Month: April 2024

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 LillSenza categoria 

Tuesday, May 7, 2024 - 11:15
(Room: TBA)
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.

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.