A regularity result for the bound states of $N$-body Schr\"odinger operators: Blow-ups and Lie manifolds

Abstract

We prove regularity estimates in weighted Sobolev spaces for the $L^2$-eigenfunctions of Schr\"odinger 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.