(Quasi)-convexification of Barta's (multi-extrema) bounding theorem:

Abstract

There has been renewed interest in the exploitation of Barta's configuration space theorem (BCST) (Barta 1937 C. R. Acad. Sci. Paris 204 472) which bounds the ground-state energy, , by using any Φ lying within the space of positive, bounded, and sufficiently smooth functions, . Mouchet's (Mouchet 2005 J. Phys. A: Math. Gen. 38 1039) 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 within . 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. EMM is an alternative eigenenergy bounding, variational procedure, overlooked by Mouchet, which also exploits the positivity of the desired physical solution. Furthermore, it is applicable to Hermitian and non-Hermitian systems with complex-number quantization parameters (Handy and Bessis 1985 Phys. Rev. Lett. 55 931, Handy et al 1988 Phys. Rev. Lett. 60 253, Handy 2001 J. Phys. A: Math. Gen. 34 5065, Handy et al 2002 J. Phys. A: Math. Gen. 35 6359). Our analysis exploits various quasi-convexity/concavity theorems common to the GEP representation. We outline the general theory, and present some illustrative examples.