Question

Is there some well-known description of an operator $D$ (pseudodifferential of order greater than 0) for which $e^{-g(x_1 p_1,...,x_n p_n)}$, where $g \colon \mathbb{R}^n_+ \to \mathbb{R}_+$ is a homogeneous of order 1 function, will be eigenfunctions for any $(p_1,...,p_n) \in \mathbb{R}^n_+$? I.e. $$ D e^{-g(p_1 x_1,...,p_n x_n)} = \lambda_p e^{-g(p_1 x_1,...,p_n x_n)}. $$