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

The spectral properties of the Schrödinger operator T_ty=−y″+q_ty in $\[$L^{2}(\mathbb{R})$$ are studied, with a potential q_t(x)=p₁(x), x<0, and q_t(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 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 p₁=p is studied. If t→0, then in any gap in the spectrum there exist both eigenvalues (≤2) and resonances (≤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., p₁=constant, are also studied. (iv) We prove that for any even 1-periodic potential p and any sequences {d_n}₁^∞, where d_n=1 or d_n=0 there exists a unique even 1-periodic potential p₁ with the same gaps and d_n eigenvalues of T₀ in the n-th gap for each n≥1.