ASYMPTOTICS OF SPECTRAL GAPS OF HILL AND 1D DIRAC OPERATORS

Berkay Anahtarcı
Mathematics, PhD Dissertation, 2014

Thesis Jury

Prof. Dr. Plamen Djakov (Thesis Advisor), Prof. Dr. Albert Erkip, Prof. Dr. Cihan Saçlıoğlu

Prof. Dr. Hüsnü Ata Erbay, Prof. Dr. Aydn Aytuna

Date & Time: December 19th, 2014 –  2:00 PM

Place: Sabancı Üniversitesi Karaköy Minerva Palas (mezzanine meeting room)

Keywords : Hill operators, Dirac operators, asymptotics

Abstract

Let L be the Hill operator or the one-dimensional Dirac operator with pi-periodic potential considered on the real line R. The spectrum of L has a band-gap structure, that is, the intervals of continuous spectrum alternate with spectral gaps. The endpoints of these gaps are eigenvalues of the same differential operator L but considered on the interval [0, pi] with periodic or antiperiodic boundary conditions.

In this thesis considering the Hill and the one-dimensional Dirac operators, we provide precise asymptotics of the spectral gaps in case of specific potentials that are linear combinations of two exponential terms.