On the asymptotics of the trace of the heat kernel for the magnetic Schrodinger operator

We consider the effect of a magnetic field for the asymptotic behavior of the trace of the heat kernel for the Schrodinger operator. We discuss the case where the operator has compact resolvents in spite of the fact that the electric potential is degenerate on some submanifold. According to the degree of the degenaracy, we obtain the classical and non-classical asymptotics.