Room P3.10, Mathematics Building

David Krejcirik, Centro de Análise Matemática, Geometria e Sistemas Dinâmicos
A lower bound to the spectral threshold in curved tubes

Motivated by the theory of quantum waveguides, we consider the Laplacian in curved tubes of arbitrary cross-section rotating together with the Frenet frame along curves in Euclidean spaces of arbitrary dimension, subject to Dirichlet boundary conditions on the cylindrical surface and Neumann conditions at the ends of the tube. We prove that the spectral threshold of the Laplacian is estimated from below by the lowest eigenvalue of the Dirichlet Laplacian in a torus determined by the geometry of the tube. This is a joint work with Pavel Exner and Pedro Freitas.