Let be a potential on ℜ that is smooth everywhere except at
a discrete set *S* of points, where it has singularities of the
form , with
for close to and
continuous on ℜ with for *p ∈ S*. Also assume
that and are smooth outside *S* and is smooth in
polar coordinates around each singular point. We either assume that
is periodic or that the set *S* is finite and extends to a
smooth function on the radial compactification of ℜ that is
bounded outside a compact set containing *S*. In the periodic case,
we let be the periodicity lattice and define *T*: = ℜ^{3}/Λ. We obtain regularity results in weighted Sobolev space for
the eigenfunctions of the Schrödinger-type operator
acting on *L*^{2}(*T*), as well as for the induced -Hamiltonians *H*_{k} obtained by restricting the action of 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.

Key Words: Regularity of eigenfunctions, Schrodinger operator, eigenvalue approximations, inverse square potential, regularity, weighted Sobolev spaces, rate of convergence of numerical methods, solid state physics.

2000 Mathematics Subject Classification: Primary: 35B65;

Secondary: 35J10, 65N12, 81Q05, 35B10, 35P10.

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