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

TL;DR

The paper derives sharp pointwise kernel bounds for the fundamental solution of a higher-order Schrödinger equation with finite rank perturbations, showing that the perturbed propagator e^{-itH_N}P_ac(H_N) obeys the same dispersive decay as the free operator in the regime t ≠ 0. The authors develop a comprehensive framework based on Stone’s formula and the Aronszajn-Krein resolvent identity, performing precise high- and low-energy analyses, including detailed resolvent expansions and oscillatory integral estimates. They extend known rank-one results to finite-rank perturbations by matrix-valued resolvent techniques and space decomposition on L^2, obtaining L^p-L^q decay in a quadrilateral region and addressing endpoint spaces via Hardy/BMO theory. The results significantly broaden the understanding of dispersive behavior for higher-order Schrödinger operators under finite-rank perturbations, with implications for nonlinear and spectral problems in any dimension and for general m > 1.

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 $$H={(-Δ)}^{m}+\displaystyle\sum_{j=1}^{N} \langle\cdotp ,{\varphi }_{j} \rangle{\varphi }_{j},$$ with each $\varphi_j$ ($1\le j\le N$) 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.

Figures (1)

• Figure 1: The $L^p-L^q$ estimates

Theorems & Definitions (46)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• Lemma 2.7