I have the following operator
$$\Phi(\chi_A)=\int \text{d}\eta\, \text{d}\zeta\,\chi_A(\eta,\zeta)e^{i(\eta \hat{P}+\zeta\hat{Q})}.$$
With $\chi_A$ the indicator function associated to a set $A\subset \mathbb{R}^2$ and $\hat{P},\hat{Q}$ hermitian operators that satisfy $[\hat{P},\hat{Q}]=1$. For which conditions on the set $A$ this operator is positive semi-definite?