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

Let V be a potential on R-3 that is smooth everywhere except at a discrete set S of points, where it has singularities of the form Z/rho(2), with rho(x) = vertical bar x - p vertical bar for x close to p and Z continuous on R-3 with Z(p) > -1/4 for p is an element of S. Also assume that rho 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 R-3 that is bounded outside a compact set containing S. In the periodic case, we let A be the periodicity lattice and define T := R-3/Lambda. We obtain regularity results in weighted Sobolev space for the eigenfunctions of the Schrodinger-type operator H = -Delta + V acting on L-2(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.