Alessandra A. Verri

Self-adjoint extensions of Coulomb systems in 1, 2 and 3 dimensions

We study the nonrelativistic quantum Coulomb hamiltonian (i.e., inverse of distance potential) in R n , n = 1,2,3. We characterize their self-adjoint extensions and, in the unidimensional case, present a discussion of controversies in the literature, particularly the question of the permeability of the origin. Potentials given by fundamental solutions of Laplace equation are also briefly considered.

On the spectrum and weakly effective operator for Dirichlet Laplacian in thin deformed tubes

Abstract We study the Laplacian in deformed thin (bounded or unbounded) tubes in R 3 , i.e., tubular regions along a curve r ( s ) whose cross sections are multiplied by an appropriate deformation function h ( s ) > 0 . One of the main requirements on h ( s ) is that it has a single point of global maximum. We find the asymptotic behaviors of the eigenvalues and weakly effective operators as the diameters of the tubes tend to zero. It is shown that such behaviors are not influenced by some geometric features of the tube, such as curvature, torsion and twisting, and so a huge amount of different deformed tubes are asymptotically described by the same weakly effective operator.

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

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.

Complex

The resolvent convergence of self-adjoint operators via the technique of

\Gamma

\Gamma

-convergence and magnetic Dirichlet Laplacian in bounded thin tubes

-convergence of quadratic forms is adapted to incorporate complex Hilbert spaces. As an application, we find effective operators to the Dirichlet Laplacian with magnetic potentials in very thin bounded tubular regions in space built along smooth closed curves; relatively weak regularity is asked for the potentials, and the convergence is in the norm resolvent sense as the cross sections of the tubes go uniformly to zero.

Absolute Continuity and Band Gaps of the Spectrum of the Dirichlet Laplacian in Periodic Waveguides

Consider the Dirichlet Laplacian operator

CONVERGENCE OF SOLUTIONS TO SOME ELLIPTIC EQUATIONS IN BOUNDED NEUMANN THIN DOMAINS

-\Delta ^D