S. De Filippo

A new characterization of completely integrable systems

SummaryA characterization of separability, projectability and integrability of dynamical systems in terms of the spectral properties of invariant mixed tensor fields with vanishing Nijenhuis tensor is given. In addition, some preliminary results on the inverse problem (from Liouville integrability to Lax representation) are illustrated.RiasuntoSi presenta una caratterizzazione della separabilità, proiettabilità ed integrabilità di sistemi dinamici in termini delle proprietà spettrali di campi di tensori misti invarianti con tensore di Nijenhuis nullo. Sono inoltre illustrati alcuni risultati preliminari sul problema inverso (dall’integrabilità à la Liouville alla rappresentazione di Lax).РезюмеПредлагается характеристика разделимости, проектируемости и интегрируемости динамических систем в терминах спектральных свойств инвариантных смешанных полей, с нулевым тензором Нидженуиса. Кроме того, обсуждаются некоторые предварительные результаты, касающиеся обратной проблемы (от интегрируемости Лиувилля к представлению Лакса).

Lyapunov exponents for the n =3 discrete self-trapping equation

Abstract Lyapunov characteristic exponents for the n = 3 discrete self-trapping equation are numerically computed and the connection between the vanishing ones and the existence of independent constants of motion is exhibited. The results obtained by this approach are in agreement with previous ones about the long-term behaviour of the system.

Numerical evidence of a sharp order window

Abstract A detailed numerical analysis for a small range of the nonlinearity parameter exhibits the existence of a sharp order window for the n = 3 discrete self-trapping equation. The analysis performed by using maximum Lyapunov characteristic exponent, power spectra, Poincare maps and correlation exponents gives a clear-cut evidence of a biperiodic dynamics on bidimensional tori.

Integrable systems on a sphere as models for quantum dots

Model potentials for quantum dots with smooth boundaries, realistic in the whole range of energies, are introduced, starting from the integrable motion of a particle on a sphere under the action of an external quadratic field. We show that in the case of rotational invariant potentials, the associated 2D Schrodinger equation has exact zero-energy eigenfunctions, in terms of which the whole discrete spectrum can be characterized.

An algebraic description of the electron—monopole dynamics

Abstract An algebraic formulation of lagrangian dynamics, suitable to be used, in a general context including situations with no global lagrangian and/or fermionic variables, is presented. As an example, an algebraic description of the electron—monopole dynamics is given.

Numerical simulation of non-unitary gravity-induced localization

The localization of a quantum state is numerically exhibited in a non-unitary Newtonian model for gravity. It is shown that an unlocalized state of a ball of mass just above the expected threshold of 1011 proton masses evolves into a mixed state with vanishing coherences above some localization lengths.

A bosonization procedure for Hamiltonian theories with fermions

Hamiltonian theories with fermions are proved to be equivalent to hierarchies of ordinary Hamiltonian theories. The corresponding Poisson brackets are defined in terms of the original super-Poisson structure, while Hamiltonian functions are simply the coefficients in the expansion of the super-Hamiltonian function as a formal power series in Grassman generators. Fermion extensions of the KdV equation are considered to illustrate the general result; its space-supersymmetric extensions are used to show in particular how supersymmetry transformations can be recast as ordinary Hamiltonian symmetries.

An alternative field-theoretic setting for anyon systems

The cyon model for anyon systems is explicitly analysed using the quantum electrodynamics of particles carrying anomalous magnetic-dipole moments in (2+1) dimensions. Statistical interaction appears together with a further interaction term whose cut-off dependence reflects the unrenormalizable character of the theory.

Semiclassical analysis of the eigenstate Wigner functions for the discrete self-trapping equation

Abstract An explicit expression is given for the exact eigenstate Wigner functions of a whole class of non-integrable quantum systems with m degrees of freedom. Their evaluation only needs finite matrix diagonalisation, without any truncation error. The application of this expression is illustrated for the m = 2 integrable case, which reproduces the expected semiclassical behaviour.

On a geometrical procedure to construct approximating maps for ordinary conservative systems

An approximating procedure for geodesic flows on Riemannian manifolds is proposed in terms of approximating singular manifolds which are locally flat. The corresponding dynamical system is then reduced to a discrete map. The reliability of the approximation is tested by evaluating the maximum Lyapunov characteristic exponent for the flow on a two-handle sphere and showing the extremely smooth tendency to the singular case, which in this case corresponds to the Sinai billiard.