 05241 Evgeny Korotyaev
 Schroedinger operator with a junction of two 1dimensional periodic potentials
(80K, latex)
Jul 14, 05

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Abstract. The spectral properties of the Schr\"odinger operator $T_ty=
y''+q_ty$ in $L^2(\R )$ 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 $t\in \R$ 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 $t\to 0$, then in
any gap in the spectrum 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\to 0 $ are determined.
iii) The eigenvalues of the halfsolid, i.e. $p_1={\rm constant}$,
are also studied. iv) We prove that for any even 1periodic
potential $p$ and any sequences $\{d_n\}_1^{\iy }$, where $d_n=1$
or $d_n=0$ there exists a unique even 1periodic potential $p_1$
with the same gaps and $d_n$ eigenvalues of $T_0$ in the nth gap
for each $n\ge 1.$
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