Difference Sturm--Liouville problems in the imaginary direction

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We consider difference operators in L² on 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.