The semiclassical zeta function for geodesic flows on negatively curved manifolds

We consider the semi-classical (or Gutzwiller-Voros) zeta function for $C^\infty$ contact Anosov flows. Analyzing the spectrum of transfer operators associated to the flow, we prove, for any $τ>0$, that its zeros are contained in the union of the $τ$-neighborhood of the imaginary axis, $|\Re(s)|<τ$, and the region $\Re(s)<-χ_0+τ$, up to finitely many exceptions, where $χ_0>0$ is the hyperbolicity exponent of the flow. Further we show that the zeros in the neighborhood of the imaginary axis satisfy an analogue of the Weyl law.

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