Pointwise estimates for the fundamental solutions of higher order schrödinger equations with finite rank perturbations

Abstract

This paper is dedicated to studying pointwise estimates of the fundamental solution for the higher order Schrödinger equation: % we investigate the fundamental solution of the higher order Schrödinger equation $$i{\partial}_{t}u(x,t)=Hu(x,t),\ \ \ t\in \mathbb{R},\ x\in {\mathbb{R}}^{n},$$ where the Hamiltonian $H$ is defined as $$H={(-Δ)}^{m}+\displaystyle\sum_{j=1}^{N} \langle\cdotp ,{φ}_{j} \rangle{φ}_{j},$$ with each $φ_j$ ($1\le j\le N$) satisfying certain smoothness and decay conditions. %Let ${P}_{ac}(H)$ denote the projection onto the absolutely continuous space of $H$. We show that for any positive integer $m>1$ and spatial dimension $n\ge 1$, %under a spectral assumption, the operator is sharp in the sense that it ${e}^{-i tH}P_{ac}(H)$ has an integral kernel $K(t,x,y)$ satisfying the following pointwise estimate: $$\left |K(t,x,y)\right |\lesssim |t|^{-\frac{n}{2m}}(1+|t|^{-\frac{1}{2m}}\left | x-y\right |)^{-\frac{n(m-1)}{2m-1}} ,\ \ t\ne 0,\ x,y\in {\mathbb{R}}^{n}.$$ This estimate is consistent with the upper bounds in the free case. As an application, we derive $L^p-L^q$ decay estimates for the propagator ${e}^{-ıtH}P_{ac}(H)$, where the pairs $(1/p, 1/q)$ lie within a quadrilateral region in the plane.