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On the existence of small-time heat expansions for operators with irregular singularities in the coefficients

The asymptotic coefficients in the expansion of Tr e−tH as t → 0+, for H a positive elliptic differential operator of even order, with smooth coefficients, on a compact manifold, are spectral invariants of H given by integrals of local functions of the coefficients of H [S, Gi]. They are of proven value in computations of indices [ABP, Gi] and determinants [CT, e.g.] and in inverse spectral problems [Gu, OPS]. They are also, in general, notoriously intractable. For large classes of differential operators with singular coefficients, certain expansions of a rather different type still exist but the complications seem to be compounded. If H is positive elliptic of order d on a manifold, with coefficients that can have singularities on submanifolds, characterized by asymptotic expansions to all orders (as in Section 1 below), the typical result is: For φ a multiplication operator by a smooth function of bounded support,