d}$ and continuous in $t\in \mathbb{R}$ [23]. We denote by $g^{t_0,t}$ the Hamiltonian flow generated by H, which is defined as the solution operator of the Cauchy problem for the Hamilton equations

$${\displaystyle \frac{d{q}_t}{dt}}={\displaystyle \frac{\partial H}{\partial p}}(q_t,p_t,t),{\displaystyle \frac{d{p}_t}{dt}}=-{\displaystyle \frac{\partial H}{\partial q}}(q_t,p_t,t),t\in [t_0,t_0+T],(q_{t_0},p_{t_0})=(q,p),$$

(7)

that is

$$(q_t(q,p),p_t(q,p))=g^{t_0,t}(q,p)$$

(8)

for some fixed time $T>0$.

For the corresponding quantum mechanical motion, let $\widehat{H}={\mathrm{Op}}_W\left(H\right)$ be the Weyl operator corresponding to the Hamiltonian function H [24]. We denote by $U(t_0,t)$ the Schrödinger propagator, which is defined as the solution operator of the Cauchy problem for the Schrödinger equation in the same time interval as above

$$iħ{\displaystyle \frac{\partial \psi }{\partial t}}=\widehat{H}\psi ,t\in [t_0,t_0+T],\psi \left(t_0\right)={\psi }_0,$$

(9)

that is

$$\psi \left(t\right)=U(t_0,t){\psi }_0.$$

(10)

By the resolution of the identity in $L^2\left({\mathbb{R}}^d\right)$, with respect to the continuous frame ${\left\{G_{(q,p)}\right\}}_{(q,p)\in {\mathbb{R}}^{2d}}$, $\mathrm{Id}={\left({\displaystyle \frac{1}{2\pi ħ}}\right)}^d\int G_{(q,p)}\langle G_{(q,p)},•\rangle dqdp$, we consider a corresponding Gaussian wave packet resolution for the Schrödinger propagator [8,22]

$$U(t_0,t)•={\left({\displaystyle \frac{1}{2\pi ħ}}\right)}^d\int U(t_0,t)G_{(q,p)}\langle G_{(q,p)},•\rangle dqdp,$$

(11)

which is the basis for the initial value representations for semi-classical Schrödinger propagators [1,6].

Setting off from the above Gaussian wave packet resolution, in the next section we construct the anisotropic Gaussian semi-classical Schrödinger propagator, $U^Z(t_0,t)$, which is parametrized by the variational matrices A and B, solutions of the related variational system, as well as the anisotropy matrix Z, solution of the related matrix Riccati equation, closely related to the nearby orbit approximation, i.e., a linearization of the Hamiltonian flow along an orbit emanating from a particula point in phase space [2,3,6,7,8,9].

The construction of the semi-classical propagator amounts to superposing anisotropic Gaussian wave packets $G_{(q,p)}^Z$, which are asymptotic solutions of the semi-classical Cauchy problem (9) with initial data ${\psi }_0=G_{(q,p)}$.

2.2. The Variational System and the Anisotropy Matrix

We proceed with the definition of the variational system and its solutions, the variational matrices, as well as the related matrix Riccati equation and its solution, the anisotropy matrix.

Definition 2.

The variational system related to the nearby orbit approximation for the Hamiltonian flow generated by a Hamiltonian function H, smooth in $(q,p)\in {\mathbb{R}}^{2d}$ and continuous in $t\in \mathbb{R}$, is as follows:

$${\displaystyle \frac{d}{dt}}\left(\begin{array}{c}A\\ B\end{array}\right)=J{\mathrm{Hess}}_{(q,p)}H(q_t,p_t,t)\left(\begin{array}{c}A\\ B\end{array}\right),$$

(12)

where

$${\mathrm{Hess}}_{(q,p)}H(q_t,p_t,t)=\left(\begin{array}{cc}{H}_{qq}& H_{qp}\\ H_{pq}& H_{pp}\end{array}\right)$$

(13)

is the Hessian matrix of H with respect to $(q,p)$, evaluated at $(q_t,p_t,t)$, and $J=\left(\begin{array}{cc}0& I\\ -I& 0\end{array}\right)$ is the $2d\times 2d$ canonical symplectic matrix, the accompanying initial conditions being $A\left(t_0\right)=I$ and $B\left(t_0\right)=iI$, where I is the $d\times d$ identity matrix. The solutions, A and B, are termed the position and momentum variational matrix, respectively.

Proposition 2.

The solution of the initial value problem for the variational system has the dynamical representation [8,9]

$$A(q,p,t_0,t)={\displaystyle \frac{\partial q_t}{\partial q}}+i{\displaystyle \frac{\partial q_t}{\partial p}},B(q,p,t_0,t)={\displaystyle \frac{\partial p_t}{\partial q}}+i{\displaystyle \frac{\partial p_t}{\partial p}}.$$

(14)

The fundamental solution of the variational system, defined as the solution of the initial value problem

$${\displaystyle \frac{d\sigma }{dt}}=J{\mathrm{Hess}}_{(q,p)}H(q_t,p_t,t)\sigma ,\sigma \left(t_0\right)=I,$$

(15)

where I is the $2d\times 2d$ identity matrix, admits the formal time ordered exponential series [25]

$$\sigma (q,p,t_0,t)=\mathcal{T}\exp \underset{t_0}{\overset{t}{\int }}J{\mathrm{Hess}}_{(q,p)}H(q_{\tau },p_{\tau },\tau )d\tau ,$$

(16)

and is given by the (symplectic) Jacobian matrix of the Hamiltonian flow

$$\sigma (q,p,t_0,t)={\displaystyle \frac{\partial (q_t,p_t)}{\partial (q,p)}}.$$

(17)

The variational matrices are expressed by means of the fundamental solution as follows

$$\left(\begin{array}{c}A\left(t\right)\\ B\left(t\right)\end{array}\right)=\sigma \left(t\right)\left(\begin{array}{c}I\\ iI\end{array}\right).$$

(18)

Proof. 

By virtue of Hamilton’s equations (7) with initial conditions $q_{t_0}=q$ and $p_{t_0}=p$, and as H is smooth in $(q,p)\in {\mathbb{R}}^{2d}$, we have

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{\partial q_t}{\partial q}}+i{\displaystyle \frac{\partial q_t}{\partial p}}\right)={\displaystyle \frac{\partial }{\partial q}}{\displaystyle \frac{d{q}_t}{dt}}+i{\displaystyle \frac{\partial }{\partial p}}{\displaystyle \frac{d{q}_t}{dt}}={\displaystyle \frac{\partial }{\partial q}}\left({\displaystyle \frac{\partial H}{\partial p}}(q_t,p_t,t)\right)+i{\displaystyle \frac{\partial }{\partial p}}\left({\displaystyle \frac{\partial H}{\partial p}}(q_t,p_t,t)\right)\\ \hfill =H_{pq}(q_t,p_t,t)\left({\displaystyle \frac{\partial q_t}{\partial q}}+i{\displaystyle \frac{\partial q_t}{\partial p}}\right)+H_{pp}(q_t,p_t,t)\left({\displaystyle \frac{\partial p_t}{\partial q}}+i{\displaystyle \frac{\partial p_t}{\partial p}}\right),\end{array}$$

(19)

while we similarly approach ${\displaystyle \frac{d}{dt}}\left({\displaystyle \frac{\partial p_t}{\partial q}}+i{\displaystyle \frac{\partial p_t}{\partial p}}\right)$; noting, additionally, that

$$\left({\displaystyle \frac{\partial q_t}{\partial q}}+i{\displaystyle \frac{\partial q_t}{\partial p}}\right){|}_{t=t_0}=I,\left({\displaystyle \frac{\partial p_t}{\partial q}}+i{\displaystyle \frac{\partial p_t}{\partial p}}\right){|}_{t=t_0}=iI,$$

(20)

initial conditions of the variational system stemming from the initial conditions of the Cauchy problem for Hamilton’s equations (7), by uniqueness of the solution of the initial value problem for the variational system, we arrive at the conclusion. The construction of the fundamental solutions follows by similar arguments. □

Definition 3.

The anisotropy matrix Z is defined in terms of the solution of the variational system, A and B, as follows

$$Z=B{A}^{-1}.$$

(21)

By Proposition 2, the anisotropy matrix Z admits the dynamical representation

$$Z\left(t\right)=\left({\displaystyle \frac{\partial p_t}{\partial q}}+i{\displaystyle \frac{\partial p_t}{\partial p}}\right){\left({\displaystyle \frac{\partial q_t}{\partial q}}+i{\displaystyle \frac{\partial q_t}{\partial p}}\right)}^{-1}.$$

(22)

In the sequel we will make use of the important fact that Z satisfies the matrix Riccati equation

$${\displaystyle \frac{d}{dt}}Z=Z{H}_{pp}Z-Z{H}_{pq}-H_{qp}Z-H_{qq},$$

(23)

and also the initial condition, $Z\left(t_0\right)=iI$. This follows by the variational system and the straigthforward calculation

$$\begin{array}{c}\hfill {\displaystyle \frac{d}{dt}}\left(B{A}^{-1}\right)={\displaystyle \frac{dB}{dt}}{A}^{-1}-B{A}^{-1}{\displaystyle \frac{dA}{dt}}{A}^{-1}\\ \hfill =\left(-H_{qq}A-H_{qp}B\right)A^{-1}-B{A}^{-1}\left(H_{pq}A+H_{pp}B\right)A^{-1}\\ \hfill =-B{A}^{-1}{H}_{pp}B{A}^{-1}-B{A}^{-1}{H}_{pq}-H_{qp}B{A}^{-1}-H_{qq}.\end{array}$$

As $\det A\left(t_0\right)$ is non-zero and smooth, by virtue of uniqueness of the solutions of the Cauchy problem of the matrix Riccati equation, Z is well defined for $t\in [t_0,t_0+T]$.

Moreover, the anisotropy matrix is symmetric, $Z^T=Z$, as the transpose matrix $Z^T$ satisfies the initial value problem for the matrix Riccati equation for Z, as $H_{qq}^T=H_{qq}$, $H_{pp}^T=H_{pp}$, $H_{qp}^T=H_{pq}$ and ${\left(iI\right)}^T=iI$.

Finally, in the subsequent propositions we summarize some algebraic relations of the variational matrices and the anisotropy matrix, that will be used in the proof of the approximation Theorem 1 below. Some of these identities have been introduced by Hagedorn as components of the the definition of the semi-classical Gaussian wave packet and the study of the semi-classical limit of quantum mechanics [4]. In the proof, we highlight their semi-classical origin, in particular, their symplectic invariance.

Proposition 3.

For $t\ge t_0$ and $(q,p)\in {\mathbb{R}}^{2d}$, the following relations on the position and momentum variational matrices, $A=A(q,p,t_0,t)$ and $B=B(q,p,t_0,t)$, hold

$$\begin{array}{c}\hfill A^{T}B-B^{T}A=0\\ \hfill \overline{A}{B}^T-A{\overline{B}}^T=2iI,A{\overline{A}}^T-\overline{A}{A}^T=0,B{\overline{B}}^T-\overline{B}{B}^T=0,A^{*}B-B^{*}A=2iI\\ \hfill \mathrm{Im}Z={\left(A{A}^{*}\right)}^{-1},\mathrm{Im}{Z}^{-1}={\left(B{B}^{*}\right)}^{-1}.\end{array}$$

(24)

A proof of the above proposition, which will be utilized in what follows in the derivation of the Van Vleck formula, is presented in Appendix A.

The variational matrices are related to the anisotropy matrix as follows.

Proposition 4.

The variational matrices, respectively, are uniquely expressed in terms of the anisotropy matrix by

$$A={\left(\mathrm{Im}Z\right)}^{-1/2},B={\left(\mathrm{Im}{Z}^{-1}\right)}^{1/2}.$$

(25)

Proof. 

From the third pair of relations in Proposition 3, it follows that $\mathrm{Im}Z\succ 0$ and subsequently ${\left(\mathrm{Im}Z\right)}^{-1}\succ 0$; by the uniqueness of positive definite square roots of positive definite matrices, there exists a unique positive definite matrix $\tilde{A}\in {\mathbb{C}}^{d\times d}$ such that ${\left(\mathrm{Im}Z\right)}^{-1}=\tilde{A}{\tilde{A}}^{*}$. However, given relation $A{A}^{*}={\left(\mathrm{Im}Z\right)}^{-1}$, the unique positive square root of ${\left(\mathrm{Im}Z\right)}^{-1}$ is $\tilde{A}=A$. The fact that $B={\left(\mathrm{Im}{Z}^{-1}\right)}^{-1/2}$ follows from $B=ZA$. □

We note that the anisotropy matrix is invariant under the right action of the group $\mathrm{SU}\left(d\right)$ on the variational matrices, in the sense that for $U\in \mathrm{SU}\left(d\right)$ the action $(A,B)\mapsto (AU,BU)$ preserves Z, as $\left(BU\right){\left(AU\right)}^{-1}=BU{U}^{-1}{A}^{-1}=Z$ as well as $a\left(t\right)={\left(\det A\right)}^{-1/2}\mapsto {\left(\det AU\right)}^{-1/2}={\left(\det A\right)}^{-1/2}$ [4]. An immediate consequence is the invariance of the relations given in Proposition 3 under the action of $\mathrm{SU}\left(d\right)$, as above.

2.3. Semi-Classical Evolution of Anisotropic Gaussian Wave Packet

We continue to the problem of semi-classical time evolution of a Gaussian wave packet in the anisotropic Gaussian approximation.

Theorem 1.

Let $t_0\in \mathbb{R}$ be a time instant, $T>0$ a fixed time interval, and H a Hamiltonian function, smooth in $(q,p)\in {\mathbb{R}}^{2d}$ and continuous in $t\in \mathbb{R}$, such that for any $\alpha ,\beta \in {\mathbb{N}}_0^d$ there exists some $c_{\alpha \beta }(t_0,T)>0$ and ${\mathcal{l}}_{|\alpha +\beta |}(t_0,T)\in \mathbb{R}$, such that, for any $(q,p)\in {\mathbb{R}}^{2d}$ and $t\in [t_0,t_0+T]$, the following estimate holds [8,9]

$$\left|{\displaystyle \frac{{\partial }^{\alpha +\beta }H}{\partial q^{\alpha }\partial p^{\beta }}}(q,p,t)\right|\le c_{\alpha \beta }(t_0,T){(1+|q|+|p\left|\right)}^{{\mathcal{l}}_{|\alpha +\beta |}(t_0,T)}.$$

(26)

The anisotropic Gaussian approximation for the semi-classical Cauchy problem (9), for given $(q,p)\in {\mathbb{R}}^{2d}$, amounts to the anisotropic semi-classical Gaussian wave packet

$$G_{(q,p)}^Z(x,t;ħ)={\left(\pi ħ\right)}^{-d/4}a\left(t\right)\exp {\displaystyle \frac{i}{ħ}}\left({\displaystyle \frac{p·q}{2}}+S\left(t\right)+p_t·(x-q_t)+{\displaystyle \frac{1}{2}}(x-q_t)·Z\left(t\right)(x-q_t)\right).$$

(27)

being an asymptotic solution of (9) in the sense that there exists some $C(q,p,t_0,T)>0$ such that

$$\parallel \left(iħ{\displaystyle \frac{\partial }{\partial t}}-\widehat{H}\right)G_{(q,p)}^Z(·,t;ħ){\parallel }_{L^2}\le C(q,p,t_0,T){ħ}^{3/2}$$

(28)

if the parameters $(q_t,p_t)$, $S\left(t\right)$, $Z\left(t\right)$ and $a\left(t\right)$, for which we assume differentiability in time for $t\ge t_0$, satisfy the characteristic system, namely

$$\begin{array}{c}\hfill {\displaystyle \frac{d{q}_t}{dt}}={\displaystyle \frac{\partial H}{\partial p}},{\displaystyle \frac{d{p}_t}{dt}}=-{\displaystyle \frac{\partial H}{\partial q}},(q_{t_0},p_{t_0})=(q,p)\\ \hfill {\displaystyle \frac{dS}{dt}}=p_t·{\displaystyle \frac{d{q}_t}{dt}}-H,S\left(t_0\right)=0\\ \hfill {\displaystyle \frac{dZ}{dt}}+Z{H}_{pp}Z+Z{H}_{pq}+H_{qp}Z+H_{qq}=0,Z\left(t_0\right)=iI\\ \hfill {\displaystyle \frac{da}{dt}}+{\displaystyle \frac{1}{2}}\mathrm{tr}(H_{pp}Z+H_{pq})a=0,a\left(t_0\right)=1,\end{array}$$

(29)

where the Hamiltonian and its derivatives are evaluated at $(q_t,p_t,t)$, whose solutions are:

1. The image of the Hamiltonian flow generated by H

$$(q_t(q,p),p_t(q,p)):=g^{t_0,t}(q,p).$$

(30)

2. The action of the orbit emanating from $(q,p)\in {\mathbb{R}}^{2d}$

$$S(q,p,t_0,t)=\underset{t_0}{\overset{t}{\int }}\left(p_{\tau }·{\displaystyle \frac{d{q}_{\tau }}{d\tau }}-H(q_{\tau },p_{\tau },\tau )\right)d\tau .$$

(31)

3. The anisotropy matrix

$$Z(q,p,t_0,t)=\left({\displaystyle \frac{\partial p_t}{\partial q}}+i{\displaystyle \frac{\partial p_t}{\partial p}}\right){\left({\displaystyle \frac{\partial q_t}{\partial q}}+i{\displaystyle \frac{\partial q_t}{\partial p}}\right)}^{-1}.$$

(32)

4. The amplitude

$$a(q,p,t_0,t)={\displaystyle \frac{1}{\sqrt{\det \left(\frac{\partial q_t}{\partial q}+i\frac{\partial q_t}{\partial p}\right)}}},$$

(33)

where the branch of the square root is that for which the initial value $a\left(t_0\right)=1$ is reached continuously.

Proof. 

We begin by deriving the commutation formula for the action of $\widehat{H}$ on the anisotropic semi-classical Gaussian wave packet $G_{(q,p)}^Z\left(t\right)$, which is well defined by virtue of the estimate (26), and given by [24]

$$\widehat{H}{G}_{(q,p)}^Z(x,t;ħ)={\left({\displaystyle \frac{1}{2\pi ħ}}\right)}^d\int e^{{\displaystyle \frac{i}{ħ}}\xi ·(x-y)}H\left({\displaystyle \frac{x+y}{2}},\xi ,t\right)G_{(q,p)}^Z(y,t;ħ)dyd\xi .$$

(34)

By integrating over the microscopic phase space co-ordinates $(y,\xi )\mapsto (Q,P)=\left({\displaystyle \frac{(x+y)/2-q_t}{\sqrt{ħ}}},{\displaystyle \frac{\xi -p_t}{\sqrt{ħ}}}\right)$, localizing the mean of the points x and y near the orbit’s position in configuration space and the the momentum $\xi$ near the orbit’s momentum in the dual space, we obtain

$$\begin{array}{c}\hfill \widehat{H}{G}_{(q,p)}^Z(x,t;ħ)=({\pi }^{-d}\int H\left(q_t+\sqrt{ħ}Q,p_t+\sqrt{ħ}P,t\right)e^{{\displaystyle \frac{2i}{\sqrt{ħ}}}P·(x-q_t)}\\ \hfill \times \exp (2iQ·ZQ-2i(P+{\displaystyle \frac{1}{\sqrt{ħ}}}Z(x-q_t))·Q)dQdP)G_{(q,p)}^Z(x,t;ħ).\end{array}$$

(35)

By Taylor expanding the Hamiltonian at the orbit $(q_t,p_t)$ to second order, we obtain the commutation formula

$$\begin{array}{c}\widehat{H}{G}_{(q,p)}^Z(x,t;ħ)=G_{(q,p)}^Z(x,t;ħ)\mathtt{(}H+{\displaystyle \frac{\partial H}{\partial q}}·(x-q_t)+{\displaystyle \frac{\partial H}{\partial q}}·Z(x-q_t)\hfill \\ +{\displaystyle \frac{1}{2}}(x-q_t)·\left(Z{H}_{pp}Z+Z{H}_{pq}+H_{qp}Z+H_{qq}\right)(x-q_t)-{\displaystyle \frac{iħ}{2}}\mathrm{tr}(H_{pp}Z+H_{pq})\\ +{\pi }^{-d}\int r_{(q_t,p_t)}(Q,P,t;ħ)\exp (2iQ·Z\left(t\right)Q-2iP·Q\\ \hfill +2i{\displaystyle \frac{x-q_t}{\sqrt{ħ}}}·P-2i{\displaystyle \frac{x-q_t}{\sqrt{ħ}}}·Z\left(t\right)Q\left)dQdP\right),\end{array}$$

(36)

where the Hamiltonian and its derivatives are evaluated at the arguments $(q_t,p_t,t)$ and $r_{(q_t,p_t)}(Q,P,t;ħ)$ is the Taylor remainder

$$r_{(q,p)}(Q,P,t;ħ)={ħ}^{3/2}\sum _{\left|\alpha \right|+\left|\beta \right|=3}{\displaystyle \frac{1}{\alpha !\beta !}}{\displaystyle \frac{{\partial }^{\alpha +\beta }H}{\partial q^{\alpha }\partial p^{\beta }}}(q+\lambda \sqrt{ħ}Q,p+\lambda \sqrt{ħ}P,t)Q^{\alpha }{P}^{\beta },$$

(37)

for some $\lambda \in ]0,1[$.

We define the remainder of the Schrödinger equation with respect to $G_{(q,p)}^Z\left(t\right)$ as

$$\begin{array}{c}{R}_{(q,p)}^Z(x,t;ħ):=\left(iħ{\displaystyle \frac{\partial }{\partial t}}-\widehat{H}\right)G_{(q,p)}^Z(x,t;ħ)\hfill \\ =(-{\displaystyle \frac{dS}{dt}}+p_t·{\displaystyle \frac{d{q}_t}{dt}}-H+\sqrt{ħ}\left(-{\displaystyle \frac{d{p}_t}{dt}}+Z{\displaystyle \frac{d{q}_t}{dt}}-{\displaystyle \frac{\partial H}{\partial q}}-Z{\displaystyle \frac{\partial H}{\partial p}}\right)·{\displaystyle \frac{x-q_t}{\sqrt{ħ}}}\\ +ħ[i{a}^{-1}{\displaystyle \frac{da}{dt}}+{\displaystyle \frac{i}{2}}\mathrm{tr}\left(H_{pp}Z+H_{qp}\right)\\ +{\displaystyle \frac{1}{2}}{\displaystyle \frac{x-q_t}{\sqrt{ħ}}}·\left({\displaystyle \frac{dZ}{dt}}+Z{H}_{pp}Z+Z{H}_{pq}+H_{qp}Z+H_{qq}\right){\displaystyle \frac{x-q_t}{\sqrt{ħ}}}\left]\right)G_{(q,p)}^Z(x,t;ħ)\\ -({\pi }^{-d}\int r_{(q_t,p_t)}(Q,P,t;ħ)\exp (2iQ·Z\left(t\right)Q-2iP·Q\\ \hfill +2i{\displaystyle \frac{x-q_t}{\sqrt{ħ}}}·P-2i{\displaystyle \frac{x-q_t}{\sqrt{ħ}}}·Z\left(t\right)Q\left)dQdP\right)G_{(q,p)}^Z(x,t;ħ).\end{array}$$

(38)

In the above semi-classical expanision of the remainder, the $O\left(1\right)$, $O\left(\sqrt{ħ}\right)$ and $O\left(ħ\right)$ terms, correspond to the equations comprising the characteristic system (29); these have been derived, e.g., by Nazaikinskii et al. [9]; however, a different choice of definition of the semi-classical Gaussian wave packet $G_{(q,p)}$ and of a different quantization of H leads to a difference in form of the $O\left(ħ\right)$ term, i.e., a difference in form of the transport equation.

Assuming the characteristic system (29) is satisfied, and introducing the configuration space co-ordinate localized at the trajectory $x=q_t$, i.e., $x\mapsto \chi ={\displaystyle \frac{x-q_t}{\sqrt{ħ}}}$, the remainder reduces to the form

$$\begin{array}{c}{R}_{(q,p)}^Z(q_t+\sqrt{ħ}\chi ,t;ħ)=-{\pi }^{-d}{ħ}^{3/2}\sum _{\left|\alpha \right|+\left|\beta \right|=3}{\displaystyle \frac{1}{\alpha !\beta !}}(\int {\displaystyle \frac{{\partial }^{\alpha +\beta }H}{\partial q^{\alpha }\partial p^{\beta }}}(q_t+\lambda \sqrt{ħ}Q,p_t+\lambda \sqrt{ħ}P,t)\hfill \\ \hfill \times Q^{\alpha }{P}^{\beta }\exp (2iQ·Z(t)Q-2iP·Q+2i\chi ·P-2i\chi ·Z(t)Q)dQdP)G_{(q,p)}^Z(q_t+\sqrt{ħ}\chi ,t;ħ).\end{array}$$

(39)

We proceed to estimate the norm of the remainder by a direct estimate of the resulting oscillatory integral

$$\begin{array}{c}\parallel R_{(q,p)}^Z{(·,t;ħ)\parallel }_{L^2}={\pi }^{-5d/4}{ħ}^{3/2}\left|a\left(t\right)\right|(\sum _{\left|\alpha \right|+\left|\beta \right|=3,|{\alpha }^{\prime }|+|{\beta }^{\prime }|=3}{\displaystyle \frac{1}{\alpha !\beta !{\alpha }^{\prime }!{\beta }^{\prime }!}}\hfill \\ \times \int f_{\alpha \beta }(Q,P,t;ħ)f_{{\alpha }^{\prime }{\beta }^{\prime }}(Q^{\prime },P^{\prime },t;ħ)\exp \left(2i\left[(Q·ZQ-Q^{\prime }·\overline{Z}{Q}^{\prime })-(P·Q-P^{\prime }·Q^{\prime })\right]\right)\\ \hfill \times \exp \left(-\chi ·Z_2\chi +2i\left[(P-ZQ)-(P^{\prime }-\overline{Z}{Q}^{\prime })\right]·\chi \right)d\chi dQdPd{Q}^{\prime }d{P}^{\prime }{)}^{1/2},\end{array}$$

where $Z_2=\mathrm{Im}Z$, and

$$f_{\alpha \beta }(Q,P,t;ħ)={\displaystyle \frac{{\partial }^{\alpha +\beta }H}{\partial q^{\alpha }\partial p^{\beta }}}(q_t+\lambda \sqrt{ħ}Q,p_t+\lambda \sqrt{ħ}P,t)Q^{\alpha }{P}^{\beta }.$$

(40)

By the Fubini theorem we may perform the integration with respect to $\chi$

$$\begin{array}{c}\hfill \int \exp (-\chi ·Z_2\chi +2i\left[(P-ZQ)-(P^{\prime }-\overline{Z}{Q}^{\prime })\right]·\chi )d\chi \\ \hfill ={\displaystyle \frac{{\pi }^{d/2}}{\sqrt{\det Z_2}}}\exp (-\left[(P-ZQ)-(P^{\prime }-\overline{Z}{Q}^{\prime })\right]·Z_2^{-1}\left[(P-ZQ)-(P^{\prime }-\overline{Z}{Q}^{\prime })\right])\end{array}$$

(41)

as $Z_2$ is invertible and positive definite, by virtue of Proposition 3, so that we obtain

$$\begin{array}{c}\parallel R_{(q,p)}^Z{(·,t;ħ)\parallel }_{L^2}={\pi }^{-d}{ħ}^{3/2}(\sum _{\left|\alpha \right|+\left|\beta \right|=3,|{\alpha }^{\prime }|+|{\beta }^{\prime }|=3}{\displaystyle \frac{1}{\alpha !\beta !{\alpha }^{\prime }!{\beta }^{\prime }!}}\hfill \\ \times \int f_{\alpha \beta }(Q,P,t;ħ)f_{{\alpha }^{\prime }{\beta }^{\prime }}(Q^{\prime },P^{\prime },t;ħ)\exp \left(2i\left[(Q·ZQ-Q^{\prime }·\overline{Z}{Q}^{\prime })-(P·Q-P^{\prime }·Q^{\prime })\right]\right)\\ \hfill \times \exp \left(-\left[(P-ZQ)-(P^{\prime }-\overline{Z}{Q}^{\prime })\right]·Z_2^{-1}\left[(P-ZQ)-(P^{\prime }-\overline{Z}{Q}^{\prime })\right]\right)dQdPd{Q}^{\prime }d{P}^{\prime }{)}^{1/2}.\end{array}$$

(42)

By the linear change of variables

$$(Q,P,Q^{\prime },P^{\prime })\mapsto \kappa (Q,P,Q^{\prime },P^{\prime })=(u,v,u^{\prime },v^{\prime })=(Z_{1}Q-P,Z_{2}Q,Z_1{Q}^{\prime }-P^{\prime },-Z_2{Q}^{\prime }),$$

(43)

where $\left|\det \kappa \right|={\left(\det Z_2\left(t\right)\right)}^{-2}$, the norm becomes

$$\begin{array}{c}\parallel R_{(q,p)}^Z{(·,t;ħ)\parallel }_{L^2}={ħ}^{3/2}{\displaystyle \frac{{\pi }^{-d}}{\det Z_2\left(t\right)}}(\sum _{\left|\alpha \right|+\left|\beta \right|=3,|{\alpha }^{\prime }|+|{\beta }^{\prime }|=3}{\displaystyle \frac{1}{\alpha !\beta !{\alpha }^{\prime }!{\beta }^{\prime }!}}\hfill \\ \hfill \times \int F_{\alpha \beta }(u,v,t;ħ)F_{{\alpha }^{\prime }{\beta }^{\prime }}(u^{\prime },v^{\prime },t;ħ)\exp (-(u,v,u^{\prime },v^{\prime })·M\left(t\right)(u,v,u^{\prime },v^{\prime }))dudvd{u}^{\prime }d{v}^{\prime }{)}^{1/2},\end{array}$$

(44)

where the amplitude is $F_{\alpha \beta }(t;ħ)=f_{\alpha \beta }\circ {\kappa }^{-1}(t;ħ)$ and the matrix of the quadratic form of the phase is

$$M\left(t\right)=\left(\begin{array}{cccc}{Z}_2^{-1}& 0& {\displaystyle \frac{1}{2}}{Z}_2^{-1}& {\displaystyle \frac{i}{2}}{Z}_2^{-1}\\ 0& Z_2^{-1}& {\displaystyle \frac{i}{2}}{Z}_2^{-1}& -{\displaystyle \frac{1}{2}}{Z}_2^{-1}\\ {\displaystyle \frac{1}{2}}{Z}_2^{-1}& {\displaystyle \frac{i}{2}}{Z}_2^{-1}& Z_2^{-1}& 0\\ {\displaystyle \frac{i}{2}}{Z}_2^{-1}& -{\displaystyle \frac{1}{2}}{Z}_2^{-1}& 0& Z_2^{-1}\end{array}\right).$$

(45)

The real part of the later

$$\mathrm{Re}M\left(t\right)=\left(\begin{array}{cccc}{Z}_2^{-1}& 0& {\displaystyle \frac{1}{2}}{Z}_2^{-1}& 0\\ 0& Z_2^{-1}& 0& -{\displaystyle \frac{1}{2}}{Z}_2^{-1}\\ {\displaystyle \frac{1}{2}}{Z}_2^{-1}& 0& Z_2^{-1}& 0\\ 0& -{\displaystyle \frac{1}{2}}{Z}_2^{-1}& 0& Z_2^{-1}\end{array}\right)$$

(46)

is positive definite, as the necessary and sufficient conditions for positive definiteness hold [26], i.e., both its upper left $d\times d$ constituent block matrix $\left(\begin{array}{cc}{Z}_2^{-1}& 0\\ 0& Z_2^{-1}\end{array}\right)$ and the corresponding Schur complement

$$\begin{array}{c}\hfill \begin{array}{r}\hfill \mathrm{Re}M/\left(\begin{array}{cc}{Z}_2^{-1}& 0\\ 0& Z_2^{-1}\end{array}\right):=\left(\begin{array}{cc}{Z}_2^{-1}& 0\\ 0& Z_2^{-1}\end{array}\right)\\ \hfill -{\left(\begin{array}{cc}\frac{1}{2}{Z}_2^{-1}& 0\\ 0& -\frac{1}{2}{Z}_2^{-1}\end{array}\right)}^T{\left(\begin{array}{cc}{Z}_2^{-1}& 0\\ 0& Z_2^{-1}\end{array}\right)}^{-1}\left(\begin{array}{cc}\frac{1}{2}{Z}_2^{-1}& 0\\ 0& -\frac{1}{2}{Z}_2^{-1}\end{array}\right)\end{array}\end{array}$$

(47)

are positive definite for $t\in [t_0,t_0+T]$.

The phase condition $\mathrm{Re}M\succ 0$ guarantees rapid decay of the exponential, while the amplitude $F_{\alpha \beta }(u,v,t;ħ)F_{{\alpha }^{\prime }{\beta }^{\prime }}(u^{\prime },v^{\prime },t;ħ)$ exhibits algebraic growth in $(u,v,u^{\prime },v^{\prime })$ for $t\in [t_0,t_0+T]$, by linearity of $\kappa$ and by estimate (26). Thus, the oscillatory integral in the expression of the norm of the remainder (44) is well defined. Finally, as $F_{\alpha \beta }(t;ħ)$ is regular in $ħ$, we conclude that

$$\parallel \left(iħ{\displaystyle \frac{\partial }{\partial t}}-\widehat{H}\right)G_{(q,p)}^Z(·,t;ħ){\parallel }_{L^2}\le C(q,p,t_0,T){ħ}^{3/2},$$

(48)

where, for some ${ħ}_0>0$

$$\begin{array}{c}C(q,p,t_0,T)={\displaystyle \frac{{\pi }^{-d}}{\det Z_2\left(t\right)}}{\sup }_{ħ\in ]0,{ħ}_0[}(\sum _{\left|\alpha \right|+\left|\beta \right|=3,|{\alpha }^{\prime }|+|{\beta }^{\prime }|=3}{\displaystyle \frac{1}{\alpha !\beta !{\alpha }^{\prime }!{\beta }^{\prime }!}}\hfill \\ \hfill \times \int F_{\alpha \beta }(u,v,t;ħ)F_{{\alpha }^{\prime }{\beta }^{\prime }}(u^{\prime },v^{\prime },t;ħ)\exp (-(u,v,u^{\prime },v^{\prime })·M\left(t\right)(u,v,u^{\prime },v^{\prime }))dudvd{u}^{\prime }d{v}^{\prime }{)}^{1/2}.\end{array}$$

(49)

□

We have proven that given that the parameters $(q_t,p_t)$, $S\left(t\right)$, $Z\left(t\right)$, and $a\left(t\right)$ satisfy the characteristic system (29), the propagated Gaussian wave packet $G_{(q,p)}^Z\left(t\right)$ is an asymptotic solution of the Cauchy problem (9) in the sense of the estimate (28). The fact that the parameters are solutions of the Cauchy problem for the characteristic system follows from direct substitution; the uniqueness and boundedness of the solutions follow from the existence and uniqueness theorem for differential equations, given the smoothness of H. Their particular form, as provided, follows from definition and by substitution. However, a comment is required on the well definedness of the dynamical representations (32) and (33) of the anisotropy matrix $Z\left(t\right)$ and the amplitude $a\left(t\right)$.

In turn, by the transport equation, as A is invertible and a is non-zero, we have, by the variational system,

$$\begin{array}{c}\hfill a{\left(t\right)}^2=\exp \left(-\underset{t_0}{\overset{t}{\int }}\mathrm{tr}\left(H_{pp}Z+H_{pq}\right)d\tau \right)=\exp \left(-\underset{t_0}{\overset{t}{\int }}\mathrm{tr}\left(H_{pp}B+H_{pq}A\right)A^{-1}d\tau \right)\\ \hfill =\exp \left(-\underset{t_0}{\overset{t}{\int }}\mathrm{tr}{\displaystyle \frac{dA}{dt}}{A}^{-1}d\tau \right)={\displaystyle \frac{1}{\det A\left(t\right)}},\end{array}$$

(50)

and so

$$a\left(t\right)={\left(\det \left({\displaystyle \frac{\partial q_t}{\partial q}}+i{\displaystyle \frac{\partial q_t}{\partial p}}\right)\right)}^{-1/2},$$

(51)

where the branch of the square root is that for which the initial value $a\left(t_0\right)=1$ is reached continuously.

It is emphasized that the form of the transport equation, as the form of its solution, depends on the choice of definition of the semi-classical Gaussian wave packet as well as the choice of quantization of H; in contrast, see the form of the transport equation derived by Nazaikinksii et al. [9] for the choice $G_{(q,p)}(x;ħ)={\left(\pi ħ\right)}^{-d/4}\exp {\displaystyle \frac{i}{ħ}}\left(p·(x-q)+{\displaystyle \frac{i}{2}}{|x-q|}^2\right)$ and $\widehat{H}={\mathrm{Op}}_{qp}\left(H\right)$.

Following the physical motivation for the defining characteristics of the semi-classical Gaussian wave packets $G_{(q,p)}$ as coherent states semi-classically micro-localized at $(q,p)$ and exhibiting optimal simultaneous position and momentum variance, saturating the Heisenberg inequalities [27], we prove a similar characterization of the $G_{(q,p)}^Z$.

Proposition 5.

The anisotropic semi-classical Gaussian wave packet $G_{(q,p)}^Z\left(t\right)$ acquires the content of the semi-classical correspondent of the pure classical state $(q_t,p_t)$ [9], in the sense that, for any $t\ge t_0$ and $(q,p)\in {\mathbb{R}}^{2d}$ it satisfies:

1. The expectation formulas for the position and momentum operators, for $j=1,\dots ,d$,

$${\langle {\widehat{q}}_j\rangle }_{G^Z}=\langle G^Z,{\widehat{q}}_j{G}^Z\rangle =q_{tj},{\langle {\widehat{p}}_j\rangle }_{G^Z}=\langle G^Z,{\widehat{p}}_j{G}^Z\rangle =p_{tj}.$$

(52)

2. The minimal variance conditions, for $j=1,\dots ,d$,

$${\left({\Delta }_{G^Z}{\widehat{q}}_j\right)}^2={\langle {\widehat{q}}_j^2\rangle }_{G^Z}-{\langle {\widehat{q}}_j\rangle }_{G^Z}^2={\displaystyle \frac{ħ}{2}},{\left({\Delta }_{G^Z}{\widehat{p}}_j\right)}^2={\langle {\widehat{p}}_j^2\rangle }_{G^Z}-{\langle {\widehat{p}}_j\rangle }_{G^Z}^2={\displaystyle \frac{ħ}{2}},$$

(53)

saturating the Heisenberg inequality, ${\Delta }_{G^Z}{\widehat{q}}_j{\Delta }_{G^Z}{\widehat{p}}_j={\displaystyle \frac{ħ}{2}}$.

Proof. 

By direct integration, both points follow directly the fact that, for $j=1,\dots ,d$ we have

$$\begin{array}{c}\hfill {\widehat{q}}_j{G}_{(q,p)}^Z(x,t;ħ)=x_j{G}_{(q,p)}^Z(x,t;ħ),{\widehat{p}}_j{G}_{(q,p)}^Z(x,t;ħ)\\ \hfill =\left(p_{tj}+\sum _{k=1}^{d}Z{\left(t\right)}_{jk}(x_k-q_{tk})\right)G_{(q,p)}^Z(x,t;ħ).\end{array}$$

(54)

□

3. The Anisotropic Gaussian Semi-Classical Schrödinger Propagator

In this section, we introduce the anisotropic Gaussian semi-classical Schrödinger propagator $U^Z(t_0,t)$ as an operator approximation to the propagator $U(t_0,t)$, by applying the result of theorem (1), i.e., that $G_{(q,p)}^Z$ is a semi-classical asymptotic solution of the Cauchy problem (9), in the phase space resolution of the Schrödinger propagator.

Definition 4.

We define the anisotropic Gaussian semi-classical Schrödinger propagator as the operator on $L^2\left({\mathbb{R}}^d\right)$

$$U^Z(t_0,t)•:={\left({\displaystyle \frac{1}{2\pi ħ}}\right)}^d\int G_{(q,p)}^Z\left(t\right)\langle G_{(q,p)},•\rangle dqdp,$$

(55)

where $G_{(q,p)}^Z\left(t\right)$ is given in (27). In particular, it is characterized by the action

$$U^Z(t_0,t)G_{(q,p)}=G_{(q,p)}^Z\left(t\right).$$

(56)

The operator $U^Z$ is a semi-classical Fourier integral operator, acting as

$$U^Z(t_0,t)\psi \left(x\right)=\int K^Z(x,y,t_0,t;ħ)\psi \left(y\right)dy,$$

(57)

its kernel given by the oscillatory integral distribution [19]

$$K^Z(x,y,t_0,t;ħ)={\left({\displaystyle \frac{1}{2\pi ħ}}\right)}^d\int {\overline{G}}_{(q,p)}(y;ħ)G_{(q,p)}^Z(x,t;ħ)dqdp.$$

(58)

We characterize $U^Z(t_0,t)$ as a semi-classical Fourier integral operator, in the sense that it semi-classically approximates the Schrödinger propagator for short times, yet with an oscillatory distribution operator kernel whose phase function is complex valued and quadratic in x. Such generalizations of conventional Fourier integral operators for the solution of the Schrödinger equation have been rigorously studied by Paul and Uribe [16], Rousse and Swart [18], Robert [8,17], and Laptev and Sigal [15] by the methods of microlocal analysis.

4. Inference of the Van Vleck Formula

The Van Vleck formula [28] is a weak semi-classical expansion of the Schrödinger propagator kernel K, which reads

$$\begin{array}{c}K(x,y,t_0,t;ħ)\sim {\left({\displaystyle \frac{1}{2\pi iħ}}\right)}^{d/2}\sum _{r=1}^N\sqrt{\left|\det {\displaystyle \frac{\partial p}{\partial q_t}}(t;y,p_r(y,x,t_0,t))\right|}\hfill \\ \hfill \times e^{{\displaystyle \frac{i}{ħ}}S(y,p_r(y,x,t_0,t),t_0,t)-{\displaystyle \frac{i\pi }{2}}{m}_r},ħ\to 0^{+}.\end{array}$$

(59)

Here, $N=N(x,y,t_0,t)$ is the number of orbits of the Hamiltonian flow [23] emanating from point $(q,p)$ at time $t_0$, terminating at point $(q_t,p_t)$ at time $t\ge t_0$, such that $q=y$ and $q_t=x$; for $r=1,\dots ,N$, $p_r=p_r(y,x,t_0,t)$ are the admissible initial momenta of the orbit and $m_r$ is the Maslov index of the orbit; $S=S(q,p,t_0,t)$ is the action of the orbit emanating from point $(q,p)$ at time $t_0$ [23]. We assume that N is finite, but, in general, an upper bound for N cannot be found [29].

The above formula is usually obtained via a formal stationary-phase asymptotic expansion applied to the Feynman path-integral representation of the Schrödinger propagator kernel [24,28,30]. In our approach, we derive the formula using the anisotropic Gaussian approximation, relying on the following complex stationary-phase theorem. This theorem, due to Nazaikinskii, Oshmyan, Sternin, and Shatalov [31], is analogous to the complex stationary-phase result of Melin and Sjöstrand [32].

Theorem 2.

Consider the oscillatory integral

$$I_{\phi ,\mathrm{\Phi }}(w;ħ)={\left({\displaystyle \frac{1}{2\pi ħ}}\right)}^d\int \phi \left(X\right)e^{{\displaystyle \frac{i}{ħ}}\mathrm{\Phi }(w,X)}dX,$$

(60)

where $\phi \in C_0^{\infty }({\mathbb{R}}^{2d},\mathbb{C})$ and $\mathrm{\Phi }\in C^{\infty }({\mathbb{R}}^m\times {\mathbb{R}}^{2d},\mathbb{C})$, where $\mathrm{\Phi }$ possesses an everywhere non-negative imaginary part on $\mathrm{supp}\phi$, $\mathrm{Im}\mathrm{\Phi }(w,X)\ge 0$, so that the equations

$$\mathrm{Im}\mathrm{\Phi }(w,X)=0,{\displaystyle \frac{\partial \mathrm{\Phi }}{\partial X}}(w,X)=0$$

(61)

have N discrete solutions on $\mathrm{supp}\phi$ for given $w\in {\mathbb{R}}^m$, denoted $X=X_r\left(w\right)$, for $r=1,\dots ,N$, while $\det {\displaystyle \frac{{\partial }^2\mathrm{\Phi }}{\partial X^2}}(w,X_r\left(w\right))\ne 0$. Then, these solutions are real and the following estimate holds

$$I_{\phi ,\mathrm{\Phi }}(w;ħ)=\sum _{r=1}^N{\displaystyle \frac{\phi \left(X_r\left(w\right)\right)}{\sqrt{\det -\frac{{\partial }^2\mathrm{\Phi }}{\partial X^2}(w,X_r\left(w\right))}}}{e}^{{\displaystyle \frac{i}{ħ}}\mathrm{\Phi }(w,X_r\left(w\right))}(1+o(ħ)),ħ\to 0^{+},$$

(62)

where $\sqrt{·}$ is the principal branch of the square root function.

Proof. 

The proof follows directly from that of Nazaikinskii, Oshmyan, Sternin, and Shatalov, in [31]. For each $w\in {\mathbb{R}}^m$, choose a neighborhood of $X_r\left(w\right),r=1,\dots ,N$, contained in $\mathrm{supp}\phi$, for $r=1,\dots ,N$, with each neighborhood chosen so as to isolate a single solution. One then repeats the argument of [31] separately for each such neighborhood. □

The possibility of deriving the Van Vleck formula from the anisotropic Gaussian approximation was already noted in [19]; the detailed proof we present here places the formula in a broader context and clarifies its relationship with other semiclassical formulae, such as trace formulae.

A relevant derivation of the Van Vleck formula on similar premises exists in the literature, by Bily and Robert [33], in the present work we present a relatively more direct, simpler and clearer construction of the anisotropic Gaussian semi-classical Schrödinger propagator on the basis of a more powerful and explicit version of the complex stationary phase lemma [31], as well as based on simple identities involving the constituent of the underlying anisotropic Gaussian semi-classical time evolution variational matrices. We also note the derivation by Blair [34], with a different approach and methodology, from a more general viewpoint involving frequency-dependent phases for larger time, $T=O\left(\log ħ\right)$.

Theorem 3.

Let H satisfy the growth condition (26), with the additional assumption that the generated flow develops no caustics in the given time interval, in the sense that $\det {\displaystyle \frac{\partial q_t}{\partial p}}(q,p)\ne 0$ for $t_0\le t\le t_0+T$. Also, for $n\in \mathbb{N}$, let the sequence of cut-off functions ${\chi }_n\in C_0^{\infty }(\mathbb{R},\mathbb{C})$ have connected support, containing the ball $B_{{\rho }_n}\left(0\right)$, for some increasing diverging real sequence $\left\{{\rho }_n\right\}$, taking the value 1 away from its boundary, and satisfying the point-wise condition ${\lim }_{n\to +\infty }{\chi }_n(q,p)=1$ for all $(q,p)\in {\mathbb{R}}^{2d}$. We define the approximate propagator $U_{{\chi }_n}^Z(t_0,t)$, with kernel

$$K_{{\chi }_n}^Z(x,y,t_0,t;ħ)={\left({\displaystyle \frac{1}{2\pi ħ}}\right)}^d\int {\chi }_n(q,p){\overline{G}}_{(q,p)}(y;ħ)G_{(q,p)}^Z(x,t;ħ)dqdp.$$

(63)

Then, the kernel $K_{{\chi }_n}^Z$ satisfies the Van Vleck formula, in the sense that, for given $x,y,t_0,t$ and for large enough n, we have

$$\begin{array}{c}\hfill K_{{\chi }_n}^Z(x,y,t_0,t;ħ)={\left({\displaystyle \frac{1}{2\pi iħ}}\right)}^{d/2}\sum _{r=1}^N{\chi }_n(y,p_r(y,x,t_0,t))\\ \hfill \times \sqrt{\left|\det {\displaystyle \frac{\partial p}{\partial q_t}}(t;y,p_r(y,x,t_0,t))\right|}{e}^{{\displaystyle \frac{i}{ħ}}S(y,p_r(y,x,t_0,t),t_0,t)-{\displaystyle \frac{i\pi }{2}}{m}_r}(1+o_n\left(ħ\right)),ħ\to 0^{+},\end{array}$$

(64)

where the N momenta $p_r(y,x,t_0,t)$ are the admissible initial momenta of the orbit emanating from point $(q,p)$ at time $t_0$, terminating at point $(q_t,p_t)$ at time $t\ge t_0$, such that $q=y$ and $q_t=x$.

Proof. 

Expressing the kernel $K_{{\chi }_n}^Z$ in the form of Theorem 2, we have

$$K_{{\chi }_n}^Z(x,y,t_0,t;ħ)={\left({\displaystyle \frac{1}{2\pi ħ}}\right)}^{3d/2}\int {\phi }_n(q,p,t_0,t)e^{{\displaystyle \frac{i}{ħ}}\mathrm{\Phi }(x,y,q,p,t_0,t)}dqdp,$$

(65)

where the phase is

$$\begin{array}{c}\hfill \mathrm{\Phi }(x,y,q,p,t_0,t)=S(q,p,t_0,t)+p_t·(x-q_t)+{\displaystyle \frac{1}{2}}(x-q_t)·Z(q,p,t_0,t)(x-q_t)\\ \hfill -p·(y-q)+{\displaystyle \frac{i}{2}}{|y-q|}^2\end{array}$$

(66)

and the amplitude is

$${\phi }_n(q,p,t_0,t)=2^{d/2}{\chi }_n(q,p){\left(\det \mathrm{Im}Z(q,p,t_0,t)\right)}^{1/4},$$

(67)

the later by virtue of the identity

$${\left(\det A\right)}^{-1/2}={\left(\det \mathrm{Im}Z\right)}^{1/4}$$

(68)

which derives from the relations (25).

Both phase and amplitude satisfy the smoothness conditions of Theorem 2. In addition, we have that

$$\mathrm{Im}\mathrm{\Phi }\left(t\right)={\displaystyle \frac{1}{2}}(x-q_t)·\mathrm{Im}Z\left(t\right)(x-q_t)+{\displaystyle \frac{1}{2}}{|y-q|}^2\ge 0$$

(69)

everywhere, as $\mathrm{Im}Z\succ 0$, so that the zero-level set of the imaginary part of the phase $\mathrm{\Phi }$, for fixed x, y and $t>t_0$, is

$$\{(q,p)\in {\mathbb{R}}^{2d}|q=y,\pi g^{t_0,t}(q,p)=x\}.$$

(70)

where $\pi :{\mathbb{R}}^{2d}\to {\mathbb{R}}^d$ is the canonical projection from phase space onto configuration space. Given the smoothness of H, the zero-level set is either finite or empty. For $(q,p)$ belonging to the zero-level set, we trivially have

$${\displaystyle \frac{\partial \mathrm{\Phi }}{\partial q}}=0,{\displaystyle \frac{\partial \mathrm{\Phi }}{\partial p}}=0$$

(71)

by virtue of the property of the action

$${\displaystyle \frac{\partial S}{\partial q}}=-p+p_t·{\displaystyle \frac{\partial q_t}{\partial q}},{\displaystyle \frac{\partial S}{\partial p}}=p_t·{\displaystyle \frac{\partial q_t}{\partial p}}.$$

(72)

The zero-level set consists of all initial phase space points whose orbits project to trajectories on configuration space joining points y and x between time instances $t_0$ and t, respectively. The equation $x=q_t(q,p)$ for $q=y$ becomes the equation for the possible initial momenta p for which a trajectory satisfies the boundary conditions $q=y$ and $q_t=x$.

As the Hamiltonian is smooth in $(q,p)$ and continuous in t, the final position is smooth in its dependence on the initial position q. Thus, given that for $t_0\le t\le t_0+T$ we have $\det {\displaystyle \frac{\partial q_t}{\partial p}}\ne 0$, the equation $q=q_t(y,p)$ possesses countably many solutions, $p_r=p_r(y,x,t_0,t)$ for $r=1,\dots ,N$, for some $N=N(x,y,t_0,t)$.

By direct computation, for $(q,p)=(y,p_r(y,x,t_0,t))$, for any $r=1,\dots ,N$, the Hessian blocks of the phase are given by

$$\begin{array}{c}\hfill {\mathrm{\Phi }}_{qq}=iI-{\left({\displaystyle \frac{\partial p_t}{\partial q}}\right)}^T{\displaystyle \frac{\partial q_t}{\partial q}}+{\left({\displaystyle \frac{\partial q_t}{\partial q}}\right)}^{T}Z{\displaystyle \frac{\partial q_t}{\partial q}},\\ \hfill {\mathrm{\Phi }}_{pp}=-{\left({\displaystyle \frac{\partial p_t}{\partial p}}\right)}^T{\displaystyle \frac{\partial q_t}{\partial p}}+{\left({\displaystyle \frac{\partial q_t}{\partial p}}\right)}^{T}Z{\displaystyle \frac{\partial q_t}{\partial p}},\\ \hfill {\mathrm{\Phi }}_{qp}=-{\left({\displaystyle \frac{\partial p_t}{\partial q}}\right)}^T{\displaystyle \frac{\partial q_t}{\partial p}}+{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{\partial q_t}{\partial q}}\right)}^{T}Z{\displaystyle \frac{\partial q_t}{\partial p}}+{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{\partial q_t}{\partial p}}\right)}^{T}Z{\displaystyle \frac{\partial q_t}{\partial q}},\\ \hfill {\mathrm{\Phi }}_{pq}=-{\left({\displaystyle \frac{\partial q_t}{\partial p}}\right)}^T{\displaystyle \frac{\partial p_t}{\partial q}}+{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{\partial q_t}{\partial p}}\right)}^{T}Z{\displaystyle \frac{\partial q_t}{\partial q}}+{\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{\partial q_t}{\partial q}}\right)}^{T}Z{\displaystyle \frac{\partial q_t}{\partial p}},\end{array}$$

(73)

while the determinant of the Hessian matrix on the zero-level set is

$${\det \mathrm{Hess}}_{(q,p)}\mathrm{\Phi }=\det \left({\displaystyle \frac{2}{i}}{\left({\displaystyle \frac{\partial q_t}{\partial q}}+i{\displaystyle \frac{\partial q_t}{\partial p}}\right)}^{-1}{\displaystyle \frac{\partial q_t}{\partial p}}\right),$$

(74)

which is non-singular, as the matrices ${\displaystyle \frac{\partial q_t}{\partial q}}+i{\displaystyle \frac{\partial q_t}{\partial p}}=A\left(t\right)$ and ${\displaystyle \frac{\partial q_t}{\partial p}}$ are non-singular within the given time interval.

Thus, we have

$$\begin{array}{c}\hfill {\displaystyle \frac{{\phi }_n(y,p_r(y,x,t_0,t),t_0,t)}{\sqrt{\det -{\mathrm{Hess}}_{(q,p)}\mathrm{\Phi }(x,y,y,p_r(y,x,t_0,t),t_0,t)}}}=2^{d/2}{\chi }_n(y,p_r(y,x,t_0,t))\\ \hfill {\left(\det \mathrm{Im}Z\right(y,p_r(y,x,t_0,t),t_0,t\left)\right)}^{1/4}{\left\{\det \left(-{\displaystyle \frac{2}{i}}{\left({\displaystyle \frac{\partial q_t}{\partial q}}+i{\displaystyle \frac{\partial q_t}{\partial p}}\right)}^{-1}{\displaystyle \frac{\partial q_t}{\partial p}}\right)\right\}}^{-1/2}\\ \hfill =i^{-d/2}{\chi }_n(y,p_r(y,x,t_0,t))\sqrt{\det {\displaystyle \frac{\partial p}{\partial q_t}}(y,p_r(y,x,t_0,t))},\end{array}$$

(75)

by virtue of relation (68).

Assuming n is large enough so that $\mathrm{supp}{\phi }_n(·,t_0,t)$ contains the zero-level set and that $T=O\left(1\right)$, conditions of Theorem 2 are met for each $t_0$ and $t\in [t_0,t_0+T]$, so that we obtain

$$\begin{array}{c}\hfill K_{{\chi }_n}^Z(x,y,t_0,t;ħ)={\left({\displaystyle \frac{1}{2\pi iħ}}\right)}^{d/2}\sum _{r=1}^N{\chi }_n(y,p_r(y,x,t_0,t))\sqrt{\left|\det {\displaystyle \frac{\partial p}{\partial q_t}}(y,p_r(y,x,t_0,t))\right|}\\ \hfill \times e^{{\displaystyle \frac{i}{ħ}}S(y,p_r(y,x,t_0,t),t_0,t)-{\displaystyle \frac{i\pi }{2}}{m}_r}(1+o_n\left(ħ\right)),ħ\to 0^{+}.\end{array}$$

(76)

In the above, the index $r=1,\dots ,N$ enumerates the orbits emanating from point $(q,p)$ at time $t_0$ with momentum $p_r(y,x,t_0,t)$, terminating at point $(q_t,p_t)$ at time $t\ge t_0$, such that $q=y$ and $q_t=x$, while

$$m_r=\mathrm{ind}\left({\left({\displaystyle \frac{\partial q_t}{\partial q}}+i{\displaystyle \frac{\partial q_t}{\partial p}}\right)}^{-1}{\displaystyle \frac{\partial p}{\partial q_t}}\right):=\sum _{\lambda }\mathrm{Arg}\left(\lambda \right),$$

(77)

for $\lambda \in \mathrm{Spec}\left({\left({\displaystyle \frac{\partial q_t}{\partial q}}+i{\displaystyle \frac{\partial q_t}{\partial p}}\right)}^{-1}{\displaystyle \frac{\partial p}{\partial q_t}}\right)$, i.e., the index of the monodromy matrix, the excess of its positive over negative eigenvalues. □

Remark 2.

As $n\to +\infty$, $K_{{\chi }_n}^Z$ clearly converges to $K^Z$, and we formally expect that the remainder term $o_n\left(ħ\right)$ vanishes, so that asymptotic equivalence (59) holds.