On critical value of the coupling constant in exterior elliptic problems

TL;DR

This paper investigates the critical coupling constant in exterior elliptic problems, analyzing how it depends on boundary conditions and potential support, and explores the discrete spectrum in non-symmetric operators with specific boundary conditions.

Contribution

It introduces a detailed analysis of the critical coupling constant in exterior elliptic problems, including its dependence on boundary conditions and potential support, and studies the discrete spectrum in non-symmetric operators.

Findings

Critical value of the coupling constant determines spectrum type.

Dependence of critical value on boundary conditions and potential support.

Discrete spectrum analysis for non-symmetric operators with FKW boundary condition.

Abstract

We consider exterior elliptic problems with coefficients stabilizing at infinity and study the critical value $\beta_{cr}$ of the coupling constant (the coefficient at the potential) that separates operators with a discrete spectrum and those without it. The dependence of $\beta_{cr}$ on the boundary condition and on the distance between the boundary and the support of the potential is described. The discrete spectrum of a non-symmetric operator with the FKW boundary condition (that appears in diffusion processes with traps) is also investigated.