Difference Sturm--Liouville problems in the imaginary direction

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.