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We prove that, under some generic non-degeneracy assumptions, real analytic centrally symmetric plane domains are determined by their Dirichlet (resp. Neumann) spectra. We prove that the conditions are open-dense for real analytic strictly convex domains. One step is to use a Maslov index calculation to show that the second derivative of the defining function of a centrally symmetric domain at the endpoints of a bouncing ball orbit is a spectral invariant. This is also true for up-down symmetric domains, removing an assumption from the proof in that case. 

We review our recent relativistic generalization of the Gutzwiller–Duistermaat–Guillemin trace formula and Weyl law on globally hyperbolic stationary space-times with compact Cauchy hypersurfaces. We also discuss anticipated generalizations to non-compact Cauchy hypersurface cases. 

LetH = H_0 + Pdenote the harmonic oscillator on\mathbb{R}^dperturbed by an isotropic pseudodifferential operatorPof order1and letU(t) = \operatorname{exp}(- it H). We prove a Gutzwiller–Duistermaat–Guillemin type trace formula for\operatorname{Tr} U(t).The singularities occur at timest \in 2 \pi \mathbb{Z}and the coefficients involve the dynamics of the Hamilton flow of the symbol\sigma(P)on the space\mathbb{CP}^{d-1}of harmonic oscillator orbits of energy1. This is a novel kind of sub-principal symbol effect on the trace. We generalize the averaging technique of Weinstein and Guillemin to this order of perturbation, and then present two completely different calculations of\operatorname{Tr} U(t). The first proof directly constructs a parametrix ofU(t)in the isotropic calculus, following earlier work of Doll–Gannot–Wunsch. The second proof conjugates the trace to the Bargmann–Fock setting, the order1of the perturbation coincides with the 'central limit scaling' studied by Zelditch–Zhou for Toeplitz operators.