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

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, usual $N$-body Hamiltonians with Coulomb-type singular are covered by our result: that case, weight is $\delta_{\mathcal{F}}(x) := \min \{ d(x, \bigcup \mathcal{F}), 1\}$, where $d(x, \mathcal{F})$ euclidean distance to union $\bigcup\mathcal{F}$ set collision planes $\bigcup\mathcal{F}$. The proof based on blow-ups manifolds corners Lie manifolds. More precisely, we start compactification $\overline{X}$ underlying space $X$ first blow-up spheres $\mathbb{S}_Y \subset \mathbb{S}_X$ infinity $Y \in \bigcup\mathcal{F}$ obtain Georgescu-Vasy compactification. Then carefully investigate how manifold structure associated data (metric, spaces, differential operators) change each blow-up. Our method applies also higher order operators, certain classes pseudodifferential matrices scalar operators.