Analysis of Schrödinger 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/p ², with p(x) = \x - p\ for x close to p and Z continuous on ℝ³ with Z(p) > -1/4 for p ε S. Also assume that p 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 A be the periodicity lattice and define T := ℝ³/A. We obtain regularity results in weighted Sobolev space for the eigenfunctions of the Schrödinger-type operator H= -△+V acting on L² (T), as well as for the induced k-Hamiltonians H_k 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.