  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.
