Generating Converging Eigenenergy Bounds for Multidimensional Systems: A New Moment Representation, Algebraic, Quantization Formalism

TL;DR

This paper introduces a novel algebraic quantization method, OPPQ-BM, that generates tight eigenenergy bounds for multidimensional quantum systems without convex optimization, surpassing previous approaches in simplicity and accuracy.

Contribution

The paper presents the OPPQ-BM, a new algebraic formalism for eigenenergy bounding that extends moment equation methods to all discrete states without convex optimization.

Findings

Successfully bounds all discrete states in multidimensional systems.

Matches or surpasses previous complex analyses like the quadratic Zeeman effect.

Offers a simpler, exact, and algebraic alternative to existing methods.

Abstract

For low dimension systems admitting a moment equation representation (MER), the development of an effective eigenenergy bounding theory applicable to all discrete states had remained elusive, until now. Whereas Handy et al (1988 Phys. Rev. Lett. 60 253) demonstrated the effectiveness of the {\it Moment Problem} based, Eigenvalue Moment Method (EMM), for generating arbitrarily tight bounds to the multidimensional, positive, bosonic ground state, its extension to arbitrary excited states seemed intractable. We have discovered a new, moment representation based, quantization formalism that achieves this. Unlike EMM, no convex optimization methods are required. The entire formulation is algebraic. As a result of our preliminary investigation, we are able to match, or surpass, the excellent, but intricate, analysis of Kravchenko et al (1996 Phys. Rev. A 54 287) with respect to the quadratic Zeeman effect, for a broad range of magnetic field strengths. Unlike their analysis, the proposed method is simple, involves no truncations, and the projection of the quantum operator is exact, within each moment subspace. Our new approach, the Orthogonal Polynomial Projection Quantization-Bounding Method (OPPQ-BM), exploits the implicit bounding capabilities of a previous method developed by Handy and Vrinceanu (2013 J. of Phys. A: Math. Theor. 46 135202). What emerges is a completely new type of analysis (i.e. constrained quadratic form minimization) that validates the importance of moment equation representations for quantizing physical systems. Whereas the underlying principles of EMM guarantee it to be more efficient than OPPQ-BM, the ability to implement algebraic computations, as opposed to pursuing nonlinear convex optimization methods (which can be relaxed through linear programming alternatives) recommends OPPQ-BM. We give an overview of the new method with applications.