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Technische UniversitÄt Graz
12 - 14 March & 19 - 21 March 2025
title: Boundary value problems and differential operators
abstract: In the lecture series "Boundary Value Problems and Differential Operators" we provide an introduction to extension theory of symmetric operators in Hilbert spaces and its diverse applications to spectral problems for self-adjoint ordinary and partial differential operators. One of the main objectives is to discuss the technique of boundary triples and Weyl functions, and to illustrate these methods for Sturm-Liouville operators, Schrödinger operators, and Dirac operators.
The course consists of the following four lectures:
I. Symmetric and self-adjoint operators in Hilbert spaces
II. Boundary triples and applications to ordinary differential operators
III. Weyl functions, Krein's resolvent formula, and spectral analysis
IV. Quasi boundary triples and applications to partial differential operators
schedule: Wednesday, March 12th, 14.00 - 16.00
Thursday, March 13th, 10:00 - 12:00
Wednesday, March 19th, 14:00 - 16:00
Thursday, March 20th, 10:00 - 12:00
University of Cambridge
12 - 16 May
title: Thermalization of open quantum many-body systems
abstract: The study of thermalization in open quantum systems is a fundamental topic in modern quantum physics, with implications for quantum information, condensed matter, and statistical mechanics.
This course provides a comprehensive introduction to the dynamics of open quantum systems, focusing on how interactions with the environment drive systems toward thermal equilibrium. We will explore key concepts such as quantum master equations, decoherence, and the role of dissipation in quantum evolution. Understanding quantum dissipative evolutions, modelled by local Lindbladians, is key to controlling this noise and enhancing the coherence of quantum memories. Recent advancements in dissipative state engineering, which leverage these evolutions to stabilize quantum states against noise, have shown promise in experimental results. Additionally, rapid decoherence can aid in preparing relevant phases of matter and estimating algorithm runtimes.
In these lectures, we will investigate thermalization in open quantum systems governed by Lindbladians, focusing on the speed of convergence to thermal equilibrium. We will derive conditions for rapid mixing and briefly review how to utilize these findings to develop efficient algorithms to prepare Gibbs states.
schedule: Monday, May 12th, 14.00 - 16.00
Wednesday, May 14th, 10:00 - 12:00
Thursday, May 15th, 14:00 - 16:00
venue: Politecnico di Milano, Via Bonardi 9, Dipartimento di Matematica, Ed. 14 "Nave", Aula Seminari III piano.
LECTURE NOTES: link.
Karlsruher Institut für Technologie
31 March - 11 April 2025
title: Quantum systems at the Brink
abstract: For quantum systems one usually distinguishes between bound states energies and the continuous spectrum. The continuous spectrum is where transport happens, the bound states energies are eigenvalues of the Hamiltonian belonging to the discrete spectrum. Usually the bound state eigenvalues are well below the continuous spectrum, with a safety distance to the continuous (or, more precisely, the essential) spectrum. In this case usual perturbation theory works and there are simple criteria for the existence or non-existence of bound states.
Now imagine you have a quantum system which depends on a parameter and you can tune this parameter such that the ground state energy hits the edge of the continuous spectrum at some critical value. Does the ground state survive or does it disappear at the critical parameter? I.e., does the ground state gets wider and wider and it disappears in the limit or does it stay finite and then explodes, if one varies the parameter just a tiny bit past its critical value?
How can one decide between these scenarios? How does one prove the existence of a ground state at the critical parameter value, when the ground state energy is at the edge of the continuous spectrum? This is obviously a highly unstable situation where perturbation theory does not apply since there is no safety distance of the ground state energy to the continuous spectrum anymore when the parameter is critical.
What is the physical mechanism which stabilizes a quantum system so that its ground state is still bound at the critical parameter?
Can one get decay estimates for the ground state even though the usual approaches to this type of questions usually need a safety distance of the ground state energy to the continuous spectrum?
In this lecture series, we develop the necessary tools to study these types of questions.
schedule: Monday, March 31st, 14.00 - 16.00
Thursday, April 3rd, 10:00 - 12:00
Friday, April 4th, 14:00 - 16:00
Monday, April 7th, 14:00 - 16:00 (NEW DATE)
Tuesday, April 8th, 14:00 - 16:00
Thursday, April 10th, 10:00 - 12:00
Uppsala Universitet
28 April - 7 May 2025
title: Mathematics of the 2D anyon gas
abstract: In the theory of quantum statistics, if one follows mathematical logic to its conclusion, one reaches the possibility of intermediate exchange statistics and "anyons", i.e. identical particles different from bosons and fermions. Over the course of about 50 years this topic has evolved from merely an exotic possibility to an almost inevitability when orientation symmetry is broken, such as in effectively two-dimensional systems subject to rotation or an external magnetic field. The signature example is the fractional quantum Hall effect, and in just the last few years very strong signatures of individual anyons have finally arrived from experiments. However, the many-body theory necessary to study precise collective properties of anyons has remained rather undeveloped until relatively recently. This mini course will focus on the mathematics of the many-anyon gas, introduce some of the main concepts involved, and thus provide a foundation for further exploration of the topic, starting from the toy model of ideal anyons, to more realistic emergent models, and also promising applications to quantum computing.
Lecture plan:
I. Quantum statistics & transmutation
II. Local exclusion & stability
III. The almost-bosonic interacting anyon gas
IV. Emergent models: FQHE & polarons
V. Non-abelian anyons & topological quantum computing
schedule: Monday, April 28th, 14.00 - 16.00
Tuesday, April 29th, 10:00 - 12:00
Wednesday, April 30, 14:00 - 16:00
Monday, May 5th, 14:00 - 17:00
Wednesday, May 7th, 14:00 - 17:00
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