Resumen: En esta charla estudiamos el operador de Dirac tridimensional con condiciones de borde de tipo MIT en un tubo delgado retorcido, donde la sección transversal  es transportada y rotada a lo largo de una recta según una función de torsión. Comenzaremos estableciendo la auto-adjunción del operador y describiendo su espectro esencial. El resultado central es la construcción de un operador efectivo unidimensional que rige la dinámica cuando el grosor  del tubo tiende a cero.

To derive an equation describing the distribution of a dilute gas of identical particles, Boltzmann assumed that the many-particle probability density function

stays close to a product for all time, provided it is a product initially; this'ld imply that in a gas of many particles, the position-velocity of the particles stay independent. 

Mark Kac introduced a probabilistic space-homogeneous model for a gas of N particles, for which Boltzmann’s assumption holds in the limit as $N\to\infty$:

For fixed time $t$, and any $k$ fixed, the $k$ marginal distribution of the $N$ particle density functions  becomes a product for large $N$. This is known as chaos (or molecular chaos), its propagation in time is known as propagation of chaos.

We retake the question, first asked in [CCLLV] For which probability distibutions g on R can we produce a sequence $\{fN\}_{N\geq 1}$, of probability distributions, with fN supported on the energy sphere $\{v1^2+v2^2+...+vN^2=N\}$ which are chaotic to $g$ ?

Using rescaled states, we expand the the class of admissible g, obtained in

[CCLLV]. We also mention some new ideas in this direction. 

References: 

Carlen, Eric A., et al. Entropy and chaos in the Kac model. Kinetic and Related Models 3.1 (2010): 85-122.

Cortez, Roberto, and Hagop Tossounian. Chaos for rescaled measures on Kac’s sphere. Electronic Journal of Probability 28 (2023): 1-29.

We consider the Dirac operator with mass perturbed by a nonnegative potential in some Lebesgue space, and study the problem of minimizing the lowest eigenvalue in the spectral gap under a constraint on the corresponding norm. This is the Dirac analogue of the classical Keller problem for Schrödinger operators. When the potential is of electrostatic type, we showed that solutions exist and are characterized by a nonlinear Dirac equation with Kerr-type nonlinearity. In the talk, I will outline the main strategies and differences for the electrostatic case and work in progress for potentials of mass type, which give rise to quite different challenges but open a road to a variational characterization of solitary waves for the Soler model. 

This is joint work Jean Dolbeault, David Gontier and Fabio Pizzichillo.

Resolvent estimates play a fundamental role in scattering and resonance theory, where they provide quantitative information on wave propagation, local energy decay, and the distribution of resonances. In the semiclassical regime, the growth of the resolvent is closely related to the properties of the underlying classical dynamics and has been the subject of extensive study for Schrödinger operators. 

In this talk, I will present new results on the semiclassical resolvent of magnetic Dirac operators. First, I will establish an exponential upper bound for the resolvent norm, without imposing any assumption on the underlying classical dynamics. This estimate provides a general control of the resolvent in the semiclassical regime. I will then discuss the influence of trapping on the cutoff resolvent. More precisely, I will show that the presence of trapped classical trajectories implies a logarithmic lower bound on the cutoff resolvent. This extends to the relativistic Dirac setting a classical phenomenon known for semiclassical Schrödinger operators, namely the existence of a logarithmic gap between non-trapping and trapping regimes. The proof of this result is based on the long-time propagation of coherent states (up to the Ehrenfest time). 

We present some recent results on PDE systems modeling the dynamics of suspension bridges. The structure is represented as a fish-bone plate with intermediate piers, consisting of a degenerate rectangular planar domain where a central beam moves vertically and a continuum of cross-sections rotate around their midpoints; the piers are modeled as rigidly fixed sections. The dynamics is then governed by a nonlinear, nonlocal coupled system of beam-wave equations describing the evolution of vertical and torsional oscillations, possibly accounting for damping effects and external forces. After establishing a rigorous functional framework and introducing a suitable notion of weak solutions, we address the long-term behavior of the system. Specifically, we discuss the dissipativity of the underlying dynamical system and investigate the existence, uniqueness, and global stability of stationary and periodic solutions, both for purely vertical and fully coupled vertical/torsional motions. We show in particular that the system may exhibit a rich and unpredictable dynamics, characterized by a global attractor with a nontrivial structure.

The talk is based on joint work with Maurizio Garrione and Filippo Gazzola (Politecnico di Milano).

Los modelos quantum double de Kitaev forman una familia prototípica de modelos exactamente solubles que exhiben orden topológico y excitaciones tipo anyón. En esta charla me centraré en describir cómo estos modelos pueden formularse de manera natural dentro del marco de las C*-álgebras, lo que permite una descripción algebraica robusta de sus observables y de las excitaciones. Finalmente, calcularé el conjunto completo de estados KMS del modelo quantum double para el caso abeliano.

We study the spectral properties of the Dirac operator $L_0$ obtained by linearizing the one-dimensional Soler model around standing waves with power nonlinearity $f(s)=s|s|^{p-1}$, $p>0$. We give a sharp characterization of the spectral gap. If $p\ge1$, the gap contains no eigenvalues other than the symmetry-induced energies $-2\omega$ and $0$. If $0<p<1$, additional eigenvalues bifurcate from the thresholds of the essential spectrum and enter the gap. We further prove that the thresholds are never eigenvalues for any $p>0$ and that the thresholds are not resonances for $p>1$.

Atomic inertial sensors constitute highly controllable and scalable platforms for precision measurements of inertial effects, particularly acceleration. In this talk, I will present and analyze a theoretical proposal for an atomtronic angular accelerometer based on an angularly ac?shaken ring lattice. I will show how supercurrents of ultracold atoms circulating in the ring can be harnessed to achieve high?precision measurements of angular acceleration.

In this system, a significant net atomic current emerges when the lattice driving frequency is tuned to integer fractions of the system’s Bloch frequency. These resonances give rise to directed transport, with the resulting supercurrents encoding information about the external angular acceleration.

Within the Bose–Hubbard model, I will discuss  two main regimes: the single-particle regime and the weakly interacting many-body regime. In the single-particle limit, I will demonstrate analytically that the resonance width scales inversely with the measurement time, thereby imposing a Fourier limited bound on the achievable precision in angular-acceleration estimation. By contrast, weak onsite interactions modify this behavior. Numerical simulations will show that interactions can lead to a pronounced sharpening of the resonances. As a result, the sensitivity to the angular acceleration can surpass the Fourier limited scaling of the non-interacting case, improving the measurement precision by several orders of magnitude.

We explore examples of Dirac operators on bounded domains exhibiting an interval of essential spectrum. In particular, we consider three-dimensional Dirac operators on Lipschitz domains with critical electrostatic and Lorentz scalar shell interactions supported on a compact smooth surface. Unlike typical bounded-domain settings where the spectrum is purely discrete, the criticality of these interactions can generate a nontrivial essential spectrum interval, whose position and length are explicitly controlled by the coupling constants and surface curvatures.

Based on joint work with J. Behrndt (TU Graz), M. Holzmann (TU Graz), and K. Pankrashkin (Univ. Oldenburg).