Let V be a potential on \RR^3 that is smooth everywhere except at a
discrete set \maS of points, where it has singularities of the form
Z/ρ2, with ρ(x)=∣x−p∣ for x close to p and Z continuous on
\RR^3 with Z(p)>−1/4 for p \in \maS. Also assume that ρ and Z
are smooth outside \maS and Z is smooth in polar coordinates around each
singular point. We either assume that V is periodic or that the set \maS is
finite and V extends to a smooth function on the radial compactification of
\RR^3 that is bounded outside a compact set containing \maS. In the
periodic case, we let Λ be the periodicity lattice and define \TT :=
\RR^3/ \Lambda. We obtain regularity results in weighted Sobolev space for the
eigenfunctions of the Schr\"odinger-type operator H=−Δ+V acting on
L^2(\TT), as well as for the induced \vt 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.Comment: 15 pages, to appear in Bull. Math. Soc. Sci. Math. Roumanie, vol. 55
(103), no. 2/201