ON THE HERMAN-KLUK SEMICLASSICAL APPROXIMATION

For a subquadratic symbol H on ℝd × ℝd = T*(ℝd), 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 Ĥ = H(x, ℏDx) (ℏ-Weyl quantization of H). Its Schwartz kernel is described by a quadratic phase and an amplitude. At every time t, when ℏ 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 ℏ 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 L2 estimates when ℏ 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 δ > 0 is a stability parameter f...