Eigenvalue Asymptotics for the Schr\"odinger Operator with a Matrix Potential in a Single Resonance Domain

TL;DR

This paper derives high-order asymptotic formulas for eigenvalues of a Schr"odinger operator with a matrix potential in a rectangular domain, focusing on the single resonance domain, advancing spectral analysis in quantum mechanics.

Contribution

It provides the first detailed asymptotic formulas of arbitrary order for eigenvalues in the single resonance domain of matrix Schr"odinger operators.

Findings

Derived asymptotic formulas of arbitrary order for eigenvalues.

Analyzed eigenvalues in the single resonance domain.

Extended spectral analysis to matrix potentials.

Abstract

We consider a Schr\"odinger Operator with a matrix potential defined in $L_2^m(F)$ by the differential expression\begin{equation*} L(\phi(x))=(-\Delta+V(x))\phi(x) \end{equation*}and the Neumann boundary condition, where $F$ is the $d$ dimensional rectangle and $V$ is a martix potential, $m\geqslant 2, d\geqslant 2$. We obtain the asymptotic formulas of arbitrary order for the single resonance eigenvalues of the Schr\"odinger operator in $L_2^m(F)$.