Dr. Lennart Ronge

We develop a formula for the diagonal values of the Hadamard coefficients associated to a normally hyperbolic operator on a globally hyperbolic spacetime in terms of the advanced and retarded Green's operators. We develop a local formula as well as formulae for integrals over (parts of) the diagonal. Furthermore, we develop analogues of the Hadamard expansion for powers of the advanced/retarded Green's operators and an analogue of a resolvent.

The Hadamard expansion describes the singularity structure of Green’s operators associated with a normally hyperbolic operator P in terms of Riesz distributions (fundamental solutions on Minkowski space, transported to the manifold via the exponential map) and Hadamard coefficients (smooth sections in two variables, corresponding to the heat Kernel coefficients in the Riemannian case). In this paper, we derive an asymptotic expansion analogous to the Hadamard expansion for powers of advanced/retarded Green’s operators associated with P, as well as expansions for advanced/retarded Green’s operators associated with P−z for z∈C. These expansions involve the same Hadamard coefficients as the original Hadamard expansion, as well as the same or analogous (with built-in z-dependence) Riesz distributions.

We study the Atiyah-Patodi-Singer (APS) index, and its equality to the spectral flow,
in an abstract, functional analytic setting. More precisely, we consider a (suitably continuous or
differentiable) family of self-adjoint Fredholm operators A(t) on a Hilbert space, parametrised
by t in a finite interval. We then consider two different operators, namely D := d/dt + A (the
abstract analogue of a Riemannian Dirac operator) and D := d/dt − iA (the abstract analogue of
a Lorentzian Dirac operator). The latter case is inspired by a recent index theorem by Bär and
Strohmaier (Amer. J. Math. 141 (2019), 1421–1455) for a Lorentzian Dirac operator equipped
with APS boundary conditions. In both cases, we prove that the Fredholm index of the operator
D equipped with APS boundary conditions is equal to the spectral flow of the family A(t) .