Regularity for eigenfunctions of Schrödinger operators
by
Bernd Ammann, Catarina Carvalho and Victor Nistor


Regularity for eigenfunctions of Schrödinger operators (.dvi, .ps,.ps.gz or .pdf)
Lett. Math. Phys. 101, 49-84 (2012)
DOI 10.1007/s11005-012-0551-z

Abstract

We prove a regularity result for the eigenfunctions of a single nucleus Schrödinger operator in weighted Sobolev (or Babuska--Kondratiev) spaces. More precisely, if Kam is the weighted Sobolev space obtained by desingularization of the set of singular points of the potential
V(x)= ∑1 ≤ j ≤ N \frac{bj}{|xj|} + ∑1 ≤ i < j ≤ N \frac{cij}{|xi-xj|},
x ∈ R3N, bj, cij ∈ R,
and (-Δ + V) u = λ u, u ∈ L2(R3N}), then u ∈ Kam for all m ∈ Z+ and all a ≤ 0.
Our result extends to the case when bj and cij suitable bounded functions.

Typo in the published version

A comment on Prop. 2.5

In the article we mostly restricted to blow-up along manifolds of positive codimension. Nevertheless most statement are still true if we allow blow-ups along manifolds of codimension 0. If X is of codimension 0 in M, then [M:X]=M\X. Thus one immediatly sees that Proposition 3.2 is still true in the case that X has the same dimension as Y. However, as this subtle case is not needed we avoided this discussion in the article.
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The Paper was written on 24.9.2010
Last update 5.4.2013