On the Herman-Kluk Semiclassical Approximation - Mathematical PhysicsReport as inadecuate

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Abstract: For a subquadratic symbol $H$ on $\R^d\times\R^d = T^*\R^d$, the quantumpropagator of the time dependent Schr\-odinger equation$i\hbar\frac{\partial\psi}{\partial t} = \hat H\psi$ is a SemiclassicalFourier-Integral Operator when $\hat H=Hx,\hbar D x$ $\hbar$-Weylquantization of $H$. Its Schwartz kernel is describe by a quadratic phase andan amplitude. At every time $t$, when $\hbar$ is small, it is -essentiallysupported- in a neighborhood of the graph of the classical flow generated by$H$, with a full uniform asymptotic expansion in $\hbar$ for the amplitude. Inthis paper our goal is to revisit this well known and fondamental result withemphasis on the flexibility for the choice of a quadratic complex phasefunction and on global $L^2$ estimates when $\hbar$ is small and time $t$ islarge. One of the simplest choice of the phase is known in chemical physics asHerman-Kluk formula. Moreover we prove that the semiclassical expansion for thepropagator is valid for $| t| << \frac{1}{4\delta}|\log\hbar|$ where $\delta>0$is a stability parameter for the classical system.

Author: Didier Robert LMJL

Source: https://arxiv.org/

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