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By making use of the potentials of the heat conduction equation the integral equations are derived which determine the heat kernel for the Laplace operator -a2Δ in the case of compound media. In each of the media the parameter a2 acquires a certain constant value. At the interface of the media the conditions are imposed which demand the continuity of the temperature and the heat flows. The integration in the equations is spread out only over the interface of the media. As a result the dimension of the initial problem is reduced by 1. The perturbation series for the integral equations derived are nothing else as the multiple scattering expansions for the relevant heat kernels. Thus a rigorous derivation of these expansions is given. In the one dimensional case the integral equations at hand are solved explicitly (Abel equations) and the exact expressions for the regarding heat kernels are obtained for diverse matching conditions. Derivation of the asymptotic expansion of the integrated heat kernel for a compound media is considered by making use of the perturbation series for the integral equations obtained. The method proposed is also applicable to the configurations when the same medium is divided, by a smooth compact surface, into internal and external regions, or when only the region inside (or outside) this surface is considered with appropriate boundary conditions. [ABSTRACT FROM AUTHOR]

Spectral zeta functions ζ(s) for the massless scalar fields obeying the Dirichlet and Neumann boundary conditions on a surface of an infinite cylinder are constructed. These functions are defined explicitly in a finite domain of the complex plane s containing the closed interval of real axis -1≤Re s≤0. Proceeding from this the spectral zeta functions for the boundary conditions given on a circle (boundary value problem on a plane) are obtained without any additional calculations. The Casimir energy for the relevant field configurations is deduced. © 2000 American Institute of Physics. [ABSTRACT FROM AUTHOR]