Dispersive estimates for the Schrödinger equation with finite rank perturbations

Abstract

In this paper, we investigate dispersive estimates for the time evolution of HamiltoniansH=−Δ+∑j=1N〈⋅,φj〉φjinRd,d≥1, where each φj satisfies certain smoothness and decay conditions. We show that, under a spectral assumption, there exists a constant C=C(N,d,φ1,…,φN)>0 such that‖e−itH‖L1−L∞≤Ct−d2,fort>0. As far as we are aware, this seems to provide the first study of L1−L∞ estimates for finite rank perturbations of the Laplacian in any dimension.We first deal with rank one perturbations (N=1). Then we turn to the general case. By using an Aronszajn-Krein type formula for finite rank perturbations, we reduce the problem to the rank one case and solve it in a unified manner. Moreover, we show that in some specific situations, the constant C(N,d,φ1,…,φN) grows polynomially in N. Finally, as an application, we are able to extend the results to N=∞ and deal with some trace class perturbations.