 02268 Denis Borisov, Pavel Exner, Rustem Gadyl'shin
 Geometric coupling thresholds in a twodimensional strip
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Jun 17, 02

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Abstract. We consider the Laplacian in a strip $\mathbb{R}\times
(0,d)$ with the boundary condition which is Dirichlet except at
the segment of a length $2a$ of one of the boundaries where it
is switched to Neumann. This operator is known to have a
nonempty and simple discrete spectrum for any $a>0$. There is a
sequence $0<a_1<a_2<\cdots$ of critical values at which new
eigenvalues emerge from the continuum when the Neumann window
expands. We find the asymptotic behavior of these eigenvalues
around the thresholds showing that the gap is in the leading
order proportional to $(aa_n)^2$ with an explicit
coefficient expressed in terms of the corresponding
thresholdenergy resonance eigenfunction.
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