Semi-classical trace formulas and heat expansions

There is a strong similarity between the expansions of the heat kernel as worked out by people in Riemannian geometry in the seventies (starting with the famous “Can one hear the shape of a drum” by Mark Kac [Kac], the Berger paper [Berger] and the Mc-Kean-Singer paper [McK-Si]) and the so-called semi-classical trace formulas developed by people in semi-classical analysis (starting with HelfferRobert [He-Ro]). In fact, this is not only a similarity, but, as we will prove, each of these expansions, even if they differ when expressed numerically for some example, can be deduced from the other one as formal expressions of the fields. Let us look first at the heat expansion on a smooth closed Riemannian manifold of dimension d, (X, g), with the (negative) Laplacian ∆g . The heat kernel e(t, x, y), with t > 0 and x, y ∈ X, is the Schwartz kernel of exp(t∆g): the solution of the heat equation ut −∆gu = 0 with initial datum u0 is given by