Difference Sturm–Liouville problems in the imaginary direction

  • Yury A. Neretin

    Universität Wien, Austria


We consider difference operators in L2L^2 on R\mathbb R of the form

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

where ii 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 L\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