We introduce a criterion for the existence of regular states in systems with a mixed phase space. If this condition is not fulfilled chaotic eigenstates substantially extend into a regular island. Wave packets started in the chaotic sea progressively flood the island. The extent of flooding by eigenstates and wave packets increases logarithmically with the size of the chaotic sea and the time, respectively. This new effect is observed for the example of island chains with just 10 islands.PACS numbers: 05.45. Mt, 03.65.Sq One of the cornerstones in the understanding of the structure of eigenstates in quantum systems is the semiclassical eigenfunction hypothesis [1]: in the semiclassical limit the eigenstates concentrate on those regions in phase space which a typical orbit explores in the longtime limit. For integrable systems these are the invariant tori. For ergodic dynamics the eigenstates become equidistributed on the energy shell [2]. Typical systems have a mixed phase space, where regular islands and chaotic regions coexist. In this case the semiclassical eigenfunction hypothesis implies that the eigenstates can be classified as being either regular or chaotic according to the phase-space region on which they concentrate. Note, that this may fail for an infinite phase space [3].In this paper we study mixed systems with a compact phase space, but away from the semiclassical limit. Here the properties of eigenstates depend on the size of phasespace structures compared to Planck's constant h. In the case of 2D maps this can be very simply stated [4]: a regular state with quantum number m = 0, 1, ... will concentrate on a torus enclosing an area (m + 1/2)h, as can be seen in Fig. 1(c).We will show that this WKB-type quantization rule is not a sufficient condition. We find a second criterion for the existence of a regular state on the m-th quantized torus,Here τ H = h/∆ ch is the Heisenberg time of the chaotic sea with mean level spacing ∆ ch and γ m is the decay rate of the regular state m if the chaotic sea were infinite. Quantized tori violating this condition will not support regular states. Instead, chaotic states will flood these regions, see Fig. 1(a). In terms of dynamics we find that wave packets started in the chaotic sea progressively flood the island as time evolves. Partial and even complete flooding is possible, depending on system properties. These findings are relevant for islands surrounded by a large chaotic sea. We numerically demonstrate the flooding and the disappearance of regular states for the important case of island chains. In typical Hamiltonian systems they appear around any regular island. On larger scales they are relevant for Hamiltonian ratchets [5], the kicked rotor with accelerator modes [6], and the experimentally [7,8,9] and theoretically [10] studied kicked atom systems. The flooding of regular islands by chaotic states is a new quantum signature of a classically mixed phase space. This phenomenon shows that not only local phasespace structures, but also global properties ...

It has been conjectured that the statistical properties of zeros of the Riemann zeta function near z = 1/2 + iE tend, as E → ∞, to the distribution of eigenvalues of large random matrices from the Unitary Ensemble. At finite E numerical results show that the nearest-neighbour spacing distribution presents deviations with respect to the conjectured asymptotic form. We give here arguments indicating that to leading order these deviations are the same as those of unitary random matrices of finite dimension N eff = log(E/2π)/ √ 12Λ, where Λ = 1.57314 . . . is a well defined constant.

We investigate the probability distribution of the quantum fluctuations of thermodynamic functions of finite, ballistic, phase-coherent Fermi gases. Depending on the chaotic or integrable nature of the underlying classical dynamics, on the thermodynamic function considered, and on temperature, we find that the probability distributions are dominated either (i) by the local fluctuations of the single-particle spectrum on the scale of the mean level spacing, or (ii) by the long-range modulations of that spectrum produced by the short periodic orbits. In case (i) the probability distributions are computed using the appropriate local universality class, uncorrelated levels for integrable systems, and random matrix theory for chaotic ones. In case (ii) all the moments of the distributions can be explicitly computed in terms of periodic orbit theory and are system-dependent, nonuniversal, functions. The dependence on temperature and on number of particles of the fluctuations is explicitly computed in all cases, and the different relevant energy scales are displayed. C 2002 Elsevier Science (USA)

We investigate the structure of eigenstates in systems with a mixed phase space in terms of their projection onto individual regular tori. Depending on dynamical tunneling rates and the Heisenberg time, regular states disappear and chaotic states flood the regular tori. For a quantitative understanding we introduce a random matrix model. The resulting statistical properties of eigenstates as a function of an effective coupling strength are in very good agreement with numerical results for a kicked system. We discuss the implications of these results for the applicability of the semiclassical eigenfunction hypothesis.Comment: 11 pages, 12 figure

Semiclassical theories such as the Thomas-Fermi and Wigner-Kirkwood methods give a good description of the smooth average part of the total energy of a Fermi gas in some external potential when the chemical potential is varied. However, in systems with a fixed number of particles N , these methods overbind the actual average of the quantum energy as N is varied. We describe a theory that accounts for this effect. Numerical illustrations are discussed for fermions trapped in a harmonic oscillator potential and in a hard-wall cavity, and for self-consistent calculations of atomic nuclei. In the latter case, the influence of deformations on the average behavior of the energy is also considered.