Positivity of a system of differential operators (Q1085318)

A necessary and sufficient condition for the positivity of the system of Weyl differential operators \(p^ w(x,D)\) with symbol \[ p(x,\xi)=\left[ \begin{matrix} ax^ 2+b\xi^ 2\\ \alpha x\xi \end{matrix} \begin{matrix} \alpha x\xi \\ cx^ 2+d\xi^ 2\end{matrix} \right],\quad a,b,c,d\geq 0 \] and \(ad+bc\neq 0\) is obtained. This condition depends only on ad, bc and \(\alpha\). In particular, it is shown that \((abcd)^{1/2}\geq \alpha^ 2/4\) is a sufficient condition. On the other hand, if \(abcd=0\), then \(p^ w(x,D)\geq 0\) if and only if \(\alpha =0\). This last result generalizes an example of Hörmander.