Linking Lyapunov exponents to low‑frequency spectral asymptotics

Derive rigorous mathematical relations connecting classical Lyapunov exponents to the low‑frequency asymptotics of spectral functions (and corresponding fidelity susceptibilities), thereby establishing how trajectory instability measures quantitatively determine long‑time observable fluctuations.

Background

The paper demonstrates numerically that, in mixed phase space, spectral functions averaged over chaotic trajectories exhibit non‑decaying low‑frequency tails (approximately 1/ω), while those averaged over regular trajectories decay as ω2. This empirical link mirrors the separation seen via Lyapunov exponents.

However, an explicit mathematical derivation relating Lyapunov exponents to the low‑frequency spectral behavior and fidelity susceptibility is currently missing, leaving a gap between trajectory‑level instability measures and ensemble‑averaged dynamical response.

References

Deriving precise mathematical connections between Lyapunov exponents and the low frequency asymptotes of spectral functions remains an unsolved problem.

— Defining classical and quantum chaos through adiabatic transformations  (2401.01927 - Lim et al., 2024) in Section 6.3 (Chaotic and Regular Regions of Phase Space)

Whether genuinely stronger low-frequency conditions impose dynamical constraints in higher dimensions remains open.

— Hyperuniform Delone Realizations and Rigidity  (2608.16547 - Björklund, 17 Aug 2026) in Section 1, Introduction (final paragraph)