Exact Christoffel-Darboux expansions: A new, multidimensional, algebraic, eigenenergy bounding method

Abstract

Abstract Although the Christoffel-Darboux representation (CDR) plays an important role within the theory of orthogonal polynomials, and many important bosonic and fermionic, multidimensional, Hermitian and Non-Hermitian, 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 (CDR and MER), 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 54 287) for a broad range of magnetic fields, without the need for any truncations or approximations. The methods developed here apply to any linear, partial differential equation eigen-parameter problem, hermitian or non-hermitian.