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Download fileAnalysis of Schrodinger operators with inverse square potentials I: regularity results in 3D
journal contribution
posted on 01.04.2015, 11:29 authored by Eugenie Hunsicker, Hengguang Li, Victor Nistor, Ville UskiLet 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.
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