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Analysis of Schrodinger operators with inverse square potentials I: regularity results in 3D

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posted on 01.04.2015, 11:29 authored by Eugenie Hunsicker, Hengguang Li, Victor Nistor, Ville Uski
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/ 2, with (x) = |x − p| for x close to p and Z continuous on R3 with Z(p) > −1/4 for p 2 S. Also assume that 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 be the periodicity lattice and define T := R3/ . We obtain regularity results in weighted Sobolev space for the eigenfunctions of the Schr¨odinger-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.

Funding

V.N. was partially supported by the NSF Grants OCI-0749202 and DMS-1016556. H.L. was partially supported by the NSF Grant DMS-1115714. E.H. was supported in part by Leverhulme Trust Project Assistance Grant F/00 261/Z.

History

School

  • Science

Department

  • Mathematical Sciences

Published in

BULLETIN MATHEMATIQUE DE LA SOCIETE DES SCIENCES MATHEMATIQUES DE ROUMANIE

Volume

55

Issue

2

Pages

157 - 178 (22)

Citation

HUNSICKER, E. ... et al, 2012. Analysis of Schrodinger operators with inverse square potentials I: regularity results in 3D. Bulletin Mathématique de la Société des Sciences Mathématiques de Roumanie, 55 (2), pp. 157 - 178.

Publisher

Société des Sciences Mathématiques de Roumanie

Version

SMUR (Submitted Manuscript Under Review)

Publisher statement

This work is made available according to the conditions of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) licence. Full details of this licence are available at: https://creativecommons.org/licenses/by-nc-nd/4.0/

Publication date

2012

Notes

This article was published in the journal, Bulletin Mathématique de la Société des Sciences Mathématiques de Roumanie and is available here with the kind permission of the publisher.

ISSN

1220-3874

Language

en