Weakly coupled bound states in quantum waveguides

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) = R x (0,1) (i.e., Omega (not equal)(superset of) Omega(0), Omega subset of R-2 open and connected, Omega = Omega(0) outside a bounded region) produces at least one positive eigenvalue below the essential spectrum [pi(2),infinity) of the Dirichlet Laplacian -Delta(Omega)(D). For \Omega\Omega(0)\ sufficiently small (\.\ abbreviating Lebesgue measure), we prove uniqueness of the ground state E-Omega of -Delta(Omega)(D) and derive the ''weak coupling'' result E-Omega = pi(2) - pi(4)\Omega\Omega(0)\(2) + O(\Omega\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 delta Omega(0), replace the Dirichlet condition by a Neumann boundary condition on an arbitrarily small segment (a,b) x {1}, a < b, of delta 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.