CHARACTERIZATION OF POTENTIAL SMOOTHNESS AND RIESZ BASIS PROPERTY OF HILL-SCHR ODINGER OPERATORS WITH SINGULAR PERIODIC POTENTIALS IN TERMS OF PERIODIC, ANTIPERIODIC AND NEUMANN SPECTRA

Ahmet Batal
Mathematics, Ph.D. Dissertation, 2014

Thesis Jury
Prof. Dr. Plamen Djakov (Thesis Supervisor), Prof. Dr. Aydın Aytuna, Prof. Dr. Cihan Saçlıoğlu, Prof. Dr. Albert Erkip, Prof. Dr.  Hüsnü Erbay

Date &Time: January 13th, 2014 - 17:00

Place: Sabancı Üniversitesi Karaköy Minerva Palas

Keywords:  potential smoothness, Riesz basis

Abstract 

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 sufficiently 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 difference of two periodic (or anti-periodic) eigenvalues and the difference  of a periodic (or anti-periodic) eigenvlaue and the Neumann eigenvalue. We show that asymptotic decay rate of these spectral deviations determines 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 (respectivel, odd) n is finite. 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.