ar X iv : h ep - t h / 04 09 16 4 v 1 15 S ep 2 00 4 Green ’ s functions and propagation of waves in strongly inhomogeneous media

We show that Green's functions of second order differential operators with singular or unbounded coefficients can have an anomalous behaviour in comparison to the well-known properties of the Green's functions of operators with bounded coefficients. We discuss some consequences of such an anomalous short or long distance behaviour for a diffusion and wave propagation in an inhomogeneous medium. 1 Introduction A wave propagation with a constant speed is an approximation to the real situation when the wave propagates in a medium with its characteristics varying in space. A similar approximation is applied when considering a diffusion. In general, the speed of the diffusion can vary in space. If this variation is slow then one could believe that its effects are negligible. It has been observed some time ago (see [1] and references quoted there) that a diffusion in strongly in-homogeneous materials can be anomalous. This happens in particular with a heat convection in a turbulent medium [2][3]. In general, it is rather difficult to investigate these problems because exact solutions are not available and approximations to equations with varying coefficients are not reliable.We discuss here the proper time method for a representation of the Green's functions. We represent solutions of the equation AG E = 2δ (1) for the Green's function of an elliptic operator A in terms of its heat kernel. In this way the Green's function is expressed by a diffusion. A solution of such an equation allows to determine the wave propagation if we consider equation (1)either as an analog of the Helmholtz equation for the propagation of monochromatic waves (then all the coordinates in (1) are spatial) or continue 1