The semiclassical zeta function for geodesic flows on negatively curved manifolds
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We consider the semi-classical (or Gutzwiller–Voros) zeta functions for $$C^\infty $$C∞ contact Anosov flows. Analyzing the spectra of the generators of some transfer operators associated to the flow, we prove that, for arbitrarily small $$\tau >0$$τ>0, its zeros are contained in the union of the $$\tau $$τ-neighborhood of the imaginary axis, $$|\mathfrak {R}(s)|<\tau $$|R(s)|<τ, and the half-plane $$\mathfrak {R}(s)<-\chi _0+\tau $$R(s)<-χ0+τ, up to finitely many exceptions, where $$\chi _0>0$$χ0>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.
Subfields: Applied Mathematics, Mathematical Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/7474e4f6cdf3fccd30149f9b7369539ecbb8f2b8