A Limiting absorption principle for high-order Schr\"odinger operators in critical spaces

TL;DR

This paper establishes a limiting absorption principle for high-order Schrödinger operators in critical spaces, extending previous results by employing advanced harmonic analysis tools like the Stein--Tomas theorem and a sharp trace lemma.

Contribution

It introduces a novel approach to boundary operators using the restriction theorem of Fourier transform for high-order Schrödinger operators.

Findings

Proves a limiting absorption principle for a broad class of potentials.

Utilizes Stein--Tomas theorem in Lorentz spaces for analysis.

Employs a sharp trace lemma to handle boundary operators.

Abstract

In this paper, we prove a limiting absorption principle for high-order Schr\"odinger operators with a large class of potentials which generalize some results by A. Ionescu and W. Schlag. Our main idea is to handle the boundary operators by the restriction theorem of Fourier transform. Two key tools we use in this paper are the Stein--Tomas theorem in Lorentz spaces and a sharp trace lemma given by S. Agmon and L. H\"ormander