For several decades now there has been interest amongst physicists and mathematicians to study the statistics of the eigenvalue spectra of physical systems - especially those with a classical counterpart which displays chaotic behaviour. One particular goal is to try to detect in the statistical distribution of a set of energy eigenvalues an indication of whether the corresponding classical system behaves chaoticly or integrably. The answer was suggested by Berry and Tabor in 1977, and examined in depth by Bohigas, Giannoni and Schmit [ MR 85f:58034], and reveals that while classically integrable (non-chaotic) systems have a spectrum of uncorrelated eigenvalues, the spectra of classically chaotic systems show the characteristic correlations seen in the spectra of ensembles of random matrices. In particular, the semiclassical spectra of physical systems which are not symmetric under time-reversal have the same local statistics (those of the ensemble of unitary matrices, , with Haar measure) as the zeros high on the critical line of the Riemann zeta function and other -functions.
Thus for a given physical system which possesses the correct
symmetries, we can construct a zeta function which will have zeros
correlated on a local scale like those of the Riemann zeta
function. For a system with a Hamiltonian operator and a set
of eigenvalues , satisfying
A semiclassical expression for the spectral determinant can be
given using only classical attributes of the system in question.
Following Berry and Keating [
MR 92m:81053], we can write
We can now define the dynamical zeta function
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