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We consider the dilute Fermi gas in three dimensions interacting through a positive, radially symmetric, compactly supported and integrable potential in the thermodynamic limit. We establish a second order lower bound for the ground state energy density with an error term which is optimal in the sense that it matches the order of the next correction term conjectured by Huang-Yang in 1957., Comment: 38 pages

We study the ground state energy of a gas of spin $1/2$ fermions with repulsive short-range interactions. We derive an upper bound that agrees, at low density $\rho$, with the Huang-Yang conjecture. The latter captures the first three terms in an asymptotic low-density expansion, and in particular the Huang-Yang correction term of order $\rho^{7/3}$. Our 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., Comment: 48 pages

It is a well known fact that the geometry of a superconducting sample influences the distribution of the surface superconductivity for strong applied magnetic fields. For instance, the presence of corners induces geometric terms described through effective models in sector-like regions. We study the connection between two effective models for the offset of superconductivity and for surface superconductivity introduced in \cite{BNF} and \cite{CG2}, respectively. We prove that the transition between the two models is continuous with respect to the magnetic field strength, and, as a byproduct, we deduce the existence of a minimizer at the threshold for both effective problems. Furthermore, as a consequence, we disprove a conjecture stated in \cite{CG2} concerning the dependence of the corner energy on the angle close to the threshold., Comment: 24 pages, pdfLaTeX

Complex networked systems in fields such as physics, biology, and social sciences often involve interactions that extend beyond simple pairwise ones. Hypergraphs serve as powerful modeling tools for describing and analyzing the intricate behaviors of systems with multi-body interactions. Herein, we investigate a discrete-time nonlinear averaging dynamics with three-body interactions: an underlying hypergraph, comprising triples as hyperedges, delineates the structure of these interactions, while the vertices update their states through a weighted, state-dependent average of neighboring pairs' states. This dynamics captures reinforcing group effects, such as peer pressure, and exhibits higher-order dynamical effects resulting from a complex interplay between initial states, hypergraph topology, and nonlinearity of the update. Differently from linear averaging dynamics on graphs with two-body interactions, this model does not converge to the average of the initial states but rather induces a shift. By assuming random initial states and by making some regularity and density assumptions on the hypergraph, we prove that the dynamics converges to a multiplicatively-shifted average of the initial states, with high probability. We further characterize the shift as a function of two parameters describing the initial state and interaction strength, as well as the convergence time as a function of the hypergraph structure., Comment: To appear in SIAM Journal on Applied Dynamical Systems

In a system of interacting fermions, the correlation energy is defined as the difference between the energy of the ground state and the one of the free Fermi gas. We consider $N$ interacting spin $1/2$ fermions in the dilute regime, i.e., $\rho\ll 1$ where $\rho$ is the total density of the system. We rigorously derive a first order upper bound for the correlation energy with an optimal error term of the order $\mathcal{O}(\rho^{7/3})$ in the thermodynamic limit. Moreover, we improve the lower bound estimate with respect to previous results getting an error $\mathcal{O}(\rho^{2+1/5})$., Comment: 49 pages

We consider the Neumann realization of the magnetic laplacian in $\mathbb{R}^3_+$, in the case in which the magnetic field has a piecewise constant strength and a uniform direction. This operator is expected to be an effective model in studying the threshold to the normal state for a 3D superconductor exposed to a discontinuous magnetic field. We review some recent results above the infimum of the spectrum of the aforementioned operator., Comment: 11 pages, contribution to the Proceedings of the Intensive Period "INdAM Quantum Meetings (IQM22)" at Politecnico di Milano, March - May 2022

In a dilute system of N fermions with spin 1/2 in three dimensions, we study the correlation energy which is given by the difference between the ground state energy and the energy of the non-interacting Fermi sea state. We review some recent results from the paper by Falconi et al. [Ann. Henri Poincar\'e 22 (2021), 2283-2353], where we make use of the almost-bosonic nature of the low-energy excitations of the system which can be effectively described by Bogoliubov theory., Comment: 14 pages, contribution to the Proceedings of the Intensive Period "INdAM Quantum Meetings (IQM22)" at Politecnico di Milano, March - May 2022

We study a system of $ N $ trapped bosons in the Thomas-Fermi regime with an interacting pair potential of the form $ g_N N^{3\beta-1} V(N^\beta x) $, for some $ \beta\in(0,1/3) $ and $ g_N $ diverging as $ N \to \infty $. We prove that there is complete Bose-Einstein condensation at the level of the ground state and, furthermore, that, if $ \beta\in (0,1/6) $, condensation is preserved by the time evolution.

We define a Schr\"odinger operator on the half-space with a discontinuous magnetic field having a piecewise-constant strength and a uniform direction. Motivated by applications in the theory of superconductivity, we study the infimum of the spectrum of the operator. We give sufficient conditions on the strength and the direction of the magnetic field such that the aforementioned infimum is an eigenvalue of a reduced model operator on the half-plane. We use the Schr\"odinger operator on the half-space to study a new semiclassical problem in bounded domains of the space, considering a magnetic Neumann Laplacian with a piecewise-constant magnetic field. We then make precise the localization of the semiclassical ground state near specific points at the discontinuity jump of the magnetic field., Comment: 37 pages, 7 figures

Type-II superconductivity is known to persist close to the sample surface in presence of a strong magnetic field. As a consequence, the ground state energy in the Ginzburg-Landau theory is approximated by an effective one-dimensional model. As shown in [CG2], the presence of corners on the surface affects the energy of the sample with a non-trivial contribution. In [CG2], the two-dimensional model problem providing the corner energy is implicitly identified and, although no explicit dependence of the energy on the corner opening angle is derived, a conjecture about its form is proposed. We study here such a conjecture and confirm it, at least to leading order, for corners with almost flat opening angle., Comment: 22 pages, pdfLaTex

We study the ground state properties of interacting Fermi gases in the dilute regime, in three dimensions. We compute the ground state energy of the system, for positive interaction potentials. We recover a well-known expression for the ground state energy at second order in the particle density, which depends on the interaction potential only via its scattering length. The first proof of this result has been given by Lieb, Seiringer and Solovej. In this paper we give a new derivation of this formula, using a different method; it is inspired by Bogoliubov theory, and it makes use of the almost-bosonic nature of the low-energy excitations of the systems. With respect to previous work, our result applies to a more regular class of interaction potentials, but it comes with improved error estimates on the ground state energy asymptotics in the density., Comment: 52 pages

We study the Ginzburg-Landau functional describing an extreme type-II superconductor wire with cross section with finitely many corners at the boundary. We derive the ground state energy asymptotics up to $ o(1) $ errors in the surface superconductivity regime, i.e., between the second and third critical fields. We show that, compared to the case of smooth domains, each corner provides an additional contribution of order $ \mathcal{O}(1) $ depending on the corner opening angle. The corner energy is in turn obtained from an implicit model problem in an infinite wedge-like domain with fixed magnetic field. We also prove that such an auxiliary problem is well-posed and its ground state energy bounded and, finally, state a conjecture about its explicit dependence on the opening angle of the sector., Comment: revised version, presentation shortened and proof arguments streamlined; 54 pages, pdfLaTex