Exact solution of the Schr\"odinger equation with a Lennard-Jones potential

TL;DR

This paper presents an exact method for solving the Schrödinger equation with a Lennard-Jones potential, accurately determining bound states and wave functions by handling singularities rigorously.

Contribution

It introduces a novel procedure that rigorously treats singularities and computes bound state energies and wave functions using connection factors and series expansions.

Findings

Calculated bound state energies as zeros of a convergent series

Derived wave functions as Laurent and asymptotic expansions

Provided a table of critical potential intensities for zero-energy bound states

Abstract

The Schr\"odinger equation with a Lennard-Jones potential is solved by using a procedure that treats in a rigorous way the irregular singularities at the origin and at infinity. Global solutions are obtained thanks to the computation of the connection factors between Floquet and Thom\'e solutions. The energies of the bound states result as zeros of a function defined by a convergent series whose successive terms are calculated by means of recurrence relations. The procedure gives also the wave functions expressed either as a linear combination of two Laurent expansions, at moderate distances, or as an asymptotic expansion, near the singular points. A table of the critical intensities of the potential, for which a new bound state (of zero energy) appears, is also given.