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A method for computing global minima of real multivariate polynomials basedon 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 tononcommutative symmetric polynomials in variables $X$ and $Y$ satisfying$YX-XY=1$ and $X^\ast=X$, $Y^\ast=-Y$. Global minima of such polynomials aredefined and showed to be equal to minima of the spectra of the correspondingdifferential operators. We also discuss how to exploit sparsity and symmetry.Several numerical experiments are included. The last section explains how ourtheory fits into the framework of noncommutative real algebraic geometry.

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