On absence of threshold resonances for Schrödinger and Dirac operators

Fritz Gesztesy; Roger Nichols

doi:10.3934/dcdss.2020243

Article Contents

2020, Volume 13, Issue 12: 3427-3460. Doi: 10.3934/dcdss.2020243

This issue Previous Article Operators of order 2

$ n $

with interior degeneracy Next Article On hyperbolic mixed problems with dynamic and Wentzell boundary conditions

On absence of threshold resonances for Schrödinger and Dirac operators

Fritz Gesztesy^1, , and
Roger Nichols^2,3,

1.
Department of Mathematics, Baylor University, One Bear Place #97328, Waco, TX 76798-7328, USA
2.
Department of Mathematics, The University of Tennessee at Chattanooga, 415 EMCS Building, Dept. 6956
3.
615 McCallie Avenue, Chattanooga, TN 37403-2504, USA

^* Corresponding author: Fritz Gesztesy
^* Corresponding author: Fritz Gesztesy

Received: December 31, 2018

Early access: January 2020

Published: August 2020

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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 Schrödinger 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 \geqslant 3 $, the absence of resonances at $ \pm m $ for massive Dirac operators (with mass $ m > 0 $) in dimensions $ n \geqslant 5 $, and recall the well-known case of absence of zero-energy resonances for Schrödinger operators in dimension $ n \geqslant 5 $.

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Mathematics Subject Classification: Primary: 35J10, 35Q41, 45P05, 47A11, 47G10; Secondary: 35Q40, 47A10, 81Q10.

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