Asymptotic Behaviour of Resonance Eigenvalues of the Schr\"odinger operator with a Matrix Potential

TL;DR

This paper analyzes the asymptotic behavior of resonance eigenvalues of a Schrödinger operator with a matrix potential under Neumann boundary conditions in a multi-dimensional setting, focusing on eigenvalues near diffraction planes.

Contribution

It provides new insights into the asymptotic distribution of resonance eigenvalues for matrix Schrödinger operators in higher dimensions.

Findings

Eigenvalues near diffraction planes exhibit specific asymptotic patterns.

Resonance eigenvalues are characterized within the framework of perturbation theory.

Results extend understanding of spectral properties of matrix Schrödinger operators.

Abstract

We will discuss the asymptotic behaviour of the eigenvalues of Schr\"{o}dinger operator with a matrix potential defined by Neumann boundary condition in $L_2^m(F)$, where $F$ is $d$-dimensional rectangle and the potential is a $m \times m$ matrix with $m\geq 2$, $d\geq 2$ , when the eigenvalues belong to the resonance domain, roughly speaking they lie near planes of diffraction. \textbf{Keywords:} Schr\"{o}dinger operator, Neumann condition, Resonance eigenvalue, Perturbation theory. \textbf{AMS Subject Classifications:} 47F05, 35P15