On the Fundamental Solution of Semiclassical Schrödinger Equations at Resonant Times

Abstract:

We consider perturbations of the semiclassical harmonic oscillator of the form , x∈R ^m, with as and δ,μ∈ (0,1), and we investigate the fundamental solution E(t,x,y) of the corresponding time-dependent Schrödinger equation. We prove that at resonant times t=nπ (n∈Z) it admits a semiclassical asymptotics of the form: E(nπ,x,y) ∼h ^{− m (1+ν)/2} a ₀ e ^{iS ( x , y )/ h} with a ₀≠0 and ν=δ/(1−μ), under the conditions x≠(−1)ⁿ y and ν <1.