  The asymptotic expansion of the heat-kernel for small values of its argument
has been studied in many different cases and has been applied to 1-loop
calculations in Quantum Field Theory. In this thesis we consider this
asymptotic behavior for certain singular differential operators which can be
related to quantum fields on manifolds with conical singularities. Our main
result is that, due to the existence of this singularity and of infinitely many
boundary conditions of physical relevance related to the admissible behavior of
the fields on the singular point, the heat-kernel has an "unusual" asymptotic
expansion. We describe examples where the heat-kernel admits an asymptotic
expansion in powers of its argument whose exponents depend on "external"
parameters. As far as we know, this kind of asymptotics had not been found and
therefore its physical consequences are still unexplored.
