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Dirichlet dynamical zeta function for billiard flow

Vesselin Petkov

TL;DR

The paper investigates the Dirichlet dynamical zeta η_D(s) for billiard flows around multiple strictly convex obstacles, formulating η_D(s) as η_D(s)=∑ a_n e^{−λ_n s} with increasing frequencies {λ_n}. It develops a local trace formula and applies Hardy–Riesz summability to show that, under σ_c<0, if η_D(s) were entire, the Dirichlet series would be (λ,k)-summable for all k, forcing strong tail-sum constraints on the coefficients {a_n}. The main result (Theorem 1.1) provides concrete conditions on frequency gaps λ_{m_j}−λ_{m_j−1} and tail sums |∑_{n≥m_j} a_n| that preclude entireness, and Section 4 builds abundant intervals with clustering frequencies to produce these conditions. Collectively, the work links analytic continuation properties of η_D to the distribution of resonances and supports the non-entireness conjecture under verifiable spectral-data assumptions, with implications for the MLPC in scattering theory.

Abstract

We study the Dirichlet dynamical zeta function $η_D(s)$ for billiard flow corresponding to several strictly convex disjoint obstacles. For large ${\rm Re}\: s$ we have $η_D(s) =\sum_{n= 1}^{\infty} a_n e^{-λ_n s}, \: a_n \in \mathbb R$ and $η_D$ admits a meromorphic continuation to $\mathbb C$. We obtain some conditions of the frequencies $λ_n$ and some sums of coefficients $a_n$ which imply that $η_D$ cannot be prolonged as entire function.

Dirichlet dynamical zeta function for billiard flow

TL;DR

The paper investigates the Dirichlet dynamical zeta η_D(s) for billiard flows around multiple strictly convex obstacles, formulating η_D(s) as η_D(s)=∑ a_n e^{−λ_n s} with increasing frequencies {λ_n}. It develops a local trace formula and applies Hardy–Riesz summability to show that, under σ_c<0, if η_D(s) were entire, the Dirichlet series would be (λ,k)-summable for all k, forcing strong tail-sum constraints on the coefficients {a_n}. The main result (Theorem 1.1) provides concrete conditions on frequency gaps λ_{m_j}−λ_{m_j−1} and tail sums |∑_{n≥m_j} a_n| that preclude entireness, and Section 4 builds abundant intervals with clustering frequencies to produce these conditions. Collectively, the work links analytic continuation properties of η_D to the distribution of resonances and supports the non-entireness conjecture under verifiable spectral-data assumptions, with implications for the MLPC in scattering theory.

Abstract

We study the Dirichlet dynamical zeta function for billiard flow corresponding to several strictly convex disjoint obstacles. For large we have and admits a meromorphic continuation to . We obtain some conditions of the frequencies and some sums of coefficients which imply that cannot be prolonged as entire function.
Paper Structure (4 sections, 9 theorems, 68 equations)

This paper contains 4 sections, 9 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Summation by typical means of $\eta_D$
  3. Proof of Theorem $1.1$
  4. Intervals with clustering frequencies

Key Result

Proposition 1.1

The function $\eta_\mathrm D(s)$ cannot be prolonged as an entire function of $s$ if and only if there exists $\alpha_0 > 0$ such that for any $\beta > \alpha_0$ we can find sequences $(\ell_j), (m_j)$ with $\ell_j \nearrow \infty$ as $j \to \infty$ such that for all $j \geqslant 0$ one has $e^{\bet

Theorems & Definitions (10)

  • Proposition 1.1
  • Theorem 1.1
  • Corollary 1.1
  • Theorem 2.1: Theorem 41, hardy1964
  • Proposition 2.1
  • Lemma 3.1
  • proof
  • Corollary 4.1
  • Corollary 4.2
  • Proposition 4.1