Improved Convergence Proof of the Delta Expansion and Order Dependent Mappings

TL;DR

This paper enhances the mathematical proof of convergence for the delta expansion method in quantum mechanics, covering complex domains and strong coupling regimes, with broad applicability to various systems.

Contribution

It provides a more general and rigorous proof of convergence for the delta expansion, including complex and strong coupling cases, with three theorems outlining sufficient conditions.

Findings

Proven uniform convergence for single-well anharmonic oscillator.

Extended convergence to complex coupling values on the Riemann surface.

Established convergence for double-well potentials in strong coupling regimes.

Abstract

We improve and generalize in several accounts the recent rigorous proof of convergence of delta expansion - order dependent mappings (variational perturbation expansion) for the energy eigenvalues of anharmonic oscillator. For the single-well anharmonic oscillator the uniformity of convergence in $g\in[0,\infty]$ is proven. The convergence proof is extended also to complex values of $g$ lying on a wide domain of the Riemann surface of $E(g)$. Via the scaling relation \`a la Symanzik, this proves the convergence of delta expansion for the double well in the strong coupling regime (where the standard perturbation series is non Borel summable), as well as for the complex ``energy eigenvalues'' in certain metastable potentials. Sufficient conditions for the convergence of delta expansion are summarized in the form of three theorems, which should apply to a wide class of quantum mechanical and higher dimensional field theoretic systems.