Schr\"odinger operator with a junction of two 1-dimensional periodic potentials

The spectral properties of the Schr"odinger operator $T_{t}y=-y^{\mathrm{\prime }\mathrm{\prime }}+q_{t}y$ in $L^2(\mathbb{R})$ are studied, with a potential $q_t(x)=p_1(x),x<0,$ and $q_t(x)=p(x+t),x>0,$ where $p_1,p$ 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 $T_0$ 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 $t\to 0$,theninanygapinthespectrumthereexistbotheigenvalues($ \le 2 $) and resonances ($\le 2$) of $T_t$ which belong to a gap on the second sheet and their asymptotics as $t\to 0$ are determined. iii) The eigenvalues of the half-solid, i.e. $p_1=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}$, are also studied. iv) We prove that for any even 1-periodic potential $p$ and any sequences $\{d_n{\}}_1^{\text{iy}}$, 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 $n\ge 1.$