On the Herman-Kluk Semiclassical Approximation

Abstract

For a subquadratic symbol $H$ on ${\mathbb{R}}^d\times {\mathbb{R}}^d=T^{\ast }({\mathbb{R}}^d)$, the quantum propagator of the time dependent Schrödinger equation $i\hslash \frac{\partial \psi }{\partial t}=\hat{H}\psi$ is a Semiclassical Fourier-Integral Operator when $\hat{H}=H(x,\hslash D_x)$ ($\hslash$-Weyl quantization of $H$). Its Schwartz kernel is describe by a quadratic phase and an amplitude. At every time $t$, when $\hslash$ is small, it is "essentially supported" in a neighborhood of the graph of the classical flow generated by $H$, with a full uniform asymptotic expansion in $\hslash$ for the amplitude. In this paper our goal is to revisit this well known and fondamental result with emphasis on the flexibility for the choice of a quadratic complex phase function and on global $L^2$ estimates when $\hslash$ is small and time $t$ is large. One of the simplest choice of the phase is known in chemical physics as Herman-Kluk formula. Moreover we prove that the semiclassical expansion for the propagator is valid for $|t|<<\frac{1}{4\delta }|\log \hslash |$ where $\delta >0$ is a stability parameter for the classical system.

Cited by in corpus (9)

• Computing quantum dynamics in the semiclassical regime
• Semiclassical wave packet dynamics for Hartree equations
• Semi-classical Time-frequency Analysis and Applications
• Sampling strategies for the Herman-Kluk propagator of the wavefunction
• Propagation of Wave Packets for Systems Presenting Codimension 1 Crossings
• A Diabatic Surface Hopping Algorithm based on Time Dependent Perturbation Theory and Semiclassical Analysis
• Asymptotic Approximations for the Phase Space Schrodinger Equation
• Propagation of Coherent States through Conical Intersections
• Scaling asymptotics of spectral Wigner functions