Propagation of anisotropic Gelfand-Shilov wave front sets

We show a result on propagation of the anisotropic Gelfand--Shilov wave front set for linear operators with Schwartz kernel which is a Gelfand--Shilov ultradistribution of Beurling type. This anisotropic wave front set is parametrized by two positive parameters relating the space and frequency variables. The anisotropic Gelfand--Shilov wave front set of the Schwartz kernel of the operator is assumed to satisfy a graph type criterion. The result is applied to a class of evolution equations that generalizes the Schr\"odinger equation for the free particle. The Laplacian is replaced by a partial differential operator defined by a symbol which is a polynomial with real coefficients and order at least two.