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February 3 |
Luca Oddis
(University of Rome La Sapienza) |
Two anyon Schroedinger operators
Anyons are quantum particles with statistics intermediate between bosons and fermions, arising in less than 3 dimensions. There are two possible approaches: one uses multi-valued wave functions, which means topologically non trivial configuration spaces, the other one allows to work in a flat, topologically trivial space, but the non interacting Hamiltonian in this case contains a singular Aharonov-Bohm - type potential. Our analysis lies in the latter approach. The Aharonov-Bohm potential has a singular behaviour at the overlapping configurations. Thus, it is not obvious how to rigorously realize the corresponding Schroedinger operator(s). We mainly focuses on the case of two anyons, which reduces to a one-body problem. The self-adjointness question is investigated and solved via quadratic form techniques. The Dirichlet forms are defined both on the magnetic Gauge and on an appropriate weighted Sobolev space. We also derive the extensions as suitable limits of scaled regular potentials. We present some perspectives for the many-body problem. |
November 7-8 |
Meccanica quantistica e dintorni |
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April 9 |
M. Erbar (University of Bonn) |
A variational characterization of the Sine-\(\beta\) point process
I will discuss the scattering problem for a quantum particle in dimension three in the presence of a semitransparent unbounded obstacle, modeled by a surface. The generator of the dynamics is the operator formally defined as the Laplacian plus a delta-interaction supported by the surface. I will consider the case in which the surface is obtained through a local deformation of a plane, it can be identified by the graph of a compactly supported, Lipschitz continuous function. In this configuration, the reference dynamics is the one generated by the Laplacian plus a delta-interaction supported by the plane. I will discuss existence and asymptotic completeness of the wave operators, provide a representation formula for the scattering matrix, and show that the scattering matrix converges to the identity as the deformation goes to zero (with a quantitative estimate on the rate of convergence). |
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February 27 |
Claudio Cacciapuoti (Università degli Studi dell'Insubria) |
Scattering from local deformations of a semitransparent plane
I will discuss the scattering problem for a quantum particle in dimension three in the presence of a semitransparent unbounded obstacle, modeled by a surface. The generator of the dynamics is the operator formally defined as the Laplacian plus a delta-interaction supported by the surface. I will consider the case in which the surface is obtained through a local deformation of a plane, it can be identified by the graph of a compactly supported, Lipschitz continuous function. In this configuration, the reference dynamics is the one generated by the Laplacian plus a delta-interaction supported by the plane. I will discuss existence and asymptotic completeness of the wave operators, provide a representation formula for the scattering matrix, and show that the scattering matrix converges to the identity as the deformation goes to zero (with a quantitative estimate on the rate of convergence). The talk is based on the joint paper with Davide Fermi and Andrea Posilicano: J. Math. Anal. Appl. 473 (2019) 215–257. |
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January 9 |
Tong Yang (City University of Hong Kong, China) |
Global well-posedness of the Non-cutoff Boltzmann Equation with Polynomial Decay Perturbations
The Boltzmann equation without angular cutoff is considered when the initial data is a perturbation of a global Maxwellian with algebraic decay in the velocity variable. Global solution is proved by combining the analysis in moment propagation, spectrum of the linearized operator and the smoothing effect of the linearized operator when initial data in Sobolev space with negative index. This is a joint work with Ricardo Alonso, Yoshinori Morimoto and Weiran Sun. |
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November 21 |
Mathieu Lewin (CNRS & Université Paris Dauphine, France) |
Nonlinear Gibbs measures and their derivation from many-body quantum mechanics
In this talk I will define and discuss some probability measures in infinite dimensions, which play an important role in (S)PDE, in Quantum Field Theory and for Bose-Einstein condensates. Those are Gibbs measures associated with the Gross-Pitaevskii and Hartree energies. In dimensions larger than or equal to 2, the measures are concentrated on distribution spaces, and the nonlinear term has to be renormalized. I will then present some recent results in collaboration with Phan Thanh Nam and Nicolas Rougerie about the derivation of these measures from many-body quantum mechanics in a mean-field type limit. |
November 14 |
Massimo Moscolari ("Sapienza" Universitá di Roma) |
Beyond Diophantine Wannier diagrams: gap labelling for Bloch-Landau Hamiltonians
In 1978 Wannier discovered a Diophantine relation expressing the integrated density of states of a gapped group of bands of the Hofstadter Hamiltonian as a linear function of the magnetic field flux with integer slope. I will show how to extend this relation to a gap labelling theorem for any 2D Bloch-Landau Hamiltonian operator and to certain non-covariant systems having slowly varying magnetic fields. The integer slope will be interpreted as the Chern character of the projection onto the space of occupied states. This result will be seen in the perspective of a non periodic generalization of the localization dichotomy for gapped quantum systems, which in the periodic case has been proved in 2016 by Monaco, Panati, Pisante and Teufel. The talk is based on a joint work with H. Cornean and D. Monaco. |
October 24 |
Søren Fournais (Aarhus Universitet) |
A simple 2nd order lower bound to the energy of dilute Bose gases
We consider a system of non-relativistic bosons interacting through a regular, positive potential \( v \) with scattering length \( a \). We give a simple proof that the ground state energy density satisfies the bound \( e(\rho) \geq 4\pi a \rho^2 (1- C \sqrt{\rho a^3}) \). This talk is based on joint work with Birger Brietzke and Jan Philip Solovej. |
October 8–17 |
Søren Fournais (Aarhus Universitet) |
On upper bounds to the energy of the dilute Bose gas (course)
In this course we will review recent work on the different upper bounds for the ground state energy of a gas of interacting bosons in the thermodynamic limit. We start by considering quasi-free states to realize that these reproduce the leading order asymptotics of the energy but not the expected first correction term. This part is based on work by Erdos-Schlein-Yau and Napiorkowski-Reuvers-Solovej. Then we pass to the more involved trial states by Yau-Yin and prove that such a state can indeed give the expected correction term. The course will essentially be elementary. Only a basic knowledge of mathematical quantum mechanics will be needed. |
September 19 |
Zied Ammari (Université de Rennes 1, France) |
Existence and uniqueness of solutions for the Hartree and
Gross-Pitaevskii hierarchy equations
The Gross-Pitaevskii and Hartree hierarchies are infinite systems of coupled PDEs derived from the mean field theory of Bose gases. Due to their physical and mathematical relevance, the issues of well-posedness and uniqueness for these equations have recently been studied thoroughly using specific nonlinear and combinatorial techniques. In this talk, I will introduce a new approach based on a duality between these hierarchies and some Liouville's equations. And I will explain how such point of view yields several new results. Joint work with Quentin Liard and Clément Rouffort. |
June 27 |
Daniele Dimonte (SISSA, Trieste) |
Effective Dynamics of Bose-Einstein Condensates in the Thomas-Fermi Limit
We discuss the time-dependent Gross-Pitaevskii (GP) approximation for interacting bosons in the Thomas-Fermi (TF) regime, i.e., for a mean-field interaction potential with shrinking support and whose scattering length diverges in the large N limit (but the gas remains dilute). Under these assumptions, the many-body Schrödinger dynamics is expected to be approximated by a one-particle nonlinear Schrödinger equation: the reliability of this approximation is the problem we plan to investigate. While the mean-field or GP limits have already been studied in detail, and the approximation proven to be correct on suitable time scales, the TF limit has never been studied, in spite of its large relevance in experimental physics. We show that the question of the dynamic approximation is more subtle in the TF regime, in particular at large time scales. The results are proven by exploiting the method developed by P. Pickl. Joint work in progress with M. Correggi, D. Mitrouskas and P. Pickl. |
June 27 |
Alessandro Olgiati (SISSA, Trieste) |
Effective description of mixtures of condensates
I will present rigorous results on the effective properties of many-body systems consisting of multiple bosonic species. This will be based on joint works with A. Michelangeli and P.T. Nam. Concerning stationary properties, we were able to prove that, both in the mean field and Gross-Pitaevskii regime, the leading order of the ground state energy is captured by the minimum of a suitable effective functional. Moreover, in the ground state, all species of the mixture exhibit condensation. For mixtures in the mean field regime, we are also able to justify Bogoliubov theory, hence computing the next-to-leading correction to the ground state energy, and proving a norm approximation for the ground state. The above properties justify crucial assumptions for the theorems on the dynamical evolution that I will present. These are results of persistence of condensation for mixtures, which I will state and discuss both in the mean-eld and Gross-Pitaevskii regime. |
June 27 |
Raffaele Scandone (SISSA, Trieste) |
On Schrödinger Operators with Point Interactions
A central topic in mathematical physics is the investigation of quantum systems subject to very short range potentials, virtually supported on a nite set of points. In this talk, after a preliminary recall of the denition and the main features of Schrödinger operators with point interactions, I will discuss some recent results on their scattering and smoothing properties. I will also discuss the characterization of the Sobolev norms induced by such operators. Based on joint works with G. Dell'Antonio, V. Georgiev, F. Iandoli, A. Michelangeli, and K.Yajima. |
June 27 |
Luca Oddis ("Sapienza" Universitá di Roma) |
Quadratic Forms for Two-Anyon Systems
We review the main issues concerning the well-posedness (as suitable self-adjoint operators) of the Hamiltonians of two non interacting anyons, i.e., exotic particles obeying to fractional statistics in two dimensions. We show that such operators can be identified with a one-parameter family of self-adjoint extensions of a suitable symmetric operator with Aharonov-Bohm-like magnetic potential. We also derive the explicit expressions of the corresponding quadratic forms and prove their closure and boundedness from below. Joint work in progress with M. Correggi. |
May 9 |
Lorenzo Tentarelli ("Sapienza" Universitá di Roma) |
De Giorgi's approach to hyperbolic Cauchy problems
We discuss an extension of some results, obtained by E. Serra and P. Tilli ('12,'16), concerning an original conjecture by E. De Giorgi ('96) on a purely minimization approach to the Cauchy problem for the defocusing nonlinear wave equation. Precisely, we show how to extend the techniques developed by Serra and Tilli to the case of nonhomogeneous second order hyperbolic equations (possibly in presence of dissipative terms). This is a joint work with P. Tilli. |
April 18 |
Fabio Toninelli (Université Claude Bernard Lyon 1, France) |
Lozenge tiling dynamics and hydrodynamic equation
We study a reversible continuous-time Markov dynamics on lozenge tilings of the torus, introduced by Luby et al. Single updates consist in concatenations of \( n \) elementary lozenge rotations at adjacent vertices, with rate \( 1/n \). The dynamics can also be seen as a reversible stochastic evolution of a 2+1-dimensional interface. The dynamics is known to enjoy especially nice features: a certain Hamming distance between configurations contracts with time on average and the relaxation time of the Markov chain is diffusive. We present another remarkable feature of this dynamics, namely we derive, in the diffusive time scale, a fully explicit hydrodynamic limit equation for the height function (in the form of a non-linear parabolic PDE). The mobility coefficient \( \mu \) in the equation has non-trivial but explicit dependence on the interface slope and, interestingly, is directly related to the system’s surface free energy. The derivation of the hydrodynamic limit is based on the so-called \( H^{-1} \) method due to Yau and Funaki-Spohn. Based on joint work with Benoit Laslier (Paris 7). |
April 11 |
Vojkan Jaksic (McGill University, Canada) |
Time and Entropy
This talk concerns mathematical theory of the so-called Fluctuation Relation (FR) and Fluctuation Theorem (FT) in context of dynamical systems relevant to physics. The FR refers to a certain universal identity linked to statistics of entropy production generated by a reversal operation and FT to the related mathematical large deviations result. The discovery of FR goes back to numerical experiments and Evans, Cohen and Morris (1993) and theoretical works of Evans and Searles (1994), Gallavotti and Cohen (1995). These discoveries generated an enormous body of numerical, theoretical and experimental works which have fundamentally altered our understanding of non-equilibrium physics, with applications extending to chemistry and biology. In this talk I will introduce modern theory of FR and FT on an example and comment on a current research program on this topic. |
March 7 |
Alessandra Occelli (Universität Bonn, Germany) |
On time correlations for last passage percolation models
We study time correlations of last passage percolation (LPP), a model in the Kardar-Parisi-Zhang universality class, with three different geometries: step, flat and stationary. We prove the convergence of the covariances of the LPP at two different times to a limiting expression given in terms of Airy processes. Furthermore, we prove the behaviour of the covariances when the two times are close to each other, conjectured in a work of Ferrari and Spohn. |
February 27 |
Eric Cances (Ecole des Ponts ParisTech & INRIA, France) |
Computational non-commutative geometry for materials science: The example of multilayer 2D materials
Descrizione:Computational non-commutative geometry for materials science: The example of multilayer 2D materials After recalling the standard mathematical formalism used to model disordered systems such as random composite materials (mesoscale disorder), doped semiconductors, alloys, or amorphous materials (atomic-scale disorder), I will present a tight-binding model for computing the electrical conductivity of incommensurate multilayer 2D materials. All these models fall into the scope of the mathematical framework, based on non-commutative geometry, introduced by Bellissard to study the physical properties of aperiodic systems. Surprisingly, this rather abstract theoretical framework leads to completely new numerical schemes allowing one to perform simulations out of the scope of usual methods. |
December 20 |
Andrea Mantile (Universitè de Reims Champagne-Ardenne, France) |
On the simultaneous identification of scattering parameters for classical waves
We prove uniqueness in inverse acoustic scattering in the case the density of the medium has an unbounded gradient across \( \Sigma \subset \Omega \), where \( \Omega \) is a 3D-Lipschitz domain. The corresponding direct problem is related to the stationary waves scattering for 3D Schrödinger operators with \( \delta\)-type singular perturbations supported on \(\partial\Omega\) and of strength \(\alpha \in L^p(\partial\Omega), p > 2\). This is a multiple scattering problem from obstacles and potentials whose solutions depend on the obstacles locations and shapes, the related transmission impedances and the background potentials. The inverse problem then consists in determining these scattering parameters from a complete set of far-field data at a fixed energy. In this framework, we show that the acoustic far-field pattern can be defined in terms of the scattering amplitude for the corresponding Schrödinger operator. A uniqueness result is then obtained by using new estimates for complex geometrical optics solutions (recently provided by B. Haberman for the Calderon's problem). This is a joint work with: M. Sini and A. Posilicano. |
December 13 |
Tomasz Komorowski (Maria Curie Sklodowska University, Lublin, Poland) |
Homogenization of a semilinear advection equation
I will discuss the problem of homogenization for the Cauchy problem for a semilinear advection equation, where the drift coefficient is given by an Ornstein-Uhlenbeck type stationary in time and space random field with incompressible realizations and the non-linearity is given by another random stationary field. We show the existence of the limit of solutions under the diffusive time-space scaling. Our main tool is the method of characteristics used to represent the solution of the equation. This is a joint work with L. Ryzhik (Stanford Univ.) and Yu Gu (Carnegie Mellon Univ.) |
December 6 |
Giovanna Marcelli ("Sapienza" University of Rome) |
Spin Conductance and Spin Conductivity in Topological Insulators: Analysis of Kubo-like terms
The last few decades witnessed an increasing interest, among solid state physicists, for physical phenomena having a topological origin. This interest traces back to the milestone paper by Thouless, Kohmoto, Nightingale and den Nijson the Quantum Hall Effect (QHE), and involves the seminal papers by Fu, Kane and Mele concerning the Quantum Spin Hall Effect (QSHE) to further developments in the flourishing field of topological insulators. As well known, in the QHE a topological invariant (Chern number) is related to an observable quantity, the charge (Hall) conductance. By analogy, in the context of the QSHE, one would like to connect the relevant topological invariant (Fu-Kane-Mele index) to a macroscopically observable quantity. The natural candidates are spin conductance and spin conductivity, which in general are not equivalent. As a paradigmatic case, we will analyse Kubo-like terms for spin conductance and spin conductivity in a discrete two-dimensional model. In view of the continuity equation for spin transport, derived from the first principles of Quantum Mechanics, our physical intuition suggests that spin conductance equals the spin conductivity whenever the spin torque mesoscopic mass vanishes. Indeed, we will prove the previous statement, as far as Kubo-like terms are concerned. To achieve the goal we first introduce the definition of the principal value trace and of the \(j\)-principal value trace (for \(j \in \{ 1,2 \}\)), and then develop a suitable machinery to compute them. The seminar is based on joint work with Gianluca Panati and Clément Tauber. |
November 29 |
Lorenzo Pinna ("Sapienza" University of Rome) |
Spin-boson models: controllability and Rotating Wave Approximation
Spin-boson models describe the interaction between a 2-level quantum system and finitely many distinguished modes of a bosonic field. In this talk I will discuss two prototypical examples, the Rabi model and the Jaynes-Cummings model, which despite their age are still very popular in several fields of quantum physics. Notably, in the context of cavity Quantum Electro Dynamics (QED) they provide an approximate yet accurate description of the dynamics of a 2-level atom in a resonant microwave cavity, as in recent experiments of S.Haroche. In the first part of the talk I will focus on the controllability properties of these models, analyzing two different types of control operators acting on the bosonic part, corresponding - in the application to cavity QED - to an external electric and magnetic field, respectively. I will review some recent results and prove the approximate controllability of the Jaynes-Cummings model with these controls. In the second part, I will consider the Rotating Wave Approximation (RWA), which consists of neglecting high oscillating terms of the Rabi Hamiltonian in the weak coupling regime, to obtain the Jaynes-Cummings Hamiltonian as an approximation. I will discuss this problem as an adiabatic limit to prove that the evolution operators of the two dynamics are norm close within a particular physical regime. |
November 24 |
Giulia Basti ("Sapienza" University of Rome) |
Efimov Effect for a system of two identical fermions and a different particle
In 1970 the physicist V. Efimov pointed out that a system of three different particles, such that the two-particle interactions are short-range and resonant, have an infinite number of bound states. This phenomenon is known as Efimov Effect and it is a paradigmatic example of the so-called universality of low-energy physics. We consider a system composed by two identical fermions of unitary mass and a third particle of mass \( m \). We assume that the interactions are short-range and that the two-particle subsystems do not have bound states. Moreover, we suppose that the subsystems composed by one of the fermions and the third particle have a zero-energy resonance. Under these assumptions we prove the existence of a mass threshold \( m_* \) such that if \( m < m_* \) then the number \( N(z) \) of eigenvalues of the three-particle Hamiltonian smaller than \( z<0 \) is infinite and \( N(z)\sim C(m)|log|z|| \) as \( z\to 0 \). On the other hand for \( m > m_*\) we show that the number of negative eigenvalues stays finite. |
November 24 |
Emanuela L. Giacomelli (University Tübingen, Germany) |
Surface Superconductivity in Presence of Corners
We consider an extreme type-II superconducting wire with non-smooth cross section, i.e., with one or more corners at the boundary, in the framework of the Ginzburg-Landau theory. We prove the existence of an interval of values of the applied field, where superconductivity is spread uniformly along the boundary of the sample. More precisely the energy is not affected to leading order by the presence of corners and the modulus of the Ginzburg-Landau minimizer is approximately constant along the transversal direction. The critical fields delimiting this surface superconductivity regime coincide with the ones in absence of boundary singularities. We will also discuss some recent results. In particular, we introduce a new effective problem near the corner that allows us to prove a refined asymptotics and to isolate the contributions to the energy density due to the presence of corners. The explicit expression of the effective energy is yet to be found but we formulate a conjecture on it based on the behavior for almost flat angles. Indeed, for corners with angles close to \( \pi \), we are able to explicitly compute the leading order of the corners effective problem and show that it sums up to the smooth boundary contribution to reconstruct the same asymptotics as in smooth domains. Joint work with Michele Correggi. |
October 25 |
Marco Falconi (Universität Zürich, Switzerland) |
Semiclassical properties of physical states
In this talk I will review the properties that classical (macroscopic) configurations of a physical system inherit from the underlying quantum (microscopic) configurations. A priori information of this type proves to be crucial in studying effective theories. As an example, consider the well-studied problem of deriving effective dynamical theories in many-body quantum mechanics. Clearly, this is possible only for classical configurations concentrated in the domain of well-posedness of the effective theory. Therefore, it is important to give sufficient microscopic conditions that would ensure macroscopic concentration on such domain. Another interesting consequence of this micro-to-macro analysis is that semiclassical states cannot be more entangled than the corresponding quantum states. |
October 4 |
Peter Pickl (LMU München, Germany) |
Bogoliubov corrections and trace norm convergence for the Hartree dynamics
We consider the dynamics of a large number N of interacting, nonrelativistic bosons in the mean field limit. In order to describe the fluctuations around the mean field Hartree state, we introduce an auxiliary Hamiltonian on the N-particle space that is very similar to the one obtained from Bogoliubov theory. We show convergence of the auxiliary time evolution to the fully interacting dynamics in the norm of the N-particle space. The results will then be generalized to gases of large volume, i.e. the case where volume and density tend to infinity simultaneously. This is joint work with D. Mitrouskas, S. Petrat and a. Soffer. |
September 20 |
Stefano Olla (Université Paris-Dauphine & CNRS, France) |
Kinetic and hydrodynamic limits for chains of harmonic oscillators
I will present two new results about macroscopic behavior of chains of harmonic oscillators. 1) The strain, momentum and energy of a chain of harmonic oscillators with random masses, even out of thermal equilibrium, converge to the solution of the Euler equation under hyperbolic space time scaling (in collaboration with Francois Huveneers and Cedric Bernardin). 2) Macroscopic scattering (in a hyperbolic scaling) caused by a Langevin thermostat in contact with a point of the harmonic chain (in collaboration with Tomasz Komorowski, Lenya Ryzhik and Herbert Spohn). |
September 13 |
Paolo Antonelli (GGSI, L'Aquila) |
Emergent dynamics for a model of quantum synchronization
In this talk I will discuss a model (introduced by Lohe, J. Phys. A 2010) describing quantum synchronization. More specifically, this is a system of coupled nonlinear Schrödinger equations where the wave functions have the tendency to align their phases. I will describe the link with the classical Kuramoto model for synchronization, then show the emergence of collective behaviors in the model. |
June 14 |
Robert Seiringer (Institute of Science and Technology, Austria) |
Stability of quantum many-body systems with point interactions
We present a proof that a system of \( N \) fermions interacting with an additional particle via point interactions is stable if the ratio of the mass of the additional particle to the one of the fermions is larger than some critical \( m_* \). The value of \( m_* \) is independent of \( N \) and turns out to be less than 1. This fact is of relevance for the stability of fermionic gases in the unitary limit. We also present a rigorous version of the Tan relations valid for all wave functions in the domain of the Hamiltonian of this model. |
May 24 |
Stefan Teufel (Universität Tübingen, Germany) |
Particle Creation at a Point Source by Means of Interior-Boundary Conditions
We consider a way of defining quantum Hamiltonians involving particle creation and annihilation based on an interior-boundary condition (IBC) on the wave function, where the wave function is the particle-position representation of a vector in Fock space, and the IBC relates (essentially) the values of the wave function at any two configurations that differ only by the creation of a particle. Here we prove, for a model of particle creation at one or more point sources using the Laplace operator as the free Hamiltonian, that a Hamiltonian can indeed be rigorously defined in this way without the need for any ultraviolet regularization, and that it is self-adjoint. We prove further that introducing an ultraviolet cut-off (thus smearing out particles over a positive radius) and applying a certain known renormalization procedure (taking the limit of removing the cut-off while subtracting a constant that tends to infinity) yields, up to addition of a finite constant, the Hamiltonian defined by the IBC. |
April 19 |
Stefano Marchesani (University of Oxford, UK) |
Hydrodynamic limits, weak solutions and thermodynamics
I will present a way to derive, via hydrodynamic limits, weak solutions to the 1D isothermal Euler equations in Lagrangian coordinates. This is obtained from a microscopic anharmonic chain with momentum preserving noise and hyperbolic scaling. Boundary conditions are added so we can define thermodynamic transformations between macroscopic equilibrium states: in particular we study the first and the second law of thermodynamics for the macroscopic system. |
April 12 |
Giada Basile (Università degli Studi di Roma "La Sapienza") |
A gradient flow approach for linear Boltzmann equations
We introduce a gradient flow formulation of linear Boltzmann equations. Under a diffusive scaling we derive a diffusion equation by using the machinery of gradient flows. This is a joint work with Dario Benedetto and Lorenzo Bertini. |
April 5 |
Carlo Marchioro (Università degli Studi di Roma "La Sapienza") |
Scudo magnetico per l'equazione di Vlasov-Poisson
Riferirò su una ricerca che sto svolgendo con Silvia Caprino (Roma2) e Guido Cavallaro (questo dipartimento). Studiamo un modello matematico di protezione di una navicella spaziale da un vento di particelle cariche tramite uno scudo magnetico. Più precisamente modellizziamo tale vento come un plasma di particelle cariche di una unica specie soggette alla equazione di Vlasov-Poisson, la navicella spaziale come un toro e lo scudo magnetico come un campo magnetico che diventa infinito sul bordo del toro. Discutiamo l'esistenza e l'unicità della soluzione e mostriamo che nessuna particella del plasma può entrare nel toro. Indichiamo possibili generalizzazioni. |
March 28 |
Sebastiano Peotta (Aalto University, Finland) |
Superfluidity and geometry of Bloch bands
Band structure theory and the BCS theory of superconductivity are two cornerstones of modern condensed matter physics. They have been used to explain many properties of crystalline solids and have found important practical applications. It is believed that the interplay between the atomic lattice and the attractive force between electrons, whose origin is still a matter of debate, is at the root of the phenomenon of high-Tc superconductivity. In weakly-coupled superconductors the effect of the lattice amounts to a simple renormalization of the electron mass and of the density of states. On the contrary, in high-Tc superconductors the coherence length is of the order of the lattice spacing and new phenomena may occur. An extreme example in this sense are 'flat bands', namely bands where the electron effective mass diverges. In this talk I will present our ongoing work on the problem of superconductivity and superfluidity in flat band systems with special emphasis on the transport properties. Naively one can expect that in a flat band, where the charge carriers are very heavy, transport is absent or at least strongly suppressed. However, we have recently shown that in the flat band limit the superfluid weight Ds is not controlled by the effective mass but rather by a geometric invariant of the band, the quantum metric, which in a sense measures the overlap between neighbouring lattice wave functions. The quantum metric is intimately related to a topological invariant, the Chern number, and as a consequence we obtain the inequality Ds ? |C| between superfluid weight and Chern number C. We show that this geometric effect is important in a number of lattice models of current interest for material science and ultracold gases. [1] N. B. Kopnin, T. T. Heikkilä, and G. E. Volovik, Phys. Rev. B 83 , 220503(R) (2011). [2] SP and P. Törmä, Nature Communications 6 , 8944 (2015). [3] A. Julku, SP, T. Vanhala, D.-H. Kim, and P. Törmä, Phys. Rev. Lett. 117, 045303 (2016). [4] M. Tovmasyan, SP, P. Törmä, and S. D. Huber, Phys. Rev. B 94, 245149 (2016). [5] L. Liang, T. I. Vanhala, SP, T. Siro, A. Harju, and Päivi Törmä, Phys. Rev. B 95, 024515 (2017). [6] N. Marzari and D. Vanderbilt, Phys. Rev. B 56 , 12847 (1997). |
March 15 |
Guo Chuan Thiang (University of Adelaide, Australia) |
The differential topology of semimetals
The "Weyl fermion" was discovered in a topological semimetal in 2015. Its mathematical characterisation turns out to involve deep and subtle results in differential topology. I will outline this theory, and explain some connections to Euler structures, torsion of manifolds, and Seiberg-Witten invariants. I also propose interesting generalisations with torsion topological charges arising from Kervaire semicharacteristics and "Quaternionic" characteristic classes. |
March 8 |
Kenji Yajima (Gakushuin University, Tokio, Japan) |
L^p-boundedness of wave operators for three dimensional Schrödinger operators with point interactions
We show that wave operators for three dimensional Schroedinger operators with point interactions are bounded in \(L^p\) for \(1 < p < 3\) but not for \(p = 1\) or \(p \geq 3\). This is a joint work with G. Dell’Antonio, A. Michelangeli and R. Scandone. |
February 22 |
Cristian Giardinà (Università degli Studi di Modena e Reggio Emilia) |
The Ising model on random graphs
Random graphs are useful models for complex networks appearing in empirical studies of networks. Several structural properties have been identified in this context, including scale-free and small-world properties. In this talk I will describe the Ising model on random graphs satisfying these properties. The Ising model is a stochastic model introduced in statistical physics to model phase transitions. Thus two sources of randomness are intertwined in the Ising model on random graphs. I will investigate their interplay studying the Boltzmann-Gibbs measure for a fixed random graph realization or when the average over graphs (quenched or annealed) is performed. I shall focus on universality, proving law of large numbers and central limit theorems in the uniqueness phase, as well as non-classical limit theorems at criticality. |
February 22 |
Cedric Bernardin (Université Nice Sophia Antipolis & CNRS, France) |
From diffusion to fractional superdiffusion in a Hamiltonian lattice field model with noise
We consider a Hamiltonian lattice field model perturbed by an energy conserving noise and show that after a space-time rescaling the energy-energy correlation function is given by the solution of a skew-fractional heat equation with exponent 3/4. We then investigate the validity of this result under some deterministic or stochastic perturbations. |
February 15 |
Fumio Hiroshima (Kyushu University, Fukuoka, Japan) |
Analysis of time operators
The time operator is informally defined as a symmetric operator satisfying CCR with a given self-adjoint operator H, i.e. \( [H,T] = i \). However it is nontrivial to construct T associated with H, and W. Pauli mentioned at 1933 that there was no quantum time operators associated with H having eigenvalues or being semi-bounded. In this talk we define 5 classes of time operators from mathematical point of view, and show that a time operator exists for a Schroedinger operator \( H = -\Delta+V \) with some \( V \). Note that H is semi-bounded and has infinitely many eigenvalues. |
January 25 |
Michel Fruchart (Instituut-Lorentz, Universiteit Leiden, Netherlands) |
Topological states in wave propagation: an introduction
Topological insulators are materials where the bulk propagation of waves is twisted in a particular, topological way. At an interface between systems with different topologies, edge states with very special properties appear, that may for example serve as robust unidirectional wave guides. Fascinating from the point of view of mathematics, such systems are also highly relevant from the experimental point of view, and were realized (indeed) in solid-sate electronic systems, but also with light, sound, or cold atoms. I will review such phenomena, with an highlight on the bulk classification of topological "effective evolutions" which describe the propagation of waves in such peculiar media. |
January 18 |
Benedetto Scoppola (Università degli Studi di Roma "Tor Vergata") |
Lattice gas Sherrington-Kirkpatrick system
We study a system in which the hamiltonian has exactly the form of the Sherrington-Kirkpatrick spin glass system but each spin takes value in the set \( \{0,1\} \) instead of the set \( \{-1,1\} \). We investigate the properties of the ground state of the system, we prove the existence of the limit of the ground energy per particle, and we discuss some rigorous bounds of its value. Joint work with Alessio Troiani. |
January 11 |
Aldo Procacci (Universidade Federal de Minas Gerais, Brazil) |
Convergence of Mayer and virial expansions and the Penrose tree-graph identity
We establish new lower bounds for the convergence radius of the Mayer series and the Virial series of a continuous particle system interacting via a stable and tempered pair potential. Our bounds considerably improve those given by Penrose and Ruelle in 1963 for the Mayer series and by Lebowitz and Penrose in 1964 for the Virial series. To get our results we exploit the tree-graph identity given by Penrose in 1967 using a new partition scheme based on minimum spanning trees. Joint work with Sergio Yuhjtman. |