Bound states in a locally deformed waveguide: the critical case

Abstract

We consider the Dirichlet Laplacian for a strip in ${\mathbb{R}}^2$ 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 $\Vert f^{\prime }\Vert <1.56a^{-1}\Vert f\Vert$; 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.

LaTeX file, 9 pages, to appear in Lett. Math. Phys

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