The semiclassical zeta function for geodesic flows on negatively curved manifolds
Résumé
We consider the semi-classical (or Gutzwiller-Voros) zeta functions for $C^\infty $ contact Anosov flows. Analyzing the spectra of the generators of some transfer operators associated to the flow, we prove that, for arbitrarily small $\tau >0$, its zeros are contained in the union of the $\tau $-neighborhood of the imaginary axis, $|\mathfrak {R}(s)|<\tau $, and the half-plane $\mathfrak {R}(s)<-\chi _0+\tau $, up to finitely many exceptions, where $\chi _0>0$ is the hyperbolicity exponent of the flow. Further we show that the density of the zeros along the imaginary axis satisfy an analogue of the Weyl law.