Schrödinger operator with a junction of two 1-dimensional periodic potentials

Article type: Research Article

Authors: Korotyaev, Evgeny

Affiliations: Institut für Mathematik, Humboldt Universität zu Berlin, Rudower Chaussee 25, 12489, Berlin, Germany E-mail: [email protected]

Abstract: The spectral properties of the Schrödinger operator Tty=−y″+qty in \[$L^{2}(\mathbb{R})$ are studied, with a potential qt(x)=p1(x), x<0, and qt(x)=p(x+t), x>0, where p1,p are periodic potentials and \[$t\in \mathbb{R}$ is a parameter of dislocation. Under some conditions there exist simultaneously gaps in the continuous spectrum of T0 and eigenvalues in these gaps. The main goal of this paper is to study the discrete spectrum and the resonances of Tt. The following results are obtained: (i) In any gap of Tt there exist 0,1 or 2 eigenvalues. Potentials with 0, 1 or 2 eigenvalues in the gap are constructed. (ii) The dislocation, i.e., the case p1=p is studied. If t→0, then in any gap in the spectrum there exist both eigenvalues (≤2) and resonances (≤2) of Tt which belong to a gap on the second sheet and their asymptotics as t→0 are determined. (iii) The eigenvalues of the half-solid, i.e., p1=constant, are also studied. (iv) We prove that for any even 1-periodic potential p and any sequences {dn}1∞, where dn=1 or dn=0 there exists a unique even 1-periodic potential p1 with the same gaps and dn eigenvalues of T0 in the n-th gap for each n≥1.

Journal: Asymptotic Analysis, vol. 45, no. 1-2, pp. 73-97, 2005