One dimensional Hydrogen problem with general connection condition at the origin and non-Rydberg spectra

TL;DR

This paper investigates the one-dimensional hydrogen atom with general boundary conditions at the origin, revealing non-Rydberg spectra and clarifying the physical meaning of connection parameters using self-adjoint extension and regularization methods.

Contribution

It introduces a comprehensive analysis of general connection conditions at the origin, including exotic cases, and relates different mathematical approaches to understanding non-standard spectra.

Findings

Established relations between self-adjoint extension and cutoff regularization methods.

Demonstrated realization of non-Rydberg spectra with exotic connection conditions.

Clarified physical interpretation of connection parameters in one-dimensional Coulomb problems.

Abstract

We consider the solution of the quantum Coulomb problem in one dimension with the most general connection condition at the origin. The divergence of the derivative of the wave function at the origin invalidates the standard current conservation approach. We explore two approaches, Wronskian self-adjoint extension method and cutoff regularization method, and establish their mutual relations, thereby clarifying the physical contents of the connection parameters. We show how to realize exotic non-Loudon connection conditions, entailing the realization of non-Rydberg spectrum.