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 Ω Ω obtained by adding an arbitrarily small “bump” to the tube Ω 0 = R × ( 0 , 1 ) Ω ₀=R× (0,1) (i.e., Ω ⫌ Ω 0 Ω Ω ₀ , Ω ⊂ R 2 Ω ⊂ R² open and connected, Ω = Ω 0 Ω =Ω ₀ outside a bounded region) produces at least one positive eigenvalue below the essential spectrum [ π 2 , ∞ ) [π ²,∞ ) of the Dirichlet Laplacian − Δ Ω D -Δ DΩ . For | Ω ∖