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