Perturbation-based Non-perturbative Method

TL;DR

This paper introduces a nonperturbative eigenproblem solving method that transforms the problem into a perturbation framework, then analytically continues solutions to obtain accurate results for all energy levels.

Contribution

It proposes a novel approach that combines perturbation theory with analytic continuation to solve eigenproblems nonperturbatively for a wide range of potentials.

Findings

Applicable to almost all potentials

Provides accurate nonperturbative energy approximations

Transforms eigenproblems into perturbation problems for easier solutions

Abstract

This paper presents a nonperturbative method for solving eigenproblems. This method applies to almost all potentials and provides nonperturbative approximations for any energy level. The method converts an eigenproblem into a perturbation problem, obtains perturbation solutions through standard perturbation theory, and then analytically continues the perturbative solution into a nonperturbative solution. Concretely, we follow three main steps: (1) Introduce an auxiliary potential that can be solved exactly and treat the potential to be solved as a perturbation on this auxiliary system. (2) Use perturbation theory to obtain an approximate polynomial of the eigenproblem. (3) Use a rational approximation to analytically continue this approximate polynomial into the nonperturbative region.