Program

Invited Speakers:

Emanuela L. Giacomelli (University of Milan): Correlation energy of the dilute spin-1/2 Fermi gas 

Abstract: In 1957, Huang and Yang predicted an asymptotic expansion for the ground state energy of a dilute Fermi gas in the thermodynamic limit, accurate up to third order. Their formula revealed remarkable universality: the correlation energy depends solely on the scattering length of the interaction, regardless of the potential's specific details.

Establishing the Huang–Yang formula requires a precise analysis of the correlation energy, which is defined as the difference between the ground state energy and that of the free Fermi gas.

In this series of lectures, we will introduce the necessary mathematical tools to carry out this analysis for a dilute gas of spin-1/2 fermions. We outline the strategy for proving matching upper and lower bounds on the ground state energy. For the upper bound, we construct suitable trial states via unitary transformations.  The trial state is constructed using an adaptation of the bosonic Bogoliubov theory to the Fermi system, where the correlation structure of fermionic particles is incorporated by quasi-bosonic Bogoliubov transformations. In the latter, it is important to consider a modified zero-energy scattering equation that takes into account the presence of the Fermi sea, in the spirit of the Bethe–Goldstone equation.

For the lower bound, rather than attempting to diagonalise the Hamiltonian via unitary operators, we employ a version of the “completing the square” method. More precisely, we decompose the correlation Hamiltonian into non-negative terms, together with the relevant constant and controllable error contributions.

Vojkan Jakšić (Politecnico di Milano): On the Minus First Law of Thermodynamics for Quantum Spin Systems

Abstract: The Minus First Law states that “an isolated system, left to itself, will evolve toward a state of thermodynamic equilibrium”. In these lectures, I will provide a pedagogical introduction to the subject and describe results obtained recently in collaboration with Anna Szczepanek, Claude-Alain Pillet, and Clement Tauber.

Prologue: What is the Minus First Law?

Talk on Remarks on Approach to Equilibrium in Quantum Spin Systems @ Rurgers University

Benjamin Schlein (University of Zurich): Bogoliubov theory in many-body quantum mechanics 

Abstract: In this mini-course, I am going to discuss some recent progress in the mathematical analysis of many-body quantum systems. I will present a recently developed rigorous version of Bogoliubov theory and I will explain how it can be used to study equilibrium and non-equilibrium properties of quantum gases.

First, I will consider a system of N bosons confined in a volume of order one in the so-called mean-field regime. In this limit, I will derive precise estimates on the ground state energy and on the low-energy excitation spectrum of the Hamilton operator.

Furthermore, I will discuss how to approximate the many-body time-evolution through an effective dynamics for the Bose-Einstein condensate and a quadratic evolution for its orthogonal excitations.

Afterwards, I will switch to a more realistic but mathematically more challenging regime, known as the Gross-Pitaevskii regime, in which N particles interact through a repulsive potential with scattering length of the order 1/N. I will discuss some mathematical tools that have been developed in the last years to control correlations among particles, that are crucial in this limit, and to obtain precise estimates on the low-energy spectrum.

Finally, I will explain how similar techniques can also be applied to infer precise estimates on the ground state energy of dilute Bose gases in the thermodynamic limit, at fixed but small density. 

Katharina Schratz (Sorbonne University): Resonances as a Computational Tool 

Abstract: A large toolbox of numerical schemes for dispersive equations has been established, based on different discretization techniques such as discretizing the variation-of-constants formula (e.g., exponential integrators) or splitting the full equation into a series of simpler subproblems (e.g., splitting methods). In many situations these classical schemes allow a precise and efficient approximation. This, however, drastically changes whenever non-smooth phenomena enter the scene such as for problems at low regularity and high oscillations. Classical schemes fail to capture the oscillatory nature of the solution, and this may lead to severe instabilities and loss of convergence. In this lecture I present a new class of resonance based schemes. The key idea in the construction of the new schemes is to tackle and deeply embed the underlying nonlinear structure of resonances into the numerical discretization. As in the continuous case, these terms are central to structure preservation and offer the new schemes strong geometric properties at low regularity.