Dispersive estimates for the Schrödinger equation with finite rank perturbations

Abstract

In this paper, we investigate dispersive estimates for the time evolution of Hamiltonians $$H=-\hat{I}”+\sum _{j=1}^N⟨\cdot ,\ddot{I}{†}_j⟩\ddot{I}{†}_j\text{in}{\mathbb{R}}^d,d\ge 1,$$ where each $\ddot{I}{†}_j$ satisfies certain smoothness and decay conditions. We show that, under a spectral assumption, there exists a constant $C=C(N,d,\ddot{I}{†}_1,\dots ,\ddot{I}{†}_N)>0$ such that $$\Vert e^{-itH}{\Vert }_{L^1-L^{\infty }}\le C{t}^{-\frac{d}{2}},\text{for}t>0.$$ 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 $C(N,d,\ddot{I}{†}_1,\dots ,\ddot{I}{†}_N)$ 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.

78 pages