The functional determinant of an eigenvalue sequence, as defined by zeta regularization, can be simply evaluated by quadratures. We apply this procedure to the Selberg trace formula for a compact Riemann surface to find a factorization of the Selberg zeta function into two functional determinants, respectively related to the Laplacian on the compact surface itself, and on the sphere. We also apply our formalism to various explicit eigenvalue sequences, reproducing in a simpler way classical results about the gamma function and the BarnesG-function. Concerning the latter, our method explains its connection to the Selberg zeta function and evaluates the related Glaisher-Kinkelin constantA.

https://projecteuclid.org/download/pdf_1/euclid.cmp/1104159315

Spectral functions, special functions and the Selberg zeta function

Comparison between formulae for the counting functions of the heights tn of the Riemann zeros and of semiclassical quantum eigenvalues En suggests that the tn are eigenvalues of an (unknown) hermitean operator H, obtained by quantizing a classical dynamical system with hamiltonian Hcl. Many features of Hcl are provided by the analogy; for example, the "Riemann dynamics" should be chaotic and have periodic orbits whose periods are multiples of logarithms of prime numbers. Statistics of the tn have a similar structure to those of the semiclassical En; in particular, they display random-matrix universality at short range, and nonuniversal behaviour over longer ranges. Very refined features of the statistics of the tn can be computed accurately from formulae with quantum analogues. The Riemann-Siegel formula for the zeta function is described in detail. Its interpretation as a relation between long and short periodic orbits gives further insights into the quantum spectral fluctuations. We speculate that the Riemann dynamics is related to the trajectories generated by the classical hamiltonian Hcl=XP.

/pdf/the-riemann-zeros-and-eigenvalue-asymptotics-29974mp4rc.pdf

The Riemann Zeros and Eigenvalue Asymptotics

Quantum systems with finite Hilbert space are considered, and phase-space methods like the Heisenberg–Weyl group, symplectic transformations and Wigner and Weyl functions are discussed. A factorization of such systems in terms of smaller subsystems, based on the Chinese remainder theorem, is studied. The general formalism is applied to the case of angular momentum. In this context, SU(2) coherent states are used for analytic representations. Links between the theory of finite quantum systems and other fields of research are discussed.

https://iopscience.iop.org/article/10.1088/0034-4885/67/3/R03/pdf

Quantum systems with finite Hilbert space

The way in which quantum eigenstates are influenced by the closed orbits of a chaotic classical system is analysed in phase space x ═ ( q, p ) through the spectral Wigner function W ( x ; E, ∊ ). This is a sum over Wigner functions of eigenstates within a range ∊ of energy E . In the classical limit, W is concentrated on the energy surface and smoothly distributed over it. Closed orbits provide oscillatory corrections (scars) for which explicit semiclassical formulae are calculated. Each scar is a fringe pattern decorating the orbit. As x moves off the energy surface the fringes form an Airy pattern with spacing of order h ⅔ . As x moves off the closed orbit the fringes form a complex gaussian with spacing h ½ .

Quantum scars of classical closed orbits in phase space

Bounded or driven classical systems often exhibit chaos (exponential instability that persists), but their quantum counterparts do not. Nevertheless, there are new regimes of quantum behaviour that emerge in the semiclassical limit and depend on whether the classical orbits are regular or chaotic, and this motivates the following definition. Definition . Quantum chaology is the study of semiclassical, but nonclassical, behaviour characteristic of systems whose classical motion exhibits chaos. This is illustrated by the statistics of energy levels. On scales comparable with the mean level spacing (of order h N for N freedoms), these fall into universality classes: for classically chaotic systems, the statistics are those of random matrices (real symmetric or complex hermitian, depending on the presence or absence of time-reversal symmetry); for classically regular ones, the statistics are Poisson. On larger scales (of order h , i. e. classically small but semiclassically large), universality breaks down. These phenomena are being explained by representing spectra in terms of classical closed orbits: universal spectral behaviour has its origin in very long orbits; non-universal behaviour depends only on short ones.

A. Voros

Papers

Chaos on the pseudosphere