# Difference Sturm–Liouville problems in the imaginary direction

### Yury A. Neretin

Universität Wien, Austria

## Abstract

We consider difference operators in $L^2$ on $\mathbb R$ of the form

$\mathcal L f(s)=p(s)f(s+i)+q(s) f(s)+r(s) f(s-i),$

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--Pollaczek, continuous Hahn, continuous dual Hahn, and Wilson hypergeometric orthogonal polynomials). We write explicit spectral decompositions for several operators $\mathcal L$ with continuous spectra. We also discuss analogs of 'boundary conditions' for such operators.

## Cite this article

Yury A. Neretin, Difference Sturm–Liouville problems in the imaginary direction. J. Spectr. Theory 3 (2013), no. 3, pp. 237–269

DOI 10.4171/JST/44