DISPERSIVE ESTIMATES FOR THE SCHRÖDINGER EQUATION WITH FINITE RANK PERTURBATIONS

作     者：Cheng, Han Huang, Shanlin Zheng, Quan 

作者机构：School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Hubei Wuhan430074 China School of Mathematics and Statistics Huazhong University of Science and Technology Hubei Wuhan430074 China 

出 版 物：《arXiv》 (arXiv)

年 卷 期：2022年

核心收录：

摘      要：In this paper, we investigate dispersive estimates for the time evolution of Hamiltonians (Equation presented) where each j satisfies certain smoothness and decay conditions. We show that, under a spectral assumption, there exists a constant C = C(N, d, 1,..., N) 0 such that (Equation presented), for t 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. The new idea in our approach is to establish the Aronszajn-Krein type formula for finite rank perturbations. This allows us to reduce the analysis to the rank one case and solve the problem in a unified manner. Moreover, we show that in some specific situations, the constant C(N, d, 1,..., N) grows polynomially in N. Finally, as an application, we are able to extend the results to N = ∞ and deal with some trace class *** Codes 35J10, 81Q15, 42B37 Copyright © 2022, The Authors. All rights reserved.