### Abstract

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/p ^{2}, with p(x) = \x - p\ for x close to p and Z continuous on ℝ^{3} 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 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}/A. We obtain regularity results in weighted Sobolev space for the eigenfunctions of the Schrödinger-type operator H= -△+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.

Original language | English (US) |
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Pages (from-to) | 157-178 |

Number of pages | 22 |

Journal | Bulletin Mathematique de la Societe des Sciences Mathematiques de Roumanie |

Volume | 55 |

Issue number | 2 |

State | Published - Dec 1 2012 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

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## Cite this

*Bulletin Mathematique de la Societe des Sciences Mathematiques de Roumanie*,

*55*(2), 157-178.