Remarks on the Fundamental Solution to Schrödinger Equation with Variable Coefficients

Abstract

We consider Schrödinger operators $H$ on $R^n$ with variable coefficients. Let $H_0=-\frac{1}{2}△$ be the free Schrödinger operator and we suppose $H$ is a "short-range" perturbation of $H_0$. Then, under the nontrapping condition, we show the time evolution operator: $e^{-itH}$ can be written as a product of the free evolution operator $e^{-it{H}_0}$ and a Fourier integral operator $W(t)$, which is associated to the canonical relation given by the classical mechanical scattering. We also prove a similar result for the wave operators. These results are analogous to results by Hassell and Wunsch, but the assumptions, the proof and the formulation of results are considerably different. The proof employs an Egorov-type theorem similar to those used in previous works by the authors combined with a Beals-type characterization of Fourier integral operators.

Cited by in corpus (2)

• Propagation of singularities for Schrödinger equations with modestly long range type potentials
• Formulas that represent Cauchy problem solution for momentum and position Schrödinger equation