TITLE:
The Banach Space of Multiply Excited Atoms
AUTHORS:
Hubert Klar
KEYWORDS:
Dominant Correlation, Nonseparable Wave Equations, Embedding of a Hilbert Space into a Banach Space
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.9 No.12,
December
31,
2021
ABSTRACT: We present, for the first time, a unified description of adiabatic collision channels and the Wannier channel in electron-atom scattering. We identify the Wannier channel as the solution of a recently presented partial differential equation of parabolic type. The kernel of that equation has been constructed near the ionization threshold. Its eigenstates are shown to be members of a Banach space. For the purpose of demonstration, this paper embeds one adiabatic channel into a Bannach space. The full set of an adiabatic spectrum will be embedded into the Wannier continuum of a Banach space in a forthcoming paper. This technique delivers amended non-adiabatic collision channnels with ebergy-dependent potentials. That dependence manifests itself as energy-dependent discontinuity at the threshold. The branch above threshold describes the double escape of electrons, whereas the branch below threshold replaces an infinity of strongly coupled adiabatic channels by one new channel. The present paper is restricted to two-electron atoms consisting only of s2 1S configurations. Our model shows new unexpected effects including an electron-electron attraction similar to a Cooper pair except that our electron pair couples only to one nucleus at rest rather than to a vibrating lattice. Our electron-electron attraction stems from a dynamic deformation of the potential surface.