CHARACTERIZATION OF POTENTIAL SMOOTHNESS OF ONE DIMENSIONAL DIRAC OPERATOR SUBJECT TO GENERAL BOUNDARY CONDITIONS AND ITS RIESZ BASIS PROPERTY

İlker Arslan
Mathematics, PhD Dissertation, 2015

Thesis Jury

Prof. Dr. Plamen Borissov Djakov (Thesis Advisor), Prof. Dr. Albert Erkip, Prof. Dr. Cihan Kemal Saçlıoğlu, Prof. Dr. Hüsnü Ata Erbay, Assist. Prof. Dr. Ahmet Batal

Date & Time: 13th of November, 2015 –  16:00 AM

Place: Karaköy Minerva Palas

Keywords : Dirac operator, potential smoothness

Abstract

            The one dimensional Dirac operators with periodic potentials subject to periodic, antiperiodic and a special family of general boundary conditions have discrete spectrums. It is known that, for large enough |n| in the disc centered at n of radius 1/4, the operator has exactly two eigenvalues  (counted according to multiplicity) which are periodic (for even n) or antiperiodic (for odd n) and one eigenvalue derived from the general boundary conditions. These eigenvalues construct a deviation which is sum of the distance between two periodic (or antiperiodic) eigenvalues and the distance between one of the periodic (or antiperiodic) eigenvalue and one eigenvalue from the general boundary conditions. We show that the smoothness of the potential could be characterized by the decay rate of this spectral deviation. Furthermore, it is shown that the Dirac operator with periodic or antiperiodic boundary condition has the Riesz basis property if and only if the absolute value of the ratio of these deviations are bounded.