Onirban Islam
A Duistermaat-Guillemin-Gutzwiller trace formula for a Dirac-type operator D on a globally hyperbolic spatially compact standard stationary spacetime \((M,g,Z)\) is achieved by generalising the recent construction by A. Strohmaier and S. Zelditch [Adv. Math. 376,107434 (2021)] to a vector bundle setting. We have analysed the spectrum of the Lie derivative \(\mathcal{L}_Z\) with respect to the global timelike Killing vector field \(Z\) on the kernel \(\mathrm{ker}\ D\) of \(D\) and found that it comprises discrete real eigenvalues. The distributional trace \(\mathrm{Tr}\ U_t\) of the time evolution operator \(U_t\) is then the trace of \(e^{t\mathcal{L}_Z}\) on ker D and \(\mathrm{Tr}\ U_t\) has singularities at the periods of induced Killing flow on the space of lightlike geodesics. This gives rise to the Weyl law asymptotic at the vanishing period. A pivotal technical ingredient to prove these results is the Feynman propagator for \(D\). In order to obtain a Fourier integral description of this propagator, we have generalised the classic work of J. Duistermaat and L. Hörmander [Acta Math. 128, 183 (1972)] on distinguished parametrices for any normally hyperbolic operator on a globally hyperbolic spacetime by propounding their microlocalisation theorem in a bundle setting. As a by-product of these analyses, another proof on the existence of Hadamard bisolutions for a normally hyperbolic operator and for a Dirac-type operator is reported.
