Bound states in a locally deformed waveguide: the critical case

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 $\mathrm{\mid }\lambda \mathrm{\mid }$ if $b<(\sqrt{3}\mathrm{/}4)a$. On the other hand, a weakly bound state exists provided $\mathrm{\parallel }{f}^{\mathrm{\prime }}\mathrm{\parallel }<1.56a^{-1}\mathrm{\parallel }f\mathrm{\parallel }$; in that case there are positive $c_1,c_2$ such that the corresponding eigenvalue satisfies $-c_1{\lambda }^4\le ϵ(\lambda )-(\pi \mathrm{/}a{)}^2\le -c_2{\lambda }^4$ for all $\mathrm{\mid }\lambda \mathrm{\mid }$ sufficiently small.