On radial Schrödinger operators with a Coulomb potential: General boundary conditions

Jan Dereziński (1) , Jérémy Faupin (2) , Quang Nhat Nguyen (3) , Serge Richard (4, 3)

1 UW - University of Warsaw
2 IECL - Institut Élie Cartan de Lorraine
3 Nagoya City University [Nagoya, Japan]
4 PSPM - Probabilités, statistique, physique mathématique

This paper presents the spectral analysis of 1-dimensional Schrödinger operator on the half-line whose potential is a linear combination of the Coulomb term 1/r and the centrifugal term 1/r^2. The coupling constants are allowed to be complex, and all possible boundary conditions at 0 are considered. The resulting closed operators are organized in three holomorphic families. These operators are closely related to the Whittaker equation. Solutions of this equation are thoroughly studied in a large appendix to this paper. Various special cases of this equation are analyzed, namely the degenerate, the Laguerre and the doubly degenerate cases. A new solution to the Whittaker equation in the doubly degenerate case is also introduced.

Domaines: Physique mathématique [math-ph]

Jan Dereziński, Jérémy Faupin, Quang Nhat Nguyen, Serge Richard. On radial Schrödinger operators with a Coulomb potential: General boundary conditions. 2020. ⟨hal-02458795⟩