Mathematical Challenges in Quantum Mechanics - Online Seminars




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 14:15 Italian time, on a monthly basis. The schedule of the upcoming seminars is: 

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 will be 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

Previous seminars of the ongoing series:

Organizing committee

Serena Cenatiempo (GSSI)  
Marco Falconi (PoliMi)  
Emanuela L. Giacomelli (LMU)  
Domenico Monaco (Sapienza)  
Marco Olivieri (Aahrus University)

Registration to MCQM Seminars
  • Alessandro Olgiati
  • Apostolos Giovanakis
  • Arnaud Triay
  • Asbjørn Bækgaard Lauritsen
  • Benjamin Hinrichs
  • Bruno Colbois
  • Camilo Gómez Araya
  • Chiara Boccato
  • Claudio Cacciapuoti
  • Cornelia Vogel
  • Cristina Caraci
  • Daniele Ferretti
  • David Miguélez Caballero
  • Davide Desio
  • Denis Périce
  • Domenico Lapadula
  • Domenico Monaco
  • Emanuela L. Giacomelli
  • Fabrizio Caragiulo
  • François Visconti
  • Gabriele Grillo
  • Gabriele Peluso
  • Ghofrane Bel Hadj Aissa
  • Gianluca Panati
  • Giovanna Marcelli
  • Giulia Basti
  • Giuseppe Gaeta
  • Giuseppe Lipardi
  • Giuseppe MARMO
  • Harman Preet Singh
  • Horia Cornean
  • Jakob Oldenburg
  • Jinyeop Lee
  • Jonas Lampart
  • Konstantin Merz
  • Lakshita Bageja
  • Leonardo Goller
  • Luca Fresta
  • Lucrezia Cossetti
  • Luis Gerardo Ayala Bertel
  • Mangaldeep Paul
  • Marco Falconi
  • Marco Olivieri
  • Massimo Moscolari
  • Matias Ginzburg
  • Matteo Gallone
  • Meriem Bahhi
  • Michele Correggi
  • Nepomuk Trauner
  • Nico Michele Schiavone
  • Nils Schopohl
  • Paolo Facchi
  • Saptarshi Mandal
  • Serena Cenatiempo
  • Sergio Navarro Obregón
  • Simone Rademacher
  • Stefano Marcantoni
  • Thiago Carvalho Corso
  • Vishnu Sanjay
  • William Borrelli
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    • 1
      MCQM PhD Lecture: Tobias König (University Frankfurt)

      Title: Functional inequalities and eigenvalue bounds for Schrödinger operators

      Abstract: Functional inequalities play an important role in proving stability of quantum-mechanical systems. The goal of this lecture is to give a gentle introduction to functional inequalities and their implication for bounds of eigenvalues of Schrödinger operators -Δ + V. The lecture will consist of two parts. In the first part, as a warm-up I will discuss the most basic instance of the above program, namely bounds on the lowest eigenvalue of a Schrödinger operator via Sobolev's or Hardy's inequality. In the second part of my lecture, the focus will be on the famous Lieb-Thirring (LT) inequality for sums of eigenvalues of Schrödinger operators, which allows to prove stability of matter. I will explain its connection to the Sobolev inequality and mention some important open questions. As a limit case of the LT inequalities, I will also discuss the Cwikel--Lieb--Rozenblum (CLR) bound for the number of negative Schrödinger eigenvalues.

    • 2
      MCQM Seminar: Dirk Hundertmark (KIT)

      Title: Cwikel's bound reloaded

      Abstract: There are several proofs by now for the famous Cwikel–Lieb– Rozenblum (CLR) bound, which is a semiclassical bound on the number of bound states for a Schrödinger operator, proven in the 1970s. Of the rather distinct proofs by Cwikel, Lieb, and Rozenblum, the one by Lieb gives the best constant, the one by Rozenblum does not seem to yield any reasonable estimate for the constants, and Cwikel’s proof is said to give a constant which is at least about 2 orders of magnitude off the truth. This situation did not change much during the last 40+ years. It turns out that this common belief, i.e, Cwikel’s approach yields bad constants, is not set in stone: We give a substantial refine- ment of Cwikel’s original approach which highlights a natural but overlooked connection of the CLR bound with bounds for maximal Fourier multipliers from harmonic analysis. Moreover, it gives an astonishingly good bound for the constant in the CLR inequality. Our proof is also quite flexible and leads to rather precise bounds for a large class of Schrödinger-type operators with generalized kinetic energies. My talk is based on a paper which was recently published in Inventiones mathematicae and is available online at or as a PDF from .
      In my talk I will explain the background and the proof in the case of a usual Schrödinger operator. I will keep the discussion free from technicalities, the main tool in the proof is the Cauchy Schwarz inequality.

    • 3
      MCQM PhD Lecture: David Gontier (Paris Dauphine)

      Title: Numerical methods for linear periodic systems

      Abstract: In this lecture, I will present numerical methods to compute the spectrum of periodic operators. Such operators have typically purely essential spectrum, and cannot be studied directly due to spectral pollution. We will review several methods for these operators, and in particular introduce the Bloch transform. We will show how to obtain the band diagrams of such operators, and how to compute physical quantities such as the energy per unit cell.

    • 4
      MCQM Seminar: Eric Cancès (Ecole des Ponts ParisTech)

      Title: Numerical methods for Density Functional Theory

      Abstract: Kohn-Sham Density Functional Theory (DFT) is the most widely used electronic structure calculation method in quantum chemistry, solid state physics, and materials science. The Kohn-Sham model has a variational structure: it consists in minimizing some energy functional over the manifold of admissible states. The corresponding Euler-Lagrange equations read as a system of nonlinear elliptic partial differential equations, and more specifically as a nonlinear elliptic eigenvalue problem. Since DFT calculations use about 15% of the total resources available in high-performance scientific computing centers, finding more efficient numerical methods for the Kohn-Sham model is of major interest. This requires a better understanding of the mathematical and numerical properties of the Kohn-Sham model. I will present some recent results in this direction, as well as some challenges.

    • 5
      MCQM PhD Lecture: Jacob Shapiro (Princeton University)

      Title: An overview of mathematical aspects of topological insulators

      Abstract: Topological insulators are recently discovered novel materials which exhibit exotic behavior: they are insulators in their bulk but excellent conductors along their boundary, and--strikingly--there is a quantum mechanical macroscopic observable which one could calculate (e.g. a zero temperature DC conductance) which exhibits quantization on Z or Z_2. The first example is the integer quantum Hall effect which dates back to the 1980s, but in 2005 more examples, associated with other symmetry classes, were discovered and later on a whole table due to Kitaev was formed, patterned after K-theory. This meeting point between quantum mechanics, functional analysis and algebraic topology is a convenient place for mathematical physicists to tackle interesting problems, some of which I shall review in this non-expert, introductory talk.

    • 6
      MCQM Seminar: Horia Cornean (Aalborg Universitet)

      Title: Bulk-edge correspondence for unbounded Dirac-Landau operators

      Abstract: We consider two-dimensional unbounded magnetic Dirac operators, either defined on the whole plane, or with infinite mass boundary conditions on a half-plane. Our main results use techniques from elliptic PDEs and integral operators, while their topological consequences are presented as corollaries of some more general identities involving magnetic derivatives of local traces of fast decaying functions of the bulk and edge operators. One of these corollaries leads to the so-called Streda formula: if the bulk operator has an isolated compact spectral island, then the integrated density of states of the corresponding bulk spectral projection varies linearly with the magnetic field as long as the gaps between the spectral island and the rest of the spectrum are not closed, and the slope of this variation is given by the Chern character of the projection. The same bulk Chern character is related to the number of edge states which appear in the gaps of the bulk operator. These results are based on joint work with M. Moscolari and K. Sørensen.

    • 7
      MCQM PhD Lecture: Léo Morin (Aarhus University)

      Title and Abstract: TBA

    • 8
      MCQM Seminar: San Vũ Ngọc

      Title and Abstract: TBA

    • 9
      MCQM PhD Lecture: TBA
    • 10
      MCQM Seminar: Caroline Lasser

      Title and Abstract: TBA