The local and global parts of the basic zeta coefficient for operators on manifolds with boundary

Abstract

For operators on a compact manifold X with boundary ∂X, the basic zeta coefficient C0(B, P1,T) is the regular value at s = 0 of the zeta function \({Tr(B P_{1,T}^{-s})}\) , where B = P+ + G is a pseudodifferential boundary operator (in the Boutet de Monvel calculus)—for example the solution operator of a classical elliptic problem—and P1,T is a realization of an elliptic differential operator P1, having a ray free of eigenvalues. Relative formulas (e.g., for the difference between the constants with two different choices of P1,T) have been known for some time and are local. We here determine C0(B, P1,T) itself (with even-order P1), showing how it is put together of local residue-type integrals (generalizing the noncommutative residues of Wodzicki, Guillemin, Fedosov–Golse–Leichtnam–Schrohe) and global canonical trace-type integrals (generalizing the canonical trace of Kontsevich and Vishik, formed of Hadamard finite parts). Our formula generalizes a formula shown recently by Paycha and Scott for manifolds without boundary. It leads in particular to new definitions of noncommutative residues of expressions involving log P1,T. Since the complex powers of P1,T lie far outside the Boutet de Monvel calculus, the standard consideration of holomorphic families is not really useful here; instead we have developed a resolvent parametric method, where results from our calculus of parameter-dependent boundary operators can be used.