Exact Christoffel-Darboux Expansions: A New, Multidimensional, Algebraic, Eigenenergy Bounding Method

TL;DR

This paper introduces the OPPQ-BM, an algebraic eigenenergy bounding method combining Christoffel-Darboux representation and moment equation representation, effectively analyzing multidimensional quantum systems without truncations.

Contribution

It develops the OPPQ-BM, a novel fusion of mathematical representations for bounding eigenenergies in multidimensional quantum systems, surpassing previous methods in accuracy and simplicity.

Findings

Successfully analyzed 1D and 2D systems including the quadratic Zeeman effect.

Matched or exceeded the accuracy of previous complex methods.

Operates without truncations or approximations.

Abstract

Although the Christoffel-Darboux representation (CDR) plays an important role within the theory of orthogonal polynomials, and many important bosonic and fermionic multidimensional Schrodinger equation systems can be transformed into a moment equation representation (MER), the union of the two into an effective, algebraic, eigenenergy bounding method has been overlooked. This particular fusion of the two representations, suitable for bounding bosonic or fermionic systems, defines the Orthonormal Polynomial Projection Quantization - Bounding Method (OPPQ-BM), as developed here. We use it to analyze several one dimensional and two dimensional systems, including the quadratic Zeeman effect for strong-superstrong magnetic fields. For this problem, we match or surpass the excellent, but intricate, results of Kravchenko et al (1996 Phys. Rev. A 54287) for a broad range of magnetic fields, without the need for any truncations or approximations.