We consider difference operators in L2 on R of the form Lf(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.Comment: 27p