Length spectrum of periodic rays for billard flow
Abstract
We study for several compact strictly convex disjoint obstacles the length spectrum $\mathcal L$ formed by the lengths of all primitive periodic reflecting rays. We prove the existence of sequences $\{\ell_j\},\: \{m_j\}$ with $\ell_j \in \mathcal L,\: m_j \in \mathbb N$ such that the condition (LB) related to the dynamical zeta function $η_D(s)$ is satisfied. This condition implies the existence of lower bounds for the number of the scattering resonances for Dirichlet Laplacian. We construct such sequences under some separation condition for a small subset of $\mathcal L$ corresponding to lengths of the periodic rays with even reflexions. Our separation condition is weaker than the assumption of exponentially separated length spectrum $\mathcal L.$ Moreover, we show that the periodic orbits in the phase space are exponentially separated.
Fields of Study & Subjects
Spectral Theory in Mathematical Physics
Quantum chaos and dynamical systems
Mathematical Dynamics and Fractals
Publication details
- Journal / source
- arXiv (Cornell University)
- Volume
- 25
- Issue
- 5
- Pages
- 163
- Publication date
- 3 February 2026
- Language
- EN
- Document type
- Preprint