On Norm Resolvent and Quadratic Form Convergences in Asymptotic Thin Spatial Waveguides

Abstract

A quantum particle is restricted to Dirichlet three-dimensional tubes built over a smooth curve r(x) ⊂ R3 through a bounded cross section that rotates along r(x).T hen the confining limit as the diameter of the tube cross section tends to zero is studied, and special attention is paid to the interplay between uniform quadratic form convergence and norm resolvent convergence of the respective Hamiltonians.In particular, it is shown a norm resolvent convergence to an effective Hamiltonian in case of null curvature and unbounded tubes, and, by means of an example, it is concluded that just norm resolvent convergence does not imply the quadratic form convergence.