On the characteristic function of random variables associated with boson Lie algebras

We compute the characteristic function of random variables de- ned as self-adjoint linear combinations of the Schrodinger algebra generators. We consider the extension of the method to innite dimensional Lie algebras. We show that, unlike the second order (Schrodinger) case, in the third order (cube of a Gaussian random variable) case the method leads to a nonlinear innite system of ODEs whose form is explicitly determined.