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Sharp regularity results for Coulombic many-electron wave functions. (English) Zbl 1075.35063
Summary: We show that electronic wave functions $$\psi$$ of atoms and molecules have a representation $$\psi = \mathcal F \phi$$, where $$\mathcal F$$ is an explicit universal factor, locally Lipschitz, and independent of the eigenvalue and the solution $$\psi$$ itself, and $$\varphi$$ has second derivatives which are locally in $$L^{\infty}$$. This representation turns out to be optimal as can already be demonstrated with the help of hydrogenic wave functions. The proofs of these results are, in an essential way, based on a new elliptic regularity result which is of independent interest. Some identities that can be interpreted as cusp conditions for second order derivatives of $$\psi$$ are derived.

##### MSC:
 35Q40 PDEs in connection with quantum mechanics 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics 81V45 Atomic physics 35J10 Schrödinger operator, Schrödinger equation
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