Open AccessOrthonormal polynomial projection quantization: an algebraic eigenenergy bounding method
Author(s) -
Carlos R. Handy
Publication year - 2022
Publication title -
acta polytechnica
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.207
H-Index - 15
eISSN - 1805-2363
pISSN - 1210-2709
DOI - 10.14311/ap.2022.62.0063
Subject(s) - hermitian matrix , semidefinite programming , mathematics , orthonormal basis , algebraic number , moment problem , eigenvalues and eigenvectors , pure mathematics , algebra over a field , mathematical analysis , quantum mechanics , mathematical optimization , principle of maximum entropy , physics , statistics
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.