Analysis of periodic Schrodinger operators: regularity and approximation of eigenfunctions.

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 Schrödinger operator H=−Δ+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 H with eigenvalueλ and let ϵ>0 be arbitrarily small. We show that the classical regularity of the eigenfunction u is u∊H 5/2−ϵ (T) in the usual Sobolev spaces, and u∊K m 3/2−ϵ (T\S) 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

Analysis of Schrodinger operators with inverse square potentials I: regularity results in 3D

Let V be a potential on R3 that is smooth everywhere except at a discrete set S of points, where it has singularities of the form Z/ 2, with (x) = |x − p| for x close to p and Z continuous on R3 with Z(p) > −1/4 for p 2 S. Also assume that and Z are smooth outside S and Z is smooth in polar coordinates around each singular point. We either assume that V is periodic or that the set S is finite and V extends to a smooth function on the radial compactification of R3 that is bounded outside a compact set containing S. In the periodic case, we let be the periodicity lattice and define T := R3/ . We obtain regularity results in weighted Sobolev space for the eigenfunctions of the Schr¨odinger-type operator H = − + V acting on L2(T), as well as for the induced k–Hamiltonians Hk obtained by restricting the action of H to Bloch waves. Under some additional assumptions, we extend these regularity and solvability results to the non-periodic case. We sketch some applications to approximation of eigenfunctions and eigenvalues that will be studied in more detail in a second paper

Analysis of Schrodinger operators with inverse square potentials II: FEM and approximation of eigenfunctions in the periodic case

In this article, we consider the problem of optimal approximation of eigenfunctions of Schrödinger operators with isolated inverse square potentials and of solutions to equations involving such operators. It is known in this situation that the finite element method performs poorly with standard meshes. We construct an alter- native class of graded meshes, and prove and numerically test optimal approximation results for the finite element method using these meshes. Our numerical tests are in good agreement with our theoretical results