Spectral gaps without the pressure condition
Copyright policy )Spectral gaps without the pressure condition
For all convex co-compact hyperbolic surfaces, we prove the existence of an essential spectral gap, that is a strip beyond the unitarity axis in which the Selberg zeta function has only finitely many zeroes. We make no assumption on the dimension $��$ of the limit set, in particular we do not require the pressure condition $��\leq {1\over 2}$. This is the first result of this kind for quantum Hamiltonians. Our proof follows the strategy developed by Dyatlov-Zahl [arXiv:1504.06589]. The main new ingredient is the fractal uncertainty principle for $��$-regular sets with $��<1$, which may be of independent interest.
39 pages, 5 figures. Added explanations of the proof (especially for Theorem 4) and revised according to referee's comments. To appear in Ann. Math
- Massachusetts Institute of Technology United States
- Institute for Advanced Study United States
Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems, Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc., Quasi-analytic and other classes of functions of one complex variable, FOS: Physical sciences, General topics in linear spectral theory for PDEs, Dynamical Systems (math.DS), Nonlinear Sciences - Chaotic Dynamics, fractal uncertainty principle, Mathematics - Spectral Theory, Mathematics - Analysis of PDEs, PDEs on manifolds, Mathematics - Classical Analysis and ODEs, spectral gap, Classical Analysis and ODEs (math.CA), FOS: Mathematics, Resonance in context of PDEs, Mathematics - Dynamical Systems, Chaotic Dynamics (nlin.CD), scattering resonance, Spectral Theory (math.SP), Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry, Analysis of PDEs (math.AP)
Inverse spectral and scattering methods for infinite-dimensional Hamiltonian and Lagrangian systems, Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc., Quasi-analytic and other classes of functions of one complex variable, FOS: Physical sciences, General topics in linear spectral theory for PDEs, Dynamical Systems (math.DS), Nonlinear Sciences - Chaotic Dynamics, fractal uncertainty principle, Mathematics - Spectral Theory, Mathematics - Analysis of PDEs, PDEs on manifolds, Mathematics - Classical Analysis and ODEs, spectral gap, Classical Analysis and ODEs (math.CA), FOS: Mathematics, Resonance in context of PDEs, Mathematics - Dynamical Systems, Chaotic Dynamics (nlin.CD), scattering resonance, Spectral Theory (math.SP), Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry, Analysis of PDEs (math.AP)
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