Analysis of periodic Schrodinger operators: Regularity and approximation of eigenfunctions

摘要

Let V be a real valued potential that is smooth everywhere on R-3, except at a periodic, discrete set S of points, where it has singularities of the Coulomb-type Z/r. We assume that the potential V is periodic with period lattice L. We study the spectrum of the Schrodinger operator H=-Delta+V acting on the space of Bloch waves with arbitrary, but fixed, wavevector k. Let T:=R-3/L. Let u be an eigenfunction of If with eigenvalue lambda and let epsilon>0 be arbitrarily small. We show that the classical regularity of the eigenfunction u is u epsilon H5/2-epsilon(T) in the usual Sobolev spaces, and u is an element of K-3/2-epsilon(m)(TS) in the weighted Sobolev spaces. The regularity index m can be as large as desired, which is crucial for numerical methods. For any choice of the Bloch wavevector k. we also show that H has compact resolvent and hence a complete eigenfunction expansion. The case of the hydrogen atom suggests that our regularity results are optimal. We present two applications to the numerical approximation of eigenvalues: using wave functions and using piecewise polynomials. (C) 2008 American Institute of Physics.