Perturbation theory for short-range weakly-attractive potentials in one dimension

TL;DR

This paper develops sixth-order perturbative formulas for bound state energies in weak, short-range one-dimensional potentials, extending previous results and validating them through applications to solvable and complex problems.

Contribution

It provides the first sixth-order perturbation expressions for bound states in one-dimensional weak potentials, building on and extending prior lower-order results.

Findings

Sixth-order energy expressions match known results up to fifth order.

New sixth-order terms calculated for the first time.

Validated formulas with applications to exactly solvable and complex potentials.

Abstract

We have obtained the perturbative expressions up to sixth order for the energy of the bound state in a one dimensional, arbitrarily weak, short range finite well, applying a method originally developed by Gat and Rosenstein Ref. [3]. The expressions up to fifth order reproduce the results already known in the literature, while the sixth order had not been calculated before. As an illustration of our formulas we have applied them to two exactly solvable problems and to a nontrivial problem.