论文标题
$ n $ n $ bodyschrödinger运营商的界限的规律性结果：爆炸和谎言歧管
A regularity result for the bound states of $N$-body Schrödinger operators: Blow-ups and Lie manifolds
论文作者
论文摘要
我们证明了Schrödinger型操作员的$ l^2 $ eigenfunctions的加权Sobolev空间的规律性估计，其电势具有反平方的奇异性和无限范围的径向限制。特别是，我们的结果涵盖了具有库仑型奇异潜力的通常的$ n $ - haby hamiltonians：在这种情况下，重量为$δ_ {\ Mathcal {f}}}（x）（x）：= \ min \ {d（x，x，x，x，bigcup \ mathcal \ mathcal {f} $ \ big \ Mathcal {f}）$是一组碰撞平面$ \ bigCup \ bigcal \ mathcal {f} $的欧盟距离$ \ bigcup \ mathcal {f} $。证明是基于对角落和谎言歧管的歧管的爆炸。更准确地说，我们从基础空间的径向压实$ \叠加{x} $开始，我们首先炸毁了球$ \ mathbb {s} _y \ subset \ subset \ subset \ subbb {s} s} _x $紧凑。然后，我们炸毁碰撞平面$ \ bigcup \ Mathcal {f} $。我们仔细研究了每个爆炸时谎言歧管结构和相关数据（度量，Sobolev空间，差异操作员）如何变化。我们的方法还适用于高阶差分运算符，某些类别的伪差操作员以及标量运营商的矩阵。
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 $δ_{\mathcal{F}}(x) := \min \{ d(x, \bigcup \mathcal{F}), 1\}$, 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.
