On Absence of Threshold Resonances for Schrodinger and Dirac Operators

TL;DR

This paper proves the absence of threshold resonances for certain Schrödinger and Dirac operators with short-range potentials across various space dimensions using a unified analytical approach.

Contribution

It introduces a unified method employing a homogeneous Lippmann-Schwinger equation to establish the absence of threshold resonances in both Schrödinger and Dirac operators.

Findings

No zero-energy resonances for Schrödinger operators in dimensions n ≥ 5.

No zero-energy resonances for massless Dirac operators in dimensions n ≥ 3.

No resonances at ±m for massive Dirac operators in dimensions n ≥ 5.

Abstract

Using a unified approach employing a homogeneous Lippmann-Schwinger-type equation satisfied by resonance functions and basic facts on Riesz potentials, we discuss the absence of threshold resonances for Dirac and Schrodinger operators with sufficiently short-range interactions in general space dimensions. More specifically, assuming a sufficient power law decay of potentials, we derive the absence of zero-energy resonances for massless Dirac operators in space dimensions $n \geq 3$, the absence of resonances at $\pm m$ for massive Dirac operators (with mass $m > 0$) in dimensions $n \geq 5$, and recall the well-known case of absence of zero-energy resonances for Schr\"odinger operators in dimension $n \geq 5$.