  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
$Z/\rho^2$, with $\rho(x) = |x - p|$ for $x$ close to $p$ and $Z$ continuous on
$\RR^3$ with $Z(p) > -1/4$ for $p \in \maS$. Also assume that $\rho$ 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 $\Lambda$ 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 $H = -\Delta + V$ 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.
