In this talk, I will discuss how Maxwell's equations emerge as an effective description of non-relativistic quantum electrodynamics (QED). Specifically, I will consider the spinless Pauli-Fierz Hamiltonian in a semiclassical mean-field limit with many fermions. I will present recent results showing that, in the large-particle-number limit and in the trace norm topology of reduced density matrices, the system's dynamics can be approximated by a fermionic variant of the Maxwell-Schrödinger equations as well as by the non-relativistic Vlasov--Maxwell system for extended charges. In both cases, the electromagnetic field behaves classically and satisfies Maxwell's equations. This talk is based on the preprints arXiv:2411.07085 and arXiv:2308.16074, the latter being a joint work with Chiara Saffirio.

We give an answer to a previously open question due to Bucur, Freitas and Kennedy on the asymptotics of the principal Laplacian eigenvalue for large negative Robin parameters on bounded Lipschitz domains. We answer the question negatively by constructing an explicit counterexample. This is joint work with Konstantin Pankrashkin.

The Thouless theory of quantum pumps establishes quantized transport per cycle and determines the conditions for that. When the description is shifted to a moving frame, as prompted by recent experiments on ultracold gases, transported and residing charges mix. Their transformation is encoded in Galilean space and time, but underlying it is one of vector bundles that may be described in a number of ways, that may or may not rely on Bloch theory. In one of them, the transformation mixes strong and weak topological indices of a bundle on a 2-torus; in another one, a 3-torus is at center stage. (Joint work with T. Esslinger and F. Santi)

In 1957, Huang and Yang predidcted an asymptotic formula for the ground state energy of a dilute Fermi gas in the thermodynamic limit up to the third order in the expansion. This formula highlights a remarkable universality, showing that the correlation energy depends solely on the interactions scattering length. In a joint work with E. Giacomelli, P. T. Nam and Robert Seiringer we give a proof of this conjecture.

The emergence of relativistic quantum mechanics has been one of the most remarkable developments in quantum physics over the past decades. Since the early work of Dirac, relativity has always been a part of the quantum physical picture. While non-relativistic theories are highly successful in describing quantum systems where particles move at speeds much slower than the speed of light $c$, relativistic effects can become significant in high-precision calculations. Now it became clear that relativistic effects had an essential influence on a number of physical and chemical properties. In this talk, we discuss recent advances on the study of relativistic effect in Dirac-Fock ground-state energy. We prove that the error bound between Dirac-Fock ground-state energy and its non-relativistic counterpart (i.e., Hartree—Fock ground-state energy) is of the order $O(c^{-2})$, and, when the potential between electrons and nuclei is regular, we obtain the well-known leading order relativistic correction term (i.e. the sum of the mass-velocity term, the Darwin term and the spin-orbit term).

We prove matching upper and lower bounds for the transition density of Hardy perturbations of subordinated Bessel heat kernels. The analysis relies on suitably constructed supermedian functions, in particular ground states. These heat kernel bounds provide new, and likely optimal, upper bounds for the ground state density of relativistic atoms close to the nucleus, in each angular momentum channel separately. This talk is based on joint works with Krzysztof Bogdan and Tomasz Jakubowski.

The Hartree and Hartree-Fock equations are fundamental tools in atomic and molecular physics, providing effective descriptions of large fermionic systems. In this talk, we present a rigorous derivation of the time-dependent Hartree equations from the microscopic evolution of N fermions, considering initial states that occupy a volume of order one. Unlike previous results, where the derivation was closely linked to a semiclassical limit, we impose neither a semiclassical scaling in the Hamiltonian nor a semiclassical structure on the initial states. This talk is based on joint work with D.V. Hoang and P. Pickl.

In this talk, I will talk about a recent work in the derivation of a system for two-species fermions with N particles. More precisely, we establish a two-species Husimi measure and construct a two-species Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy with error terms. We provide proofs of the smallness of these errors in the semiclassical and mean-field limits, fixing $\hbar = N - 1/3$. After that, we rigorously establish the uniqueness of this hierarchy. This allows us to conclude that the two-species Husimi measure from the solution of $N$-particle Schrödinger equation approximates the solution of the Vlasov equation for two-species systems. The talk is based on a joint work with Hongshuo Chen, Jinyeop Lee, and Zhiwei Sun.