ON THE HERMAN-KLUK SEMICLASSICAL APPROXIMATION

For a subquadratic symbol H on R-d X R-d = T*(R-d), the quantum propagator of the time dependent Schrodinger equation ih partial derivative psi/partial derivative t = (H) over cap psi is a Semiclassical Fourier-Integral Operator when (H) over cap = H(x, hD(x)) (h-Weyl quantization of H). Its Schwartz kernel is described by a quadratic phase and an amplitude. At every time t, when h 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 h 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-2 estimates when h 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 broken vertical bar t broken vertical bar << 1/4 delta broken vertical bar log h broken vertical bar where delta > 0 is a stability parameter for the classical system.