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Analytic structure of many-body Coulombic wave functions. (English) Zbl 1171.35110
Summary: We investigate the analytic structure of solutions of non-relativistic Schrödinger equations describing Coulombic many-particle systems. We prove the following: Let \(\psi (\mathbf x)\) with \({{\mathbf x} = (x_{1},\dots, x_{N})\in \mathbb {R}^{3N}}\) denote an \(N\)-electron wavefunction of such a system with one nucleus fixed at the origin. Then in a neighbourhood of a coalescence point, for which \(x_{1} = 0\) and the other electron coordinates do not coincide, and differ from \(0\), \(\psi \) can be represented locally as \(\psi (\mathbf x) = \psi^{(1)}(\mathbf x) + |x _{1}|\psi^{(2)}(\mathbf x)\) with \(\psi^{(1)}, \psi^{(2)}\) real analytic. A similar representation holds near two-electron coalescence points. The Kustaanheimo-Stiefel transform and analytic hypoellipticity play an essential role in the proof.

MSC:
35Q55 NLS equations (nonlinear Schrödinger equations)
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35A30 Geometric theory, characteristics, transformations in context of PDEs
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