1997/04/01 by E. Brézin, S. Hikami · 7 citations
Chemistry · Mathematics · Physics and Astronomy · #Molecular spectroscopy and chirality #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics
paper · doi:10.1103/physreve.55.4067
openalex publication_date 1997/04/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/31
In the theory of disordered systems the spectral form factor S(\ensuremathτ), the Fourier transform of the two-level correlation function with respect to the difference of energies, is linear for \ensuremathτ\mathrm\ensuremathτc and constant for \ensuremathτ>\mathrm\ensuremathτc. Near zero and near \mathrm\ensuremathτc it exhibits oscillations which have been discussed in several recent papers. In problems of mesoscopic fluctuations and quantum chaos a comparison is often made with a random matrix theory. It turns out that, even in the simplest Gaussian unitary ensemble, these oscillations have not yet been studied there. For random matrices, the two-level correlation function \ensuremathρ(\ensuremathλ1,\ensuremathλ2) exhibits several well-known universal properties in the large-N limit. Its Fourier transform is linear as a consequence of the short-distance universality of \ensuremathρ(\ensuremathλ1,\ensuremathλ2). However the crossover near zero and \mathrm\ensuremathτc requires one to study these correlations for finite N. For this purpose we use an exact contour-integral representation of the two-level correlation function which allows us to characterize these crossover oscillatory properties. This representation is then extended to the case in which the Hamiltonian is the sum of a deterministic part H0 and of a Gaussian random potential V. Finally, we consider the extension to the time-dependent case.