Titles and Abstracts

A. Giuliani(Università di Roma Tre)

Universal conductivity for the Hubbard's model on hexagonal lattice (Joint work with V. Mastropietro and M. Port)

Graphene attracts great attention throughout the community of solid state, both from experimental and theoretical points of view. A still controversial issue is the conductivity: there is ample experimental evidence that even for a semi-full band, which means zero density for charge carriers, graphene shows a minimal conductivity different from zero. The value of this conductivity was calculated theoretically in terms of an effective theory of Dirac particles in 2 +1 dimensions. However, such a theory is affected by an ultraviolet divergence which makes the results ambiguous and highly dependent from the choice of the regularization. Therefore, the debate on the value of conductivity in the presence of interactions is still open: is it universal or not? In this seminar I will discuss a recent result obtained in collaboration with V. Mastropietro and M. Port. We show the universality of the minimal conductivity for the Hubbard model on a 2D hexagonal lattice with semi-full band. For sufficiently small values of the interaction U, we find that the minimal conductivity is independent of U.

N. Visciglia (Università degli Studi di Pisa)

On the construction of invariant measures for KdV and Benjamin-Ono equations

We present a joint work with N. Tzvetkov devoted to the construction of invariant measures for the Benjamin-Ono equation. First we present shortly the approach of Zhidkov to prove existence of invariant measures for the KdV equations. We show the main difficulties that this method meets in the case of the Benjamin-Ono equation and we show the main strategy we use to overcome those difficulties.

M.Correggi (Università degli Studi di Trento)

On the stability of quantum many-body systems with point interactions (Joint work with G. Dell'Antonio e A. Michelangeli)

We first review the question of stability for quantum systems of particles interacting through zero-range potentials: We start by discussing the rigorous definition of the Hamiltonian and then study its spectral properties and in particular its boundedness from below. In the second part of the talk we focus on a special system given by N fermions interacting with a test particle of mass m and show very recent results about the dependence of the stability on the parameters N and m.

A. Zampini (Max Planck Institut für Mathematik, Bonn)

Laplacians on Hopf fibrations with quantum groups symmetries (Joint work with G. Landi)

Given the Hopf U(1)-fibration on the standard Podles' sphere, I shall describe in this talk the main results of a joint research project with G.Landi, concerning the possibility to introduce suitable Laplacians, to couple them with gauge connections, and to describe their spectra.

A. Mantile (Université du Sud, Toulon-Var)

Adiabatic evolution of 1D shape resonances: an artificial interface conditions approach (Joint work with A. Faraj and F. Nier)

In connection with the transport problem through resonant heterostructures, we reconsider the adiabatic evolution of resonant states. Our approach consists in introducing, in addition to the complex deformation, a further modification formed by artificial interface conditions. This allows to obtain a family of maximal accretives generators and to develop the adiabatic theory for the modified system. At the same time, we show that this modification produces small perturbations of the relevant spectral quantities concerned with the non-linear modelling. A first application is given for a time dependent explicitly solvable model. In this simplified framework, we obtain a reduced equation for the adiabatic evolution of the sheet density of charges accumulating in a quantum well.

D. Noja (Università di Milano Bicocca)

Nonlinear Schrödinger equations on star graphs (Joint work with R. Adami, C. Cacciapuoti and D. Finco)

We define the Schrödinger equation with focusing, cubic nonlinearity on one-vertex graphs. We prove global well-posedness in the energy domain and conservation laws for some self-adjoint boundary conditions at the vertex, i.e. Kirchhoff boundary condition and the so called δ and &delta′ boundary conditions. Moreover, in the same setting we study the collision of a fast solitary wave with the vertex and we show that it splits in reflected and transmitted components. The outgoing waves preserve a soliton character over a time which depends on the logarithm of the velocity of the ingoing solitary wave. Over the same timescale the reflection and transmission coefficients of the outgoing waves coincide with the corresponding coefficients of the linear problem. In the analysis of the problem we follow ideas borrowed from the seminal paper about scattering of fast solitons by a delta interaction on the line, by Holmer, Marzuola and Zworski; the present paper represents an extension of their work to the case of graphs and, as a byproduct, it shows how to extend the analysis of soliton scattering by other point interactions on the line, interpreted as a degenerate graph.

M. Pulvirenti (Università di Roma 1 Sapienza)

Strong semiclassical approximation of the Wigner functions for the Hartree dynamics (Joint work with A. Athanassoulis, T. Paul and F. Pezzotti)

We consider a quantum nonlinear self-consistent dynamics in a smooth setting (for instance for a smooth mixture of coherent states) and show that the time-evolved Wigner functions converge, in the L2 sense, to the corresponding solution of the Vlasov equation, when the Planck constant goes to zero. In contrast with previous results formulated in terms of compactness and weak convergence arguments, we improve (loosing generality) the convergence and give explicit estimates of the error. In the proof we crucially make use of the Husimi transform as a natural bridge between the Wigner function formalism and the classical picture in the phase space.

S. Teufel (Universitët Tübingen)

Effective Hamiltonians for constrained quantum systems

We consider the Laplace-Beltrami operator on a Riemannian manifold A with a potential that localizes certain states close to a submanifold C. When scaling the potential in the directions normal to C by a parameter ε<<1 the solutions concentrate in an ε-neighborhood of C. This situation occurs for example in quantum wave guides and for the motion of nuclei in electronic potential surfaces in quantum molecular dynamics. We derive an effective Schroedinger operator on the submanifold C that approximates the spectrum and the dynamics of the full operator on A up to terms of order ε2. In the talk I will focus on new results concerning Dirichlet-tubes, i.e. the potential can only take the values 0 and ∞.

G. De nittis (PhD Candidate at SISSA)

Hunting colored (quantum) butterflies: a geometric derivation of the TKNN-equations

This talk will constitute the PhD defense of the speaker

E. Costa (PhD Candidate at SISSA)

The Schroedinger equation on thin Dirichlet waveguides: the quantum graph approximation

This talk will constitute the PhD defense of the speaker