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

2025/01/05 by Chen, Xinyi, Cheng, Han, Huang, Shanlin
#Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2501.02562

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∂_tu(x,t)=Hu(x,t), t∈ ℝ, x∈ ℝⁿ, where the Hamiltonian H is defined as H=(-Δ)^m+∑_j=1^N ⟨\cdotp ,φ_j ⟩φ_j, with each φ_j (1≤ j≤ 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≥ 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: |K(t,x,y) |\lesssim |t|^-(n)/(2m)(1+|t|^-(1)/(2m) | x-y |)^{-(n(m-1))/(2m-1)} , t≠ 0, x,y∈ ℝⁿ. 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^-ıtHP_ac(H), where the pairs (1/p, 1/q) lie within a quadrilateral region in the plane.

Related

• Pointwise estimates for the fundamental solutions of higher order Schrödinger equations in low odd dimensions
• Pointwise estimates for the fundamental solutions of higher order Schrödinger equations in odd dimensions II: high dimensional case
• Dispersive estimates for the Schrödinger equation with finite rank perturbations
• Dispersive estimates for higher order Schrödinger operators with scaling-critical potentials
• Pointwise Lower Bounds on the Heat Kernels of Higher Order Schrödinger Operators