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Mathematical Challenges in Quantum Mechanics - Online Seminars
from
Wednesday, November 9, 2022 (11:00 AM)
to
Wednesday, May 24, 2023 (1:00 PM)
Monday, November 7, 2022
Tuesday, November 8, 2022
Wednesday, November 9, 2022
11:30 AM
MCQM PhD Lecture: Tobias König (University Frankfurt)
MCQM PhD Lecture: Tobias König (University Frankfurt)
11:30 AM - 12:30 PM
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.
2:15 PM
MCQM Seminar: Dirk Hundertmark (KIT)
MCQM Seminar: Dirk Hundertmark (KIT)
2:15 PM - 3:15 PM
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 https://doi.org/10.1007/s00222-022-01144-7 or as a PDF from https://link.springer.com/content/pdf/10.1007/s00222-022-01144-7.pdf . 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.
Thursday, November 10, 2022
Friday, November 11, 2022
Saturday, November 12, 2022
Sunday, November 13, 2022
Monday, November 14, 2022
Tuesday, November 15, 2022
Wednesday, November 16, 2022
Thursday, November 17, 2022
Friday, November 18, 2022
Saturday, November 19, 2022
Sunday, November 20, 2022
Monday, November 21, 2022
Tuesday, November 22, 2022
Wednesday, November 23, 2022
Thursday, November 24, 2022
Friday, November 25, 2022
Saturday, November 26, 2022
Sunday, November 27, 2022
Monday, November 28, 2022
Tuesday, November 29, 2022
Wednesday, November 30, 2022
Thursday, December 1, 2022
Friday, December 2, 2022
Saturday, December 3, 2022
Sunday, December 4, 2022
Monday, December 5, 2022
Tuesday, December 6, 2022
Wednesday, December 7, 2022
Thursday, December 8, 2022
Friday, December 9, 2022
Saturday, December 10, 2022
Sunday, December 11, 2022
Monday, December 12, 2022
Tuesday, December 13, 2022
Wednesday, December 14, 2022
11:30 AM
MCQM PhD Lecture: David Gontier (Paris Dauphine)
MCQM PhD Lecture: David Gontier (Paris Dauphine)
11:30 AM - 12:30 PM
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.
2:15 PM
MCQM Seminar: Eric Cancès (Ecole des Ponts ParisTech)
MCQM Seminar: Eric Cancès (Ecole des Ponts ParisTech)
2:15 PM - 3:15 PM
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.
Thursday, December 15, 2022
Friday, December 16, 2022
Saturday, December 17, 2022
Sunday, December 18, 2022
Monday, December 19, 2022
Tuesday, December 20, 2022
Wednesday, December 21, 2022
Thursday, December 22, 2022
Friday, December 23, 2022
Saturday, December 24, 2022
Sunday, December 25, 2022
Monday, December 26, 2022
Tuesday, December 27, 2022
Wednesday, December 28, 2022
Thursday, December 29, 2022
Friday, December 30, 2022
Saturday, December 31, 2022
Sunday, January 1, 2023
Monday, January 2, 2023
Tuesday, January 3, 2023
Wednesday, January 4, 2023
Thursday, January 5, 2023
Friday, January 6, 2023
Saturday, January 7, 2023
Sunday, January 8, 2023
Monday, January 9, 2023
Tuesday, January 10, 2023
5:30 PM
MCQM PhD Lecture: Jacob Shapiro (Princeton University)
MCQM PhD Lecture: Jacob Shapiro (Princeton University)
5:30 PM - 6:30 PM
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.
Wednesday, January 11, 2023
2:15 PM
MCQM Seminar: Horia Cornean (Aalborg Universitet)
MCQM Seminar: Horia Cornean (Aalborg Universitet)
2:15 PM - 3:15 PM
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.
Thursday, January 12, 2023
Friday, January 13, 2023
Saturday, January 14, 2023
Sunday, January 15, 2023
Monday, January 16, 2023
Tuesday, January 17, 2023
Wednesday, January 18, 2023
Thursday, January 19, 2023
Friday, January 20, 2023
Saturday, January 21, 2023
Sunday, January 22, 2023
Monday, January 23, 2023
Tuesday, January 24, 2023
Wednesday, January 25, 2023
Thursday, January 26, 2023
Friday, January 27, 2023
Saturday, January 28, 2023
Sunday, January 29, 2023
Monday, January 30, 2023
Tuesday, January 31, 2023
Wednesday, February 1, 2023
Thursday, February 2, 2023
Friday, February 3, 2023
Saturday, February 4, 2023
Sunday, February 5, 2023
Monday, February 6, 2023
Tuesday, February 7, 2023
Wednesday, February 8, 2023
Thursday, February 9, 2023
Friday, February 10, 2023
Saturday, February 11, 2023
Sunday, February 12, 2023
Monday, February 13, 2023
Tuesday, February 14, 2023
Wednesday, February 15, 2023
11:30 AM
MCQM PhD Lecture: Léo Morin (Aarhus University)
MCQM PhD Lecture: Léo Morin (Aarhus University)
11:30 AM - 12:30 PM
Title and Abstract: TBA
2:15 PM
MCQM Seminar: San Vũ Ngọc
MCQM Seminar: San Vũ Ngọc
2:15 PM - 3:15 PM
Title and Abstract: TBA
Thursday, February 16, 2023
Friday, February 17, 2023
Saturday, February 18, 2023
Sunday, February 19, 2023
Monday, February 20, 2023
Tuesday, February 21, 2023
Wednesday, February 22, 2023
Thursday, February 23, 2023
Friday, February 24, 2023
Saturday, February 25, 2023
Sunday, February 26, 2023
Monday, February 27, 2023
Tuesday, February 28, 2023
Wednesday, March 1, 2023
Thursday, March 2, 2023
Friday, March 3, 2023
Saturday, March 4, 2023
Sunday, March 5, 2023
Monday, March 6, 2023
Tuesday, March 7, 2023
Wednesday, March 8, 2023
Thursday, March 9, 2023
Friday, March 10, 2023
Saturday, March 11, 2023
Sunday, March 12, 2023
Monday, March 13, 2023
Tuesday, March 14, 2023
Wednesday, March 15, 2023
11:30 AM
MCQM PhD Lecture: TBA
MCQM PhD Lecture: TBA
11:30 AM - 12:30 PM
2:15 PM
MCQM Seminar: Caroline Lasser
MCQM Seminar: Caroline Lasser
2:15 PM - 3:15 PM
Title and Abstract: TBA
Thursday, March 16, 2023
Friday, March 17, 2023
Saturday, March 18, 2023
Sunday, March 19, 2023
Monday, March 20, 2023
Tuesday, March 21, 2023
Wednesday, March 22, 2023
Thursday, March 23, 2023
Friday, March 24, 2023
Saturday, March 25, 2023
Sunday, March 26, 2023
Monday, March 27, 2023
Tuesday, March 28, 2023
Wednesday, March 29, 2023
Thursday, March 30, 2023
Friday, March 31, 2023
Saturday, April 1, 2023
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