Weakly coupled bound states in quantum waveguides

Abstract:

We study the eigenvalue spectrum of Dirichlet Laplacians which model quantum waveguides associated with tubular regions outside of a bounded domain. Intuitively, our principal new result in two dimensions asserts that any domain $\Omega$ obtained by adding an arbitrarily small “bump” to the tube $\Omega _{0}=\mathbb {R}\times (0,1)$ (i.e., $\Omega \supsetneqq \Omega _{0}$, $\Omega \subset \mathbb {R}^{2}$ open and connected, $\Omega =\Omega _{0}$ outside a bounded region) produces at least one positive eigenvalue below the essential spectrum $[\pi ^{2},\infty )$ of the Dirichlet Laplacian $-\Delta ^{D}_{\Omega }$. For $|\Omega \backslash \Omega _{0}|$ sufficiently small ($| . |$ abbreviating Lebesgue measure), we prove uniqueness of the ground state $E_{\Omega }$ of $-\Delta ^{D}_{\Omega }$ and derive the “weak coupling” result $E_{\Omega }=\pi ^{2}-\pi ^{4}|\Omega \backslash \Omega _{0}|^{2} +O(|\Omega \backslash \Omega _{0}|^{3})$ using a Birman-Schwinger-type analysis. As a corollary of these results we obtain the following surprising fact: Starting from the tube $\Omega _{0}$ with Dirichlet boundary conditions at $\partial \Omega _{0}$, replace the Dirichlet condition by a Neumann boundary condition on an arbitrarily small segment $(a,b)\times \{1\}$, $a<b$, of $\partial \Omega _{0}$. If $H(a,b)$ denotes the resulting Laplace operator in $L^{2}(\Omega _{0})$, then $H(a,b)$ has a discrete eigenvalue in $[\pi ^{2} /4,\pi ^{2})$ no matter how small $|b-a|>0$ is.