# Regularity for eigenfunctions of Schrödinger operators

by

Bernd Ammann, Catarina Carvalho and Victor Nistor

**Regularity for eigenfunctions of Schrödinger operators**
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*Lett. Math. Phys.* **101**, *49-84* (2012)

DOI 10.1007/s11005-012-0551-z

We prove a regularity result for the eigenfunctions of a single nucleus Schrödinger operator in weighted Sobolev (or Babuska--Kondratiev) spaces. More precisely, if K_{a}^{m} is the weighted Sobolev space obtained by desingularization of the set of singular points of the potential

V(x)= ∑_{1 ≤ j ≤ N} \frac{b_{j}}{|x_{j}|} + ∑_{1 ≤ i < j ≤ N} \frac{c_{ij}}{|x_{i}-x_{j}|},

x ∈ R^{3N}, b_{j}, c_{ij} ∈ R, and (-Δ + V) u = λ u, u ∈ L^{2}(R^{3N}}), then u ∈ K_{a}^{m} for all m ∈ Z^{+} and all a ≤ 0.

Our result extends to the case when b_{j} and c_{ij} suitable bounded functions.
### Typo in the published version

- In Definition 2.3: In the phrase "(which
is equivalent to saying that $\bar x$ is in the interior of
$X\cap H_1\cap \ldots \cap H_s$)" one should replace
"$X\cap H_1\cap \ldots \cap H_s$" by
"$H_1\cap \ldots \cap H_s$".

### 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