The Coulomb potential V(r)=1/r and other radial problems on the Bethe lattice

TL;DR

This paper provides an exact solution for a particle with Coulomb potential on the Bethe lattice and introduces a numerical mapping method for generalized radial potentials, aiding analysis when analytical solutions are unavailable.

Contribution

It offers the first exact Green's function and spectrum for Coulomb potential on the Bethe lattice, and develops a numerical mapping technique for complex radial problems.

Findings

Exact Green's function derived for Coulomb potential

Full energy spectrum obtained

Numerical mapping method for generalized potentials

Abstract

We study the problem of a particle hopping on the Bethe lattice in the presence of a Coulomb potential. We obtain an exact solution to the particle's Green's function along with the full energy spectrum. In addition, we present a mapping of a generalized radial potential problem defined on the Bethe lattice to an infinite number of one dimensional problems that are easily accessible numerically. The latter method is particularly useful when the problem admits no analytical solution.