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