(Quasi)-Convexification of Barta's (Multi-Extrema) Bounding Theorem

TL;DR

This paper enhances Barta's configuration space theorem for bounding ground state energies by transforming it into a moments' representation and applying a generalized Eigenvalue Moment Method, addressing key difficulties like multi-extrema and inefficiency.

Contribution

It introduces a systematic transformation of BCST into a moments' form and employs a generalized Eigenvalue Moment Method to improve bound calculations.

Findings

Addresses multi-extrema issues in BCST

Improves efficiency for stiff problems

Provides a systematic bound improvement procedure

Abstract

There has been renewed interest in the exploitation of Barta's configuration space theorem (BCST, (1937)) which bounds the ground state energy. Mouchet's (2005) BCST analysis is based on gradient optimization (GO). However, it overlooks significant difficulties: (i) appearance of multi-extrema; (ii) inefficiency of GO for stiff (singular perturbation/strong coupling) problems; (iii) the nonexistence of a systematic procedure for arbitrarily improving the bounds. These deficiencies can be corrected by transforming BCST into a moments' representation equivalent, and exploiting a generalization of the Eigenvalue Moment Method (EMM), within the context of the well known Generalized Eigenvalue Problem (GEP), as developed here.