On the Spectrum of the Schrödinger Operator with Periodic Surface Potential

We consider a discrete Schrödinger operator H=−Δ+V acting in l2(\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb{Z}$$ \end{document}d), with periodic potential V supported by the subspace ‘surface’ {0}×\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathbb{Z}$$ \end{document}d2. We prove that the spectrum of H is purely absolutely continuous, and that surface waves oscillate in the longitudinal directions to the ‘surface’. We also find an explicit formula for the generalized spectral shift function introduced by the author in Helv. Phys. Acta.72 (1999), 93–122.