### Abstract

Let V be a real valued potential that is smooth everywhere on ℝ^{3}, except at a periodic, discrete set S of points, where it has singularities of the Coulomb-type Z/r. We assume that the potential V is periodic with period lattice L. We study the spectrum of the Schrödinger operator H=-Δ+V acting on the space of Bloch waves with arbitrary, but fixed, wavevector k. Let := ℝ^{3} / ℒ. Let u be an eigenfunction of H with eigenvalue λ and let ∈ > 0 be arbitrarily small. We show that the classical regularity of the eigenfunction u is u ∈ H^{5/2-∈} () in the usual Sobolev spaces, and u ∈ Κ _{3/2-∈}^{m} (\S) in the weighted Sobolev spaces. The regularity index m can be as large as desired, which is crucial for numerical methods. For any choice of the Bloch wavevector k, we also show that H has compact resolvent and hence a complete eigenfunction expansion. The case of the hydrogen atom suggests that our regularity results are optimal. We present two applications to the numerical approximation of eigenvalues: using wave functions and using piecewise polynomials.

Original language | English (US) |
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Article number | 083501 |

Journal | Journal of Mathematical Physics |

Volume | 49 |

Issue number | 8 |

DOIs | |

State | Published - 2008 |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

*Journal of Mathematical Physics*,

*49*(8), [083501]. https://doi.org/10.1063/1.2957940