Difference Sturm–Liouville problems in the imaginary direction

Abstract

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

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.