On the general theory of bound state spectra in the Coulomb few- and many-body systems

TL;DR

This paper introduces two analytical methods for solving the Schrödinger equation in Coulomb systems, enabling explicit formulas for bound state energies in complex many-particle scenarios.

Contribution

It develops two novel approaches—matrix factorization and hyper-radial algebra representations—for analytical solutions in Coulomb many-body systems.

Findings

Derived closed-form formulas for bound state energies.

Applicable to arbitrary few- and many-particle Coulomb systems.

Validated methods for analytical solutions.

Abstract

Based on the fact that the Hamiltonians of the Coulomb many-particle systems are always factorized we develop the two different approaches for analytical solution of the Schr\"{o}dinger equation written for arbitrary few- and many-particle Coulomb systems. The first approach is the matrix factorization method. Another method is based on the $D^{+}-$series of representations of the hyper-radial O(2,1)-algebra. The both these methods allow us to obtain the closed analytical formulas for the bound state energies in an arbitrary many-particle Coulomb system.