Bound States in a Locally Deformed Waveguide: The Critical Case

Abstract

We consider the Dirichlet Laplacian for astrip in $\mathbb{R}^2$ with one straight boundary and a width$a(1 + \lambda f(x))$, where $f$ is a smooth function of acompact support with a length 2b. We show that in the criticalcase, $\int {_{ - b}^b } f(x)dx = 0$, the operator has nobound statesfor small $\left| \lambda \right|{\text{ if }}b < (\sqrt 3 /4)a$.On the otherhand, a weakly bound state existsprovided $\left\| {f\prime } \right\| < 1.59a^{ - 1} \left\| f \right\|$. In thatcase, there are positive c ₁,c ₂ suchthat the corresponding eigenvalue satisfies $- c_1 \lambda ^4 \leqslant \varepsilon (\lambda ) - (\pi /a)^2 \leqslant - c_2 \lambda ^4$ for all$\left| \lambda \right|$ sufficiently small.