Difference Sturm--Liouville problems in the imaginary direction

TL;DR

This paper studies difference Sturm--Liouville problems involving complex shifts, providing explicit spectral decompositions for operators with continuous spectra and discussing boundary condition analogs in the complex domain.

Contribution

It introduces explicit spectral decompositions for difference operators with continuous spectra and explores boundary condition analogs in the complex setting.

Findings

Explicit spectral decompositions for several operators with continuous spectra.

Analysis of boundary condition analogs for difference operators in the complex plane.

Extension of classical Sturm--Liouville theory to complex difference operators.

Abstract

We consider difference operators in $L^2$ on $\R$ of the form $$ L f(s)=p(s)f(s+i)+q(s) f(s)+r(s) f(s-i) ,$$ where $i$ is the imaginary unit. The domain of definiteness are functions holomorphic in a strip with some conditions of decreasing at infinity. Problems of such type with discrete spectra are well known (Meixner--Pollaszek, continuous Hahn, continuous dual Hahn, and Wilson hypergeometric orthogonal polynomials). We write explicit spectral decompositions for several operators $L$ with continuous spectra. We also discuss analogs of 'boundary conditions' for such operators.