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 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.
|Original language||English (US)|
|Number of pages||22|
|Journal||Bulletin Mathematique de la Societe des Sciences Mathematiques de Roumanie|
|State||Published - Dec 1 2012|
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