Orthonormal polynomial projection quantization: an algebraic eigenenergy bounding method

Abstract

The ability to generate tight eigenenergy bounds for low dimension bosonic or ferminonic, hermitian or non-hermitian, Schrödinger operator problems is an important objective in the computation of quantum systems. Very few methods can simultaneously generate lower and upper bounds. One of these is the Eigenvalue Moment Method (EMM) originally introduced by Handy and Besssis, exploiting the use of semidefinite programming/nonlinear-convex optimization (SDP) techniques as applied to the positivity properties of the multidimensional bosonic ground state for a large class of important physical systems (i.e. those admitting a moments’ representation). A recent breakthrough has been achieved through another, simpler, moment representation based quantization formalism, the Orthonormal Polynomial Projection Quantization Bounding Method (OPPQ-BM). It is purely algebraic and does not require any SDP analysis. We discuss its essential structure in the context of several one dimensional examples.