2009/04/10 by Nalini Anantharaman, Anantharaman, Nalini
Engineering · Mathematics · #35P20 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Mathematical Dynamics and Fractals #Spectral Theory in Mathematical Physics #Stability and Controllability of Differential Equations #math.AP #math.DG #msc:35P20
paper · pdf · doi:10.48550/arxiv.0904.1736
arxiv created 2009/04/10 · openalex publication_date 2009/04/10 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove a Weyl-type fractal upper bound for the spectrum of the damped wave equation, on a negatively curved compact manifold. It is known that most of the eigenvalues have an imaginary part close to the average of the damping function. We count the number of eigenvalues in a given horizontal strip deviating from this typical behaviour; the exponent that appears naturally is the `entropy' that gives the deviation rate from the Birkhoff ergodic theorem for the geodesic flow. A Weyl-type lower bound is still far from reach; but in the particular case of arithmetic surfaces, and for a strong enough damping, we can use the trace formula to prove a result going in this direction.