Perturbation Theory for Arbitrary Coupling Strength ?

TL;DR

The paper introduces a novel mean field perturbation theory (MFPT) for quantum systems that is valid for any coupling strength, overcoming limitations of standard perturbation methods by providing convergent, non-analytic results.

Contribution

It develops a self-consistent, feedback-based perturbation framework that handles arbitrary interaction strengths and non-analytic properties, demonstrated with anharmonic oscillators.

Findings

Borel-summability of MFPT shown for anharmonic oscillators

Accurate ground state results for arbitrary coupling strengths

Unambiguous spectrum calculations for double-well oscillator

Abstract

We present a \emph{new} formulation of perturbation theory for quantum systems, designated here as: `mean field perturbation theory'(MFPT), which is free from power-series-expansion in any physical parameter, including the coupling strength. Its application is thereby extended to deal with interactions of \textit{arbitrary} strength and to compute system-properties having non-analytic dependence on the coupling, thus overcoming the primary limitations of the `standard formulation of perturbation theory' ( SFPT). MFPT is defined by developing perturbation about a chosen input Hamiltonian, which is exactly solvable but which acquires the non-linearity and the analytic structure~(in the coupling-strength)~of the original interaction through a self-consistent, feedback mechanism. We demonstrate Borel-summability of MFPT for the case of the quartic- and sextic-anharmonic oscillators and the quartic double-well oscillator (QDWO) by obtaining uniformly accurate results for the ground state of the above systems for arbitrary physical values of the coupling strength. The results obtained for the QDWO may be of particular significance since `renormalon'-free, unambiguous results are achieved for its spectrum in contrast to the well-known failure of SFPT in this case. \pacs{11.15.Bt,11.10.Jj,11.25.Db,12.38.Cy,03.65.Ge}