Name: Mathematical Challenges in Quantum Mechanics - Online Seminars
Start: 2022-11-09T11:00:00+01:00
End: 2023-05-24T13:00:00+02:00
Location: Online

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
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Wednesday, November 16, 2022
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Monday, November 21, 2022
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Monday, November 28, 2022
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Monday, December 5, 2022
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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
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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
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Monday, May 22, 2023
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