Resonance theory for perturbed Hill operator

Article type: Research Article

Authors: Korotyaev, Evgeny

Affiliations: Saint-Petersburg University, Saint Petersburg, Russia. E-mail: korotyaev@gmail.com

Abstract: We consider the Schrödinger operator Hy=−y″+(p+q)y with a periodic potential p plus a compactly supported potential q on the real line. The spectrum of H consists of an absolutely continuous part plus a finite number of simple eigenvalues below the spectrum and in each spectral gap γn≠∅,n≥1. We prove the following results: (1) the distribution of resonances in the disk with large radius is determined, (2) the asymptotics of eigenvalues and antibound states are determined at high energy gaps, (3) if H has infinitely many open gaps in the continuous spectrum, then for any sequence (κ)1∞,κn∈{0,2}, there exists a compactly supported potential q with ∫Rq dx=0 such that H has κn eigenvalues and 2−κn antibound states (resonances) in each gap γn for n large enough.

Keywords: resonances, scattering, periodic potential, S-matrix

DOI: 10.3233/ASY-2011-1050

Journal: Asymptotic Analysis, vol. 74, no. 3-4, pp. 199-227, 2011