Dependence of eigenvalues on the diffusion operators with random jumps from the boundary

Abstract

This paper deals with a non-self-adjoint eigenvalue problem{a(x)y″(x)+b(x)y′(x)=λy(x),y(0)=∫01y(x)dν0(x), y(1)=∫01y(x)dν1(x), which is associated with the generator of one dimensional diffusions with random jumps from the boundary. We focus on the dependence of spectral gap, eigenvalues and eigenfunctions on the coefficients a, b and the probability distributions ν0, ν1. To prove this, we show that all the eigenvalues are confined to a parabolic neighborhood of the real axis. Moreover, we also prove that zero is an algebraically simple eigenvalue of the problem.