Asymptotically straight quantum waveguides with axial symmetry in two and three dimensions

Abstract Consider a configuration space Ω whose boundary is made up of an asymptotically straight curve and its reflection (in two dimensions) or rotation (in three dimensions) about an axis of symmetry. We prove that by imposing an infinite net-area condition on the profile curve along just one end of Ω, the Dirichlet Laplacian <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>-</m:mo> <m:mi mathvariant="normal">Δ</m:mi> </m:mrow> </m:math> {-\Delta} on Ω has at least one isolated eigenvalue (hence at least one bound state) below a natural threshold. The varying eigenvalue of the cross-section along the axis of symmetry plays a central role in our analysis, and integrals of the Bessel function are also of crucial importance.