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 R obtained by adding an arbitrarily small "bump" to the tube a0 = R x ( 0 , l ) (i.e., 2 no, R C W2 open and connected, R = Ro outside a bounded region) produces a t least one positive eigenvalue below the essential spectrum [7r2, m) of the Dirichlet Laplacian -A:. For /R\Ro/ sufficiently small ( 1 . / abbreviating Lebesgue measure), we prove uniqueness of the ground state En of -A: and derive the "weak coupling" result En = 7r2 + 0(/R\C20/3) using a Birman-Schwinger-type T ~ / Q \ R O / ~ analysis. As a corollary of these results we obtain the following surprising fact: Starting from the tube flo with Dirichlet boundary conditions at aRo, replace the Dirichlet condition by a Neumann boundary condition on an arbitrarily small segment (a, b) x (1). a < b, of aRo. If H(a , b) denotes the resulting Laplace operator in L ' (R~) , then H(a , b) has a discrete eigenvalue in [7r2/4,7r2) no matter how small lb a1 > 0 is. Our goal in this paper is to study the bound state spectra of the Dirichlet Laplacian -A: for open regions R c Rn which are tubes outside of a bounded region (quantum waveguides). (Following the traditional notation in quantum physics, we denote the Laplacian by -A as opposed to A in the following.) In particular, let RocR2 be defined by Ro=R x (0 , l ) . Consider open connected sets R such that: (i) For some R > 0, R n{x E R2 I 1x1 > R) =Ron{x E R2 I 1x1 > R). (ii) R, cR, R, #a. Because of condition (i), (1) aess(-~:) = fless(-Ag,) = [.ir2, 00). Then one of our main goals will be to prove Received by the editors November 13, 1995. 1991 Mathematics Subject Classification. Primary 81Q10, 35P15; Secondary 47A10, 35510.