Schrr Odinger Operator with a Junction of Two 1-dimensional Periodic Potentials

The spectral properties of the Schrr odinger operator T t y = ?y 00 + q t y in L 2 (R) are studied, with a potential q t = p 1 (x); x < 0; and q t = p(x + t); x > 0; where p 1 ; p are periodic potentials and t 2 R is a parameter of the dislocation. Under some conditions there exist simultaneously gaps in the continuous specrum of T 0 and eigenvalues in these gaps. The main goal of this paper is to study the discrete spectrum and the resonances of T t. The following results are obtained: i) in any gap of T t 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 p 1 = p is studied. If t ! 0 then in any nite gap there exist both eigenvalues (6 2) and resonances (6 2) of T t 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. p 1 = constant, are also studied. iv) We prove that for any even 1-periodic potential p and any sequences fd n g 1 1 , where d n = 1 or d n = 0 there exists a unique even 1-periodic potential p 1 with the same gaps and d n eigenvalues of T 0 in the n-th gap for each n > 1: