On two perturbation series for short-range attractive potentials

TL;DR

This paper compares two perturbation series for finite attractive wells, analyzing their effectiveness and limitations through exactly solvable models, highlighting issues with coefficients blowing up in one approach.

Contribution

It introduces and evaluates an alternative perturbation expansion based on Rayleigh-Schrödinger theory for short-range attractive potentials.

Findings

The traditional power series in strength parameter can be effective.

The Rayleigh-Schrödinger based series has limitations with boundary conditions.

Coefficients in the Rayleigh-Schrödinger series can diverge for large system sizes.

Abstract

We compare two alternative expansions for finite attractive wells. One of them is known from long ago and is given in terms of powers of the strength parameter. The other one is based on the solution of the equations of the Rayleigh-Schr\"{o}dinger perturbation theory in a basis set of functions of period $L$. The analysis of exactly solvable models shows that although the exact solution of the problem with periodic boundary conditions yields the correct result when $L\rightarrow \infty$ the coefficients of the series for this same problem blow up and fail to produce the correct asymptotic expansion.