# On the Herman-Kluk Semiclassical Approximation Robert, Didier

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Author
Robert, Didier
Appears In
Reviews in Mathematical Physics
Subjects
mathematical physics; [phys:mphy] physics/mathematical physics; [math:math_mp] mathematics/mathematical physics
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Summary
To access, purchase, authenticate, or subscribe to the full-text of this article, please visit this link: http://dx.doi.org/10.1142/S0129055X1000417X Byline: DIDIER ROBERT For a subquadratic symbol H on a.sup.d x a.sup.d = T*(a.sup.d), the quantum propagator of the time dependent Schrodinger equation $i\hbar\frac{\partial\psi}{\partial t} = \hat H\psi$ is a Semiclassical Fourier-Integral Operator when A[currency] = H(x, aD.sub.x) (a-Weyl quantization of H). Its Schwartz kernel is described by a quadratic phase and an amplitude. At every time t, when a 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 a for the amplitude. In this paper, our goal is to revisit this well-known and fundamental result with emphasis on the flexibility for the choice of a quadratic complex phase function and on global L.sup.2 estimates when a 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 $|t|\ll \frac{1}{4\delta}|{{\rm log}}\, \hbar|$ where I[acute accent] > 0 is a stability parameter for the classical system.
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http://trove.nla.gov.au/work/187905
Work ID
187905

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