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Mathematical Challenges in Quantum Mechanics - Online Seminars

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Description

The series of online seminars Mathematical Challenges in Quantum Mechanics (MCQM) focuses on current topics of mathematical physics, with special attention to the mathematical aspects of quantum mechanics. It aims to bring together the Italian community working on mathematical methods for quantum theory.      

The seminars take place on Wednesday afternoon, usually at 14:30 Italian time, on a monthly basis. The schedule for the 2024/25 seminar series is as follows:       
 

Each seminar will be accompanied by a lecture addressed to PhD students and young researchers, aimed at introducing the broad topic of the seminar and presenting some perspectives of the open problems and mathematical challenges in the field. The lecture is usually held on the same day of the seminar, at 11.30 Italian time.

Titles and abstracts of the MCQM Seminars and PhD lectures are available at this page and this page respectively. To receive the zoom link to attend the seminars and/or the PhD lectures please register here. 

The online seminars are part of a long-running series of events organized over the years by the Italian community working on mathematical methods for quantum physics, including the following School and Workshops: 

MCQM - Mathematical Challenges in Quantum Mechanics  
MCQM22 - Como, June 13-18, 2022 
MCQM18 - Rome, February 19-24, 2018  
MCQM16 - Bressanone, February 8-13, 2016


Organizing committee

Serena Cenatiempo (GSSI)     
Marco Falconi (PoliMi)     
Emanuela L. Giacomelli (UniMI)     
Domenico Monaco (Sapienza)     
Marco Olivieri (Copenhagen)     

 

MCQM PhD Lectures 2022/23
MCQM PhD Lectures 2023/24
MCQM PhD Lectures 2024/25
MCQM Seminars 2022/23
Registration
Registration to MCQM Seminars
Register
Participants
  • Wednesday, 6 November
    • 11:30 → 12:30
      MCQM PhD Lecture: Benjamin Alvarez 1h

      Title: An introduction to non relativistic QED

      Abstract: This talk is dedicated to presenting the fundamental notions to adress non relativistic QED models. The general framework of quantum field theory will be presented together with the Hamiltonian of non relativistic QED. In addition, we aim to give a taste of important works that have been done in the field together with some open problems.

    • 14:30 → 15:30
      MCQM Seminar: Volker Bach 1h

      Title: TBA

      Abstract: TBA

  • Wednesday, 18 December
    • 11:30 → 12:30
      MCQM PhD Lecture: Angelo Lucia 1h

      Title: Locality and spectral gaps in quantum spin systems

      Abstract: In this lecture, I will give a brief introduction to the mathematical framework used to describe quantum spin systems on lattices. The main goal will be to connect the study of models defined over finite volumes to an idealized infinite volume limit called the thermodynamic limit. A key role will be played by locality estimates known as Lieb-Robinson bounds. I will then focus on the spectral gap of the model, the difference between the two lowest energy levels, and I will explain how it is related to the problem of phase classification of quantum phases of matter.

    • 14:30 → 15:30
      MCQM Seminar: David Pérez García 1h

      Title: Spectral gaps in many body quantum systems

      Abstract: How does the spectral gap of a quantum many body system scale with the system size? This turns out to be a very relevant question both in condensed matter physics and in quantum computing. However, there are very few available techniques to give bounds of that scaling. In the first part of this talk, we will review recent results which explain why: it is undecidable to know if the spectral gap will vanish or not with the system size. In the second part, we will review complementary results which show that, despite this general impossibility result, one can still prove spectral gap estimates in a rather wide family of systems. As a consequence, one obtains new bounds on the lifetime of quantum memories.

  • Wednesday, 15 January
    • 11:30 → 12:30
      MCQM PhD Lecture: Benjamin Hinrichs 1h

      Title: An Invitation to Path Measure Methods for Polaron Models

      Abstract: This talk gives an introduction to the connection between the Hamiltonian formulation of quantum mechanics and its probabilistic counterpart. We first discuss this in terms of Feynman-Kac formulas for Schrödinger operators together with some applications. Then, we move to their analogues for polaron models, i.e., models of quantum particles linearly coupled to a quantum field. We also sketch how the named applications extend to the polaron, e.g., when studying ground states or the effective mass.

    • 14:30 → 15:30
      MCQM Seminar: Robert Seiringer 1h

      Title: Spectral analysis of polaron models in the strong coupling limit

      Abstract: The Fröhlich polaron and related models of quantum field theory have played a prominent role in mathematical physics over several decades. In this talk, we shall explain recent bounds on the quantum corrections to the (classical) Pekar approximation of the ground state energy of the Fröhlich polaron model in the strong coupling limit, and their consequence on the existence of excited states and the polaron's effective mass.

  • Wednesday, 12 March
    • 11:30 → 12:30
      MCQM PhD Lecture: Andrew Rout 1h

      Title: A (brief) introduction to Gibbs measures for the nonlinear Schrödinger equation

      Abstract: In this talk I will give an introduction to Gibbs measures for the nonlinear Schrödinger equation. The construction of global solutions to dispersive PDEs usually relies on the conservation of quantities like the energy and the mass. For lower regularity functions, these quantities are infinite, so cannot be used. Instead one introduces an invariant measure, which can act as a substitute for the conserved quantities.
      I give the heuristic ideas for the construction the Gibbs measure, and also sketch the details of the rigorous construction. I will also discuss how to use the Gibbs measure to construct global solutions to the nonlinear Schrödinger equation.

  • Wednesday, 9 April
    • 11:30 → 12:30
      MCQM PhD Lecture: Konstantin Merz 1h

      Title: The Ionization Conjecture in Quantum Mechanics and Density Functional Theory: An Introduction

      Abstract: Quantum mechanics accurately describes physics on atomic length scales. However, many fundamental questions about the structure of matter remain unanswered. A prominent example is the Ionization Conjecture. It asserts that an atom with nuclear charge Z can bind at most Z+1 electrons. Although this has been experimentally documented since the 1970s, a mathematical proof is not in sight. In this lecture, we introduce the Ionization Conjecture and review some landmark results and recent progress. Although the conjecture for the full many-body problem seems out of reach, significant progress has been made in certain effective single-particle models. These models are easier to study yet still capture key aspects of the full many-body system. Among these are density functional theories such as the classic Thomas–Fermi model, which effectively describe the energy and particle distribution of large atoms and molecules. Remarkably, despite describing the bulk of the electrons, these models also provide insights into the outermost electrons—the key objects in the Ionization Conjecture.

    • 14:30 → 15:30
      MCQM Seminar: Rafael Benguria 1h

      Title: Bound on the Excess Charge of Generalized Thomas-Fermi-Weizsäcker functionals

      Abstract: We bound the number of electrons Q that an atom can bind in excess of neutrality for density functionals generalizing the classical Thomas-Fermi-Weizsäcker functional: instead of the classical power 5/3 more general powers p are considered. For 3/2 < p <2 we prove the excess charge conjecture, i.e., that Q is uniformly bounded in the atomic number Z. The case p = 3/2 is critical: the behavior changes from a uniform bound in Z to a linear bound at the critical coupling 4 √ π of the nonlinear term. We also improve the linear bound for all p ≥ 6/5.

      This is joint work with Heinz Siedentop, LMU, Munich.

  • Wednesday, 14 May
    • 11:30 → 12:30
      MCQM PhD Lecture: Alessio Ranallo 1h

      Title: TBA

      Abstract: TBA

    • 14:30 → 15:30
      MCQM Seminar: Nicola Pinamonti 1h

      Title: Interacting regularised scalar quantum field theories: convergence of the S-matrix

      Abstract: We discuss the construction of self-interacting scalar quantum field theories on d-dimensional Minkowski spacetime, focusing on a class of interaction Lagrangians given by suitable functions of the scalar field. Using perturbation theory, we express interacting field observables as formal power series expansions over the free theory. Central to this construction is the time-ordered exponential of the interaction Lagrangian—the S-matrix—which itself is a power series in the coupling constant.
      We introduce a regularization procedure that renders the S-matrix convergent to well-defined unitary operators. This regularization involves two parameters: one controlling the suppression of high-frequency modes in the propagators, and another regulating large field contributions in the interaction Lagrangian. We analyze the removal of these regularization parameters in lower-dimensional theories and for specific interaction Lagrangians.
      In particular, we show that for a φ4​ theory in three spacetime dimensions, form sequences of regularized S-matrices obtained scaling the regularization parameters to zero we can extract convergent subsequences in the weak-* topology. The asymptotic expansions of all the possible limit points of the extracted subsequences agree and match the predictions of perturbation theory. We also outline how these results extend to the four-dimensional case, where similar behavior is observed.

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