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H0 has compact resolvents (c.f. for example, Reed and Simon [8]). Odencrantz [7] studied the asymptotic behavior of Tr[exp(−tH)− exp(−tH0)] as t ↓ 0 in the case where V (z) ≈ |z|2p (p > 0) and the curl of A(z) is a uniform magnetic field. Matsumoto [5] extended the result to the case with more general magnetic field. However, (1.2) is not a necessary condition in order that H0 has compact resolvents. In spite of the lack of (1.2), there are some cases where H0 has compact resolvents. The motivation of this paper originates in the works of Simon [11], Robert [9] and Aramaki [3]. In order to explain this, we put

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