The twilight zone in the parametric evolution of eigenstates: beyond perturbation theory and semiclassics

Considering a quantized chaotic system, we analyze the evolution of its eigenstates as a result of varying a control parameter. As the induced perturbation becomes larger, there is a crossover from a perturbative to a non-perturbative regime, which is reflected in the structural changes of the local density of states. For the first time the {\em full} scenario is explored for a physical system: an Aharonov-Bohm cylindrical billiard. As we vary the magnetic flux, we discover an intermediate twilight regime where perturbative and semiclassical features co-exist. This is in contrast with the {\em simple} crossover from a Lorentzian to a semicircle line-shape which is found in random-matrix models.Comment: 4 pages, 4 figures, improved versio

Contractions of low-dimensional nilpotent Jordan algebras

In this paper we classify the laws of three-dimensional and four-dimensional nilpotent Jordan algebras over the field of complex numbers. We describe the irreducible components of their algebraic varieties and extend contractions and deformations among them. In particular, we prove that J2 and J3 are irreducible and that J4 is the union of the Zariski closures of two rigid Jordan algebras.Comment: 12 pages, 3 figure

Several open problems in operator theory

We report on the meeting Operators in Banach spaces recently held in Castro Urdiales as a homage to Pietro Aiena, and we collect the questions proposed by the participants during the Open Problems Session.Ministerio de Ciencia e Innovación (MICINN) de España. Grant MTM2011-2653 y MTM2010-20190.peerReviewe

Parametric invariant Random Matrix Model and the emergence of multifractality

We propose a random matrix modeling for the parametric evolution of eigenstates. The model is inspired by a large class of quantized chaotic systems. Its unique feature is having parametric invariance while still possessing the non-perturbative crossover that has been discussed by Wigner 50 years ago. Of particular interest is the emergence of an additional crossover to multifractality.Comment: 7 pages, 6 figures, expanded versio

Mesoscopic non-equilibrium thermodynamics approach to non-Debye dielectric relaxation

Mesoscopic non-equilibrium thermodynamics is used to formulate a model describing non-homogeneous and non-Debye dielectric relaxation. The model is presented in terms of a Fokker-Planck equation for the probability distribution of non-interacting polar molecules in contact with a heat bath and in the presence of an external time-dependent electric field. Memory effects are introduced in the Fokker-Planck description through integral relations containing memory kernels, which in turn are used to establish a connection with fractional Fokker-Planck descriptions. The model is developed in terms of the evolution equations for the first two moments of the distribution function. These equations are solved by following a perturbative method from which the expressions for the complex susceptibilities are obtained as a functions of the frequency and the wave number. Different memory kernels are considered and used to compare with experiments of dielectric relaxation in glassy systems. For the case of Cole-Cole relaxation, we infer the distribution of relaxation times and its relation with an effective distribution of dipolar moments that can be attributed to different segmental motions of the polymer chains in a melt.Comment: 33 pages, 6 figure