Multiplicities of Eigenvalues of the Diffusion Operator with Random Jumps from the Boundary

TL;DR

This paper investigates the eigenvalue multiplicities of a non-self-adjoint diffusion operator with boundary jumps, establishing a relationship between algebraic multiplicity and zero order of a characteristic function, aiding in concrete eigenvalue analysis.

Contribution

It introduces a method to determine eigenvalue multiplicities for diffusion operators with boundary jumps by linking algebraic multiplicity to the zero order of the characteristic function.

Findings

Algebraic multiplicity equals zero order of the characteristic function

Provides a practical way to compute eigenvalue multiplicities

Applicable to specific diffusion operators with boundary jumps

Abstract

This paper deals with a non-self-adjoint differential operator which is associated with a diffusion process with random jumps from the boundary. Our main result is that the algebraic multiplicity of an eigenvalue is equal to its order as a zero of the characteristic function $\Delta(\lambda) $. This can be used to determine the multiplicities of eigenvalues for concrete operators.