  In this paper we study the interior transmission problem and transmission
eigenvalues for multiplicative perturbations of linear partial differential
operator of order $\ge 2$ with constant real coefficients. Under suitable
growth conditions on the symbol of the operator and the perturbation, we show
the discreteness of the set of transmission eigenvalues and derive sufficient
conditions on the existence of transmission eigenvalues. We apply these
techniques to the case of the biharmonic operator and the Dirac system. In the
hypoelliptic case we present a connection to scattering theory.
