30-31 May 2024: Andrea Posilicano @ Polimi

Marco FalconiSeminar PoliMi

Thursday, May 30, 2024 and Friday, May 31, 2024 - 10:00
Sala Consiglio VII Piano, D-Mat
Politecnico di Milano
Ed. 14 "Nave", Campus Leonardo

SPEAKER: Andrea Posilicano (Università dell’Insubria)

Self-adjoint extensions by a Krein-type resolvent formula

We present a simple recipe to build all the self-adjoint extensions of a symmetric operator S which is the restriction of a given self-adjoint one. This provides the resolvent of the extensions and requires knowledge of neither the defect spaces nor the adjoint of S. Some applications to quantum mechanical models are given.

This initiative is part of the “PhD Lectures” activity of the project “Departments of Excellence 2023-2027” of the Department of Mathematics of Politecnico di Milano. This activity consists of seminars open to PhD students, followed by meetings with the speaker to discuss and go into detail on the topics presented at the talk.

05/06/2024 : Umberto Morellini @UniMi

Ngọc Nhi NguyễnSeminar UniMi 
Wednesday, June 5, 2024 - 11:15
Sala di Rappresentanza
Dipartimento di Matematica
Università degli Studi di Milano
Via Cesare Saldini 50

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SPEAKER: Umberto Morellini (Université Paris Dauphine, CEREMADE, PSL) ________________________________________________________________________

The free energy of Dirac’s vacuum in purely magnetic fields

The Dirac vacuum is a non-linear polarisable medium rather than an empty space. This non-linear behaviour starts to be significant for extremely large electromagnetic fields such as the magnetic field on the surface of certain neutron stars. Even though the null temperature case was deeply studied in the past decades, the problem at non-zero temperature needs to be better understood.
In this talk, we will present the first rigorous derivation of the one-loop effective magnetic Lagrangian at positive temperature, a non-linear functional describing the free energy of quantum vacuum in a classical magnetic field. After introducing our model, we will properly define the free energy functional using the Pauli-Villars regularisation technique in order to remove the worst ultraviolet divergences, which represent a well known issue of the theory. The study of the properties of this functional will be addressed before focusing on the limit of slowly varying classical magnetic fields. In this regime, one can prove the convergence of this functional to the Euler-Heisenberg
formula with thermal corrections, recovering the effective Lagrangian first derived by Dittrich in 1979. The talk is based on the work available at arXiv:2404.12733.

20-22-24 May 2024: Hynek Kovarik @ Polimi

Marco FalconiSeminar PoliMi

Monday, May 20, 2024 - 10:00 , Wednesday, May 22, 2024 - 11:00, Friday, May 24, 2024 - 10:00
Sala Consiglio VII Piano, D-Mat
Politecnico di Milano
Ed. 14 "Nave", Campus Leonardo

SPEAKER: Hynek Kovarik (Università di Brescia)

Trace formulas for one-dimensional Schrödinger operators

One-dimensional Schroedinger operators satisfy certain identity, called the trace formula, which relates the scattering and spectral data of the operator in question with integral means of the corresponding potential. In this mini-course we will give a sketch of the proof of this formula and discuss some of its applications.

This initiative is part of the “PhD Lectures” activity of the project “Departments of Excellence 2023-2027” of the Department of Mathematics of Politecnico di Milano. This activity consists of seminars open to PhD students, followed by meetings with the speaker to discuss and go into detail on the topics presented at the talk.

7-8 May 2024: Horia Cornean @ Polimi

Marco FalconiSeminar PoliMi

Tuesday, May 7, 2024 - 10:15 , Wednesday, May 8, 2024 - 11:00
Aula Seminari III Piano, D-Mat
Politecnico di Milano
Ed. 14 "Nave", Campus Leonardo

SPEAKER: Horia Cornean (Aalborg Universitet)

On the Landauer-Büttiker formalism

In the first part we will introduce the setting and prove some fundamental scattering results related to the existence and completeness of wave operators arising in mesoscopic systems, and also prove the “classical” Landauer-Büttiker formula for non-interacting systems. The second part will be about providing sufficient conditions such that the time evolution of a mesoscopic tight-binding open system with a local Hartree-Fock non-linearity converges to a self-consistent non-equilibrium steady state, which is independent of the initial condition from the “small sample”. We will also show that the steady charge current intensities are given by Landauer-Büttiker-like formulas, and make the connection with the case of weakly self-interacting many-body systems. In order to get a better idea of what the lectures will cover, see https://arxiv.org/abs/2309.01564 .

This initiative is part of the “PhD Lectures” activity of the project “Departments of Excellence 2023-2027” of the Department of Mathematics of Politecnico di Milano. This activity consists of seminars open to PhD students, followed by meetings with the speaker to discuss and go into detail on the topics presented at the talk.

07/05/2024: Cornelia Vogel @ UniMi

Sascha LillSeminar UniMi 

Tuesday, May 7, 2024 - 11:30
Sala di rappresentanza
Dipartimento di Matematica
Università degli Studi di Milano
Via Cesare Saldini 50

SPEAKER: Cornelia Vogel (Universität Tübingen)

Concentration of measure for thermal distributions of quantum states

We generalize Lévy’s Lemma, a concentration-of-measure result for the uniform probability distribution on high-dimensional spheres, to a more general class of measures, so-called GAP measures. For any given density matrix ρ on a separable Hilbert space H, GAP(ρ) is the most spread out probability measure on the unit sphere of H that has density matrix ρ and thus forms the natural generalization of the uniform distribution. We prove concentration-of-measure whenever the largest eigenvalue ||ρ|| of ρ is small. With the help of this result we generalize the well-known and important phenomenon of ”canonical typicality” to GAP measures. Canonical typicality is the statement that for ”most” pure states ψ of a given ensemble, the reduced density matrix of a sufficiently small subsystem is very close to a ψ-independent matrix. So far, canonical typicality is known for the uniform distribution on finite-dimensional spheres, corresponding to the micro-canonical ensemble. Our result shows that canonical typicality holds in general for systems described by a density matrix with small eigenvalues. Since certain GAP measures are quantum analogs of the canonical ensemble of classical mechanics, our results can also be regarded as a version of equivalence of ensembles. The talk is based on joint work with Stefan Teufel and Roderich Tumulka.

TalkMilanDownload

22/04/2024: Antoine Prouff @UniMi

Ngọc Nhi NguyễnSeminar UniMi 
Monday, April 22, 2024 - 11:15
Sala di Rappresentanza
Dipartimento di Matematica
Università degli Studi di Milano
Via Cesare Saldini 50

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SPEAKER: Antoine Prouff (Université Paris-Saclay, Laboratoire de Mathématique d’Orsay) _________________________________________________________________________

Egorov’s theorem in the Weyl-Hörmander calculus and application to the control of PDEs

It is known that geometric optics can be derived as the high-frequency limit of the wave equation, from both experimental and theoretical perspectives. This fact can be regarded as an instance of a “quantum-classical correspondence principle”, made rigorous by Egorov’s theorem, which relates the evolution of a linear PDE (e.g. the wave equation) to the natural underlying classical dynamics (e.g. the geodesic flow).

We will present a version of Egorov’s theorem in the Euclidean space, in the setting of the “Weyl-Hörmander calculus”. This general framework of microlocal analysis involves Riemannian metrics on the phase space adapted to the dynamics under consideration, and allows for a fairly large range of applications (study of Schrödinger, wave and transport equations).

If time allows, we will discuss in more detail an application to the observability of the Schrödinger equation with a confining potential in the Euclidean space.

15, 16 & 19/04/2024 : Jérémy Faupin + Sébastien Breteaux @ PoliMi

Marco FalconiSeminar PoliMi

Monday, April 15, 2024 - 14:15 , Tuesday, April 16, 2024 - 14:15 and Friday, April 19, 2024 - 10:30
Sala Consiglio, D-Mat
Politecnico di Milano
Ed. 14 "Nave", Campus Leonardo

SPEAKER: Jérémy Faupin & Sébastien Breteaux (Institut Élie Cartan de Lorraine – Université de Metz)

Number of bound states for fractional Schrödinger operators

Estimating the number of bound states (i.e. the number of negative eigenvalues counting multiplicities) of the two-body Schrödinger operator -Δ+V(x) on L²(ℝᵈ) constitutes a rich problem that has attracted lots of attention in the mathematical literature. This series of lectures will focus on bounds on the number of bound states for fractional Schrödinger operators (-Δ)ˢ+V(x) on L²(ℝᵈ), for any s>0 and in any spatial dimension d≥1. In the subcritical case s<d/2, we will in particular review the celebrated Cwikel-Lieb-Rozenblum bounds, while in the super-critical case s≥ d/2, we will report on a recent joint work with V. Grasselli.

This initiative is part of the “PhD Lectures” activity of the project “Departments of Excellence 2023-2027” of the Department of Mathematics of Politecnico di Milano. This activity consists of seminars open to PhD students, followed by meetings with the speaker to discuss and go into detail on the topics presented at the talk.

18/03/2024: Zhituo Wang @ UniMi

quantumseminarSeminar UniMi 
Monday, March 18, 2024 - 11:15
Aula Dottorato (first floor)
Dipartimento di Matematica
Università degli Studi di Milano
Via Cesare Saldini 50

SPEAKER: Zhituo Wang (Harbin Institute of Technology)

Constructive renormalizations of the 2-D Honeycomb-Hubbard model

In this talk I will present some recent progress on the construction of ground state of the 2-dimensional Hubbard model, which is a prototypical model for studying phase transitions in quantum many-body system. Using fermionic cluster expansions and constructive renormalization theory, we proved that the ground state of the 2-d Hubbard model on the honeycomb lattice with triangular Fermi surfaces is not a Fermi liquid in the mathematical precise sense of Salmhofer. I will also discuss the crossover phenomenon in the 2-d square Hubbard model and universalities. This presentation is based on the work arXiv:2108.10852, CMP 401, 2569–2642(2023) and arXiv:2303.13628.

04 & 07/03/2024 : Stefan Teufel @ PoliMi

Marco FalconiSeminar PoliMi

Monday, March 4, 2024 - 14:15 and Thursday, March 7, 2024 - 10:15
Sala Consiglio
Dipartimento di Matematica
Politecnico di Milano
Ed. 14 "Nave", Campus Leonardo

SPEAKER: Stefan Teufel (Eberhard Karls Universität Tübingen)

Quantisation of Hall conductivity in infinite interacting fermion systems

Understanding the exact quantisation of the experimentally observed Hall
conductivity has led to many interesting developments also in mathematical
physics and to two Nobel prizes in physics. Although the problem is now more
than 40 years old, significant progress has been made on a rigorous level for
interacting systems in the last ten years. In my talk I will briefly review
some of the mathematical highlights of this development and then present a new result: We consider infinite systems of interacting electrons on a lattice
governed by a translation-invariant Hamiltonian with a unique gapped ground
state. Using the NEASS approach, we show that the Hall conductivity of such a
system at zero temperature is given by a many-body version of the famous
double-commutator formula without power-law corrections. We also show that this formula takes only quantised values. The main novelty compared to
existing mathematical results is that we consider the conductivity instead of
the conductance, and that working directly in infinite volume simplifies and
clarifies some arguments, provided one is willing to work in the C^*-algebraic
framework used in the mathematical description of such systems.

In the first lecture I will mainly talk about the history of the problem and set up the mathematical framework. In the second lecture I will present the new results. The latter are based on joint work with Giovanna Marcelli, Tadahiro Miyao, Domenico Monaco and Marius Wesle.

This initiative is part of the “PhD Lectures” activity of the project “Departments of Excellence 2023-2027” of the Department of Mathematics of Politecnico di Milano. This activity consists of seminars open to PhD students, followed by meetings with the speaker to discuss and go into detail on the topics presented at the talk.

28/2/2024: Filippo Boni @ UnIns

quantumseminarSeminar UnIns 
Wednesday, February 28, 2024 - 9:45
Aula V2.10
Dipartimento di Scienza e Alta Tecnologia
Università degli Studi dell'Insubria
Via Valleggio 11, Como

SPEAKER: Filippo Boni (Scuola Superiore Meridionale)

Normalized ground states for Nonlinear Schrödinger equations on metric graphs

In this talk we present some results about the existence of normalized ground states on noncompact metric graphs for nonlinear Schrödinger equations involving possibly both a standard power nonlinearity and delta nonlinearities located at the vertices of the graph. In the first part, we review some results when the sole standard nonlinearity is present.
In the second part, we present more recent results, both when only delta nonlinearities at the vertices are considered and when standard and delta nonlinearities coexist. In the first case, we show that the ground state problem strongly depends on the degree of periodicity of the graph, the total number of delta nonlinearities and their dislocation in the graph. In the second case, we highlight that the existence of ground states is strongly affected by the value of the mass, the relation between the powers of the two nonlinear terms and by topological and metric properties of the graph itself. These results have been obtained in collaboration with R. Adami, S. Dovetta and E. Serra (Politecnico di Torino).