Asymptotics of spectral gaps of the 1d Schrödinger operator with Mathieu potential

Abstract

R üzerinde periyodik reel potansiyel fonksiyonu v(x) ile düşünülen, 1 boyutluSchrodinger operatörü L(y) = y00 + v(x)y özeşleniktir ve spektrumu boslukluyapdadr- surekli spektrumu spektral bosluklarla ayrlmstr. Bu tezde, L operatorunun spektral bosluklarnn asimtotik davransn inceliyoruz. Mathieu potansiyelfonksiyonu v(x) = 2a cos (2x) durumunda, Harrell-Avron-Simon'n spektralbosluklarn uzunluklaryla ilgili kesin asimtotik sonuclarna esdeger bir ispat veriyoruz.

The one-dimensional Schrödinger operator L(y) = ? y00 + v(x)y, considered on Rwith -periodic real-valued potential v(x), is self-adjoint, and its spectrum has agap-band structure- the intervals of continuous spectrum are separated by spectralgaps. In this thesis, we study the asymptotic behaviour of the spectral gaps of L.In the case of the Mathieu potential v(x) = 2a cos (2x), we give an alternative proofof the result of Harrell-Avron-Simon about the precise asymptotics of the lengths ofspectral gaps.