Characterization of potential smoothness and riesz basis property of hill-schrödinger operators with singular periodic potentials in terms of periodic, antiperiodic and neumann spectra

Official URL: http://risc01.sabanciuniv.edu/record=b1558934 (Table of Contents)

The Hill-Schrödinger operators, considered with singular complex valued periodic potentials, and subject to the periodic, anti-periodic or Neumann boundary conditions, have discrete spectra. For su ciently large integer n, the disk with radius n and with center square of n, contains two periodic (if n is even) or anti-periodic (if n is odd) eigenvalues and one Neumann eigenvalue. We construct two spectral deviations by taking the di erence of two periodic (or anti-periodic) eigenvalues and the difference of a periodic (or anti-periodic) eigenvalue and the Neumann eigenvalue. We show that asymptotic decay rates of these spectral deviations determine the smoothness of the potential of the operator, and there is a basis consisting of periodic (or anti-periodic) root functions if and only if the supremum of the absolute value of the ratio of these deviations over even (respectively, odd) n is nite. We also show that, if the potential is locally square integrable, then in the above results one can replace the Neumann eigenvalues with the eigenvalues coming from a special class of boundary conditions more general than the Neumann boundary conditions.