Completely integrable systems: a generalization

We present a slight generalization of the notion of completely integrable systems to get them being integrable by quadratures. We use this generalization to integrate dynamical systems on double Lie groups.Comment: Latex, 15 page

A class of commutative dynamics of open quantum systems

We analyze a class of dynamics of open quantum systems which is governed by the dynamical map mutually commuting at different times. Such evolution may be effectively described via spectral analysis of the corresponding time dependent generators. We consider both Markovian and non-Markovian cases.Comment: 22 page

Stochastic evolution of finite level systems: classical vs. quantum

Quantum dynamics of the density operator in the framework of a single probability vector is analyzed. In this framework quantum states define a proper convex quantum subset in an appropriate simplex. It is showed that the corresponding dynamical map preserving a quantum subset needs not be stochastic contrary to the classical evolution which preserves the entire simplex. Therefore, violation of stochasticity witnesses quantumness of evolution.Comment: 8 page

On pseudo-stochastic matrices and pseudo-positive maps

Stochastic matrices and positive maps in matrix algebras proved to be very important tools for analysing classical and quantum systems. In particular they represent a natural set of transformations for classical and quantum states, respectively. Here we introduce the notion of pseudo-stochastic matrices and consider their semigroup property. Unlike stochastic matrices, pseudo-stochastic matrices are permitted to have matrix elements which are negative while respecting the requirement that the sum of the elements of each column is one. They also allow for convex combinations, and carry a Lie group structure which permits the introduction of Lie algebra generators. The quantum analog of a pseudo-stochastic matrix exists and is called a pseudo-positive map. They have the property of transforming a subset of quantum states (characterized by maximal purity or minimal von Neumann entropy requirements) into quantum states. Examples of qubit dynamics connected with "diamond" sets of stochastic matrices and pseudo-positive maps are dealt with.Comment: 15 pages; revised versio

Hamilton-Jacobi Theory and Information Geometry

Recently, a method to dynamically define a divergence function $D$ for a given statistical manifold $(\mathcal{M}\,,g\,,T)$ by means of the Hamilton-Jacobi theory associated with a suitable Lagrangian function $\mathfrak{L}$ on $T\mathcal{M}$ has been proposed. Here we will review this construction and lay the basis for an inverse problem where we assume the divergence function $D$ to be known and we look for a Lagrangian function $\mathfrak{L}$ for which $D$ is a complete solution of the associated Hamilton-Jacobi theory. To apply these ideas to quantum systems, we have to replace probability distributions with probability amplitudes.Comment: 8 page

Non-symplectic symmetries and bi-Hamiltonian structures of the rational Harmonic Oscillator

The existence of bi-Hamiltonian structures for the rational Harmonic Oscillator (non-central harmonic oscillator with rational ratio of frequencies) is analyzed by making use of the geometric theory of symmetries. We prove that these additional structures are a consequence of the existence of dynamical symmetries of non-symplectic (non-canonical) type. The associated recursion operators are also obtained.Comment: 10 pages, submitted to J. Phys. A:Math. Ge

Towards a definition of quantum integrability

We briefly review the most relevant aspects of complete integrability for classical systems and identify those aspects which should be present in a definition of quantum integrability. We show that a naive extension of classical concepts to the quantum framework would not work because all infinite dimensional Hilbert spaces are unitarily isomorphic and, as a consequence, it would not be easy to define degrees of freedom. We argue that a geometrical formulation of quantum mechanics might provide a way out.Comment: 37 pages, AmsLatex, 1 figur

The WINGS Survey: a progress report

A two-band (B and V) wide-field imaging survey of a complete, all-sky X-ray selected sample of 78 clusters in the redshift range z=0.04-0.07 is presented. The aim of this survey is to provide the astronomical community with a complete set of homogeneous, CCD-based surface photometry and morphological data of nearby cluster galaxies located within 1.5 Mpc from the cluster center. The data collection has been completed in seven observing runs at the INT and ESO-2.2m telescopes. For each cluster, photometric data of about 2500 galaxies (down to V~23) and detailed morphological information of about 600 galaxies (down to V~21) are obtained by using specially designed automatic tools. As a natural follow up of the photometric survey, we also illustrate a long term spectroscopic program we are carrying out with the WHT-WYFFOS and AAT-2dF multifiber spectrographs. Star formation rates and histories, as well as metallicity estimates will be derived for about 350 galaxies per cluster from the line indices and equivalent widths measurements, allowing us to explore the link between the spectral properties and the morphological evolution in high- to low-density environments, and across a wide range in cluster X-ray luminosities and optical properties.Comment: 12 pages, 10 eps figures, Proceedings of the SAIt Conference 200