Analytic wave front set for solutions to Schrödinger equations

Abstract

This paper is a continuation of [A. Martinez, S. Nakamura, V. Sordoni, Analytic smoothing effect for the Schrödinger equation with long-range perturbation, Comm. Pure Appl. Math. LIX (2006) 1330–1351], where an analytic smoothing effect was proved for long-range type perturbations of the Laplacian H 0 on R n . In this paper, we consider short-range type perturbations H of the Laplacian on R n , and we characterize the analytic wave front set of the solution to the Schrödinger equation: e − i t H f , in terms of that of the free solution: e − i t H 0 f , for t < 0 in the forward non-trapping region. The same result holds for t > 0 in the backward non-trapping region. This result is an analytic analogue of results by Hassel and Wunsch [A. Hassel, J. Wunsch, The Schrödinger propagator for scattering metrics, Ann. of Math. 162 (2005) 487–523] and Nakamura [S. Nakamura, Wave front set for solutions to Schrödinger equations, J. Funct. Anal. 256 (2009) 1299–1309].