Consider the Laplace operator on some manifold, $\Delta=(\det g)^{-1/2} \frac{\partial}{\partial x^j} \left( (\det g)^{1/2}g^{jk} \frac{\partial}{\partial x^k}\right) $. I am looking for differential operators $Q$ satisfying the following relationship with the Laplace operator,
$$ [ \Delta, Q]f = \Delta (Qf) - Q \Delta f = \omega Q f~,$$
where the first equality sign serves as a definition, $\omega$ is a non-zero real number, and $f$ is a test function. Have these operators $Q$ been studied before in the literature? If so, does an explicit expression exist for such operators?
Perhaps it is worth mentioning that I am particularly interested in the above question for the Laplacian on the Poincaré disk,
$$\Delta = \frac{1}{4}(1-x^2-y^2)(\partial_x^2+\partial_y^2)~.$$
Any help with either the general case or the specific case of the Poincaré disk is most welcome.
NOTE: I asked the same question in the math stack exchange but, perhaps due to the `physics phrasing', did not receive any response so far. For this reason and because of the fact that this question is also relevant for physics, I decided to ask it here as well.
