Math. Nachr. 166 (1994) 207-215 Heat Equation Asymptotics of Elliptic Operators with Non-scalar Leading Symbol zyxw By THOMAS P. BRANSON of Iowa, PETER B. GILKEY of Oregon and ANTONI PIERZCHALSKI of Lodi (Received October 23, 1992) (Revised Version June 25, 1993) Abstract. Let zyxwvuts A4 be a compact smoot Riemannian manifold without boundary and let D zy = zy adoh, + bald, - E on the space of smooth sections of the cotangent bundle where a and zyx b are positive constants and where E is an endomorphism. We use functorial methods and the pseudo-differential operator calculus to compute the quadratic term zyxwv n,(D) in the asymptotic expansion of the heat equation trace. 1. Introduction Let M be a compact smooth Riemannian manifold without boundary of dimension m and let Vbe a smooth vector bundle over M. Let P be an elliptic second order partial differential operator on Cw(V) with leading symbol a,; we assume that the eigenvalues of a, lie in a small cone about the positive real axis. The fundamental solution of the heat equation e-tp is well-defined and of trace class for t > 0. Let I? be an auxiliary endomorphism of zy I/: As t 1 O', there is an asymptotic series of the form: (1.1) f(E, P, zyxwvutsr t) := TrLZ (E"e-") - (47~)-~/' f a,($ P) t(n-m)i2 ; n=O see [16, Lemma 1.7.71 for details. The u,(& P) vanish for n odd and are locally computable. We set E" = 1 to definef(P, t) and to recover the usual heat equation asymptotics a,(P). If CJ~ is scalar, P is said to be of Laplace type; the u,(P) have been studied in this setting by many authors and play an important role in spectral geometry; see GILKEY zy [15] for n The situation is very different if a2 is not scalar; such operators are often said to be "non-minimal". The following example is quite important in severai different contexts. Let Ap be the bundle of p forms, let d = dp:CwAP 4 CWApf1 be exterior differentiation, and let 6 = hp:CmAPf' -+ CmAp be the adjoint operator, interior differentiation. Let a and b positive constants, let E be an endomorphism of T*M, and let (1.4 The leading symbol a, of D is self-adjoint and positive definite; c2 is scalar if and only if a = b. 6 and AMSTERDAMSKI et al. [2] and AVRAMIDI [3] for n = 8. D = D(a, b, E) = ado&, + bald, - E: CwT*M --f CwT*M.