A Gutzwiller Trace formula for Dirac Operators on a Stationary Spacetime

Abstract

A Duistermaat-Guillemin-Gutzwiller trace formula for Dirac-type operators on a globally hyperbolic spatially compact standard stationary spacetime 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 with respect to the Killing vector field on the solution space of the Dirac equation and found that it consists of discrete real eigenvalues. The distributional trace of the Cauchy evolution operator 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.1 We do not require Einstein's field equations to be satisfied.2 Global existence of a Lorentz metric depends on the topology of M . In particular, such a metric exists in all non-compact manifolds and compact manifolds with vanishing Euler characteristics. 3A spin-structure always exists locally but its global existence depends on some higher orientability property of the base manifold. For example, spin c and spin structures exist if and only if the third integral and the second Stiefel-Whitney class of spacetime vanish, respectively.

A Trace Formula on Stationary Kaluza-Klein Spacetimes

“…This involves understanding the distribution of the frequency spectrum for the wave equation on a Kaluza-Klein spacetime when restricted to the isotypic subspace of an irreducible representation of the structure group, in the limit that the weight of the representation approaches infinity in the Weyl chamber. This is a direct generalization of the results from [23] and is closely related to [22], [14]. Furthermore we show how to apply these results to frequency asymptotics for the massive Klein-Gordon equation on vector bundles as one takes the representation defining the vector bundle to infinity.…”

“…Since we allow for a potential (as long as it is constant along the fibers of P and independent of t) the energy quadratic form on the space of solutions ker ω need not be positive definite, but we use standard results from harmonic analysis together with some results on Krein and Pontryagin spaces to show that it is positive definite on the isotypic subspace H m for m sufficiently large. In Section 3.1 we apply a result from [14] to obtain that µ(E, m, −) is tempered and then we provide a proof of Corollary 1.4 given Theorems 1.1,1.2,1.3. Finally, in Section 4 we simply combine the techniques from [22] and [11] to obtain our main theorems.…”

“…Instead we first notice that µ(E, m, −) defines a linear functional on the collection of all ϕ ∈ S(R) with compactly supported Fourier transform. Our goal now is to apply a result of [14] which generalizes the Weyl law of [22] to vector bundles in order to prove that µ(E, m, −) is actually tempered and hence µ(E, m, ϕ) is defined for any ϕ ∈ S(R).…”

“…We are now ready to apply the results of [14]. Really we are using a very special case of these results since we only need them to show that our multiplicity distributions µ(E, m, −) are tempered.…”

“…Theorem 3.29. [14] The operator D m,Z is self-adjoint on (ker m , Q m ) with σ(D m,Z ) ⊆ R discrete and accumulating at ±∞ with polynomial growth.…”