On the Curvature and Torsion Effects in One Dimensional Waveguides

We consider the Laplace operator in a thin tube of R with a Dirichlet condition on its boundary. We study asymptotically the spectrum of such an operator as the thickness of the tube’s cross section goes to zero. In particular we analyse how the energy levels depend simultaneously on the curvature of the tube’s central axis and on the rotation of the cross section with respect to the Frenet frame. The main argument is a Γ-convergence theorem for a suitable sequence of quadratic energies. Mathematics Subject Classification. 49R50, 35P20, 78A50, 81Q15. Received February 23, 2006. Revised July 7, 2006. Published online September 5, 2007.