Analysis of Schrödinger operators with inverse square potentials I: Regularity results in 3D

Eugenie Hunsicker, Hengguang Li, Victor Nistor, Ville Uski

Research output: Contribution to journalArticle

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Abstract

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 languageEnglish (US)
Pages (from-to)157-178
Number of pages22
JournalBulletin Mathematique de la Societe des Sciences Mathematiques de Roumanie
Volume55
Issue number2
StatePublished - Dec 1 2012

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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