Exact solution of the specific-heat-phonon spectrum inversion from the Mobius inverse formula

Periodic orbit spectrum in terms of Ruelle-Pollicott resonances

Fully chaotic Hamiltonian systems possess an infinite number of classical solutions which are periodic, e.g., a trajectory "p" returns to its initial conditions after some fixed time tau(p). Our aim is to investigate the spectrum [tau(1),tau(2), ...] of periods of the periodic orbits. An explicit formula for the density rho(tau)= Sigma(p)delta(tau-tau(p)) is derived in terms of the eigenvalues of the classical evolution operator. The density is naturally decomposed into a smooth part plus an interferent sum over oscillatory terms. The frequencies of the oscillatory terms are given by the imaginary part of the complex eigenvalues (Ruelle-Pollicott resonances). For large periods, corrections to the well-known exponential growth of the smooth part of the density are obtained. An alternative formula for rho(tau) in terms of the zeros and poles of the Ruelle zeta function is also discussed. The results are illustrated with the geodesic motion in billiards of constant negative curvature. Connections with the statistical properties of the corresponding quantum eigenvalues, random-matrix theory, and discrete maps are also considered. In particular, a random-matrix conjecture is proposed for the eigenvalues of the classical evolution operator of chaotic billiards.

Evaluation of terahertz density of vibrational states from specific-heat data: application to silica glass

The efficiency of the Tikhonov regulation method for the specific-heat phonon spectrum inversion problem is demonstrated. The regulation method is applied to the evaluation of the density of vibrational states from the low-temperature specific heat of a silica glass. The density of states obtained is found to be in good agreement with the results of inelastic neutron scattering. The extracted density of states is used for the analysis of the Raman light-vibration coupling coefficient. It is established that the coupling coefficient increases with frequency faster than approximately nu at nu<10 cm(-1). In the frequency range above 10 cm(-1) the coupling coefficient is found to be in agreement with the results of previous investigations.