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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.