Dispersive estimates for the Schr\"{o}dinger equation with finite rank perturbations

TL;DR

This paper establishes dispersive $L^1-L^{ olinebreak\infty}$ estimates for Schrödinger operators with finite rank perturbations of the Laplacian, introducing a new approach based on Aronszajn-Krein formulas applicable in any dimension.

Contribution

It provides the first $L^1-L^{\infty}$ dispersive estimates for finite rank perturbations of the Laplacian in all dimensions, using a unified approach via Aronszajn-Krein formulas.

Findings

Proves dispersive estimates for finite rank perturbations in any dimension.

Develops a unified method reducing the problem to rank one cases.

Extends results to infinite rank and trace class perturbations.

Abstract

In this paper, we investigate dispersive estimates for the time evolution of Hamiltonians $$ H=-\Delta+\sum_{j=1}^N\langle\cdot\,, \varphi_j\rangle \varphi_j\quad\,\,\,\text{in}\,\,\,\mathbb{R}^d,\,\, d\ge 1, $$ where each $\varphi_j$ satisfies certain smoothness and decay conditions. We show that, under a spectral assumption, there exists a constant $C=C(N, d, \varphi_1,\ldots, \varphi_N)>0$ such that $$ \|e^{-itH}\|_{L^1-L^{\infty}}\leq C t^{-\frac{d}{2}}, \,\,\,\text{for}\,\,\, t>0. $$ As far as we are aware, this seems to provide the first study of $L^1-L^{\infty}$ 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 $C(N, d, \varphi_1,\ldots, \varphi_N)$ grows polynomially in $N$. Finally, as an application, we are able to extend the results to $N=\infty$ and deal with some trace class perturbations.