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Sfb288 logo Sfb 288 Differential Geometry and Quantum Physics

Abstract for Sfb Preprint No. 572

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

Evgeni Korotyaev

The spectral properties of the Schr"odinger operator $T_ty= -y''+q_ty$ 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 $tin R$ 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 $t o 0$ then in any finite gap there exist both eigenvalues ($ le 2 $) and resonances ($ le 2 $) of $T_t$ which belong to a gap on the second sheet and their asymptotics as $t o 0 $ are determined. iii) The eigenvalues of the half-solid, i.e. $p_1={ m constant}$, are also studied. iv) We prove that for any even 1-periodic potential $p$ and any sequences ${d_n}_1^{iy }$, 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 $nge 1.$

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