A method for computing lowest eigenvalues of symmetric polynomial differential operators by semidefinite programming

TL;DR

This paper extends semidefinite programming methods to compute the lowest eigenvalues of symmetric polynomial differential operators, incorporating noncommutative variables, sparsity, and symmetry considerations, with practical numerical experiments.

Contribution

It introduces a novel approach for noncommutative polynomial minimization related to differential operators, expanding existing methods to noncommutative settings and spectral analysis.

Findings

Successfully computes eigenvalues of noncommutative differential operators

Demonstrates efficiency with sparsity and symmetry exploitation

Includes numerical experiments validating the method

Abstract

A method for computing global minima of real multivariate polynomials based on semidefinite programming was developed by N. Z. Shor, J. B. Lasserre and P. A. Parrilo. The aim of this article is to extend a variant of their method to noncommutative symmetric polynomials in variables $X$ and $Y$ satisfying $YX-XY=1$ and $X^\ast=X$, $Y^\ast=-Y$. Global minima of such polynomials are defined and showed to be equal to minima of the spectra of the corresponding differential operators. We also discuss how to exploit sparsity and symmetry. Several numerical experiments are included. The last section explains how our theory fits into the framework of noncommutative real algebraic geometry.