A regularity result for the bound states of $N$-body Schrödinger operators: Blow-ups and Lie manifolds

Abstract

We prove regularity estimates in weighted Sobolev spaces for the $L^2$-eigenfunctions of Schrödinger type operators whose potentials have inverse square singularities and uniform radial limits at infinity. In particular, the usual $N$-body Hamiltonians with Coulomb-type singular potentials are covered by our result: in that case, the weight is ${\delta }_{\mathcal{F}}(x):=\min \{d(x,\bigcup \mathcal{F}),1\}$, where $d(x,\bigcup \mathcal{F})$ is the usual euclidean distance to the union $\bigcup \mathcal{F}$ of the set of collision planes $\bigcup \mathcal{F}$. The proof is based on blow-ups of manifolds with corners and Lie manifolds. More precisely, we start with the radial compactification $\overline{X}$ of the underlying space $X$ and we first blow-up the spheres ${\mathbb{S}}_Y\subset {\mathbb{S}}_X$ at infinity of the collision planes $Y\in \bigcup \mathcal{F}$ to obtain the Georgescu-Vasy compactification. Then we blow-up the collision planes $\bigcup \mathcal{F}$. We carefully investigate how the Lie manifold structure and the associated data (metric, Sobolev spaces, differential operators) change with each blow-up. Our method applies also to higher order differential operators, to certain classes of pseudodifferential operators, and to matrices of scalar operators.

Small changes, last preprint version before sending to publisher. Appendices D and E only in preprint version, not published

References in corpus (1)

• Semiclassical resolvent estimates for Schroedinger operators with Coulomb singularities

Cited by in corpus (1)

• The diagonal behaviour of the one-particle Coulombic density matrix