 9615 Exner P., Vugalter S.A.
 Bound states in a locally deformed waveguide: the critical case
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Jan 22, 96

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Abstract. We consider the Dirichlet Laplacian for a strip in $\,\R^2$ with one
straight boundary and a width $\,a(1+\lambda f(x))\,$, where $\,f\,$
is a smooth function of a compact support with a length $\,2b\,$. We
show that in the critical case, $\,\int_{b}^b f(x)\,dx=0\,$, the
operator has no bound states for small $\,\lambda\,$ if
$\,b<(\sqrt{3}/4)a\,$. On the other hand, a weakly bound state
exists provided $\,\f'\< 1.56 a^{1}\f\\,$; in that case there
are positive $\,c_1, c_2\,$ such that the corresponding eigenvalue
satisfies $\,c_1\lambda^4\le \epsilon(\lambda) (\pi/a)^2 \le
c_2\lambda^4\,$ for all $\,\lambda\,$ sufficiently small.
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