We investigate the relations between normalized critical points of the nonlinear Schrödinger energy functional and critical points of the corresponding action functional on the associated Nehari ...manifold. Our first general result is that the ground state levels are strongly related by the following duality result: the (negative) energy ground state level is the Legendre–Fenchel transform of the action ground state level. Furthermore, whenever an energy ground state exists at a certain frequency, then all action ground states with that frequency have the same mass and are energy ground states too. We prove that the converse is in general false and that the action ground state level may fail to be convex. Next we analyze the differentiability of the ground state action level and we provide an explicit expression involving the mass of action ground states. Finally we show that similar results hold also for local minimizers.
We realize a phase-sensitive closed-loop control scheme to engineer the fluctuations of the pump field which drives an optomechanical system and show that the corresponding cooling dynamics can be ...significantly improved. In particular, operating in the counterintuitive "antisquashing" regime of positive feedback and increased field fluctuations, sideband cooling of a nanomechanical membrane within an optical cavity can be improved by 7.5 dB with respect to the case without feedback. Close to the quantum regime of reduced thermal noise, such feedback-controlled light would allow going well below the quantum backaction cooling limit.
We consider the minimization of the NLS energy on a metric tree, either rooted or unrooted, subject to a mass constraint. With respect to the same problem on other types of metric graphs, several new ...features appear, such as the existence of minimizers with positive energy, and the emergence of unexpected threshold phenomena. We also study the problem with a radial symmetry constraint that is in principle different from the free problem due to the failure of the Pólya–Szegő inequality for radial rearrangements. A key role is played by a new Poincaré inequality with remainder.
We compare ground states for the nonlinear Schrödinger equation on metric graphs, defined as global minimizers of the action functional constrained on the Nehari manifold, and least action solutions, ...namely minimizers of the action among all solutions to the equation. In principle, four alternative cases may take place: ground states do exist (thus coinciding with least action solutions); ground states do not exist while least action solutions do; both ground states and least action solutions do not exist and the levels of the two minimizing problems coincide; both ground states and least action solutions do not exist and the levels of the two minimizing problems are different. We show that in the context of metric graphs all four alternatives do occur. This is accomplished by a careful analysis of doubly constrained variational problems. As a by-product, we obtain new multiplicity results for positive solutions on a wide class of noncompact metric graphs.
We consider the nonlinear Schrödinger equation with pure power nonlinearity on a general compact metric graph, and in particular its stationary solutions with fixed mass. Since the the graph is ...compact, for every value of the mass there is a constant solution. Our scope is to analyze (in dependence of the mass) the variational properties of this solution, as a critical point of the energy functional: local and global minimality, and (orbital) stability. We consider both the subcritical regime and the critical one, in which the features of the graph become relevant. We describe how the above properties change according to the topology and the metric properties of the graph.
De la discorde à l’entente Serra, Enrico
Éditions du Comité des travaux historiques et scientifiques,
11/2023
Book
Odprti dostop
Camille Barrère fut ambassadeur de France à Rome pendant un quart de siècle (de 1897 à 1924) : c’était un temps où les ambassadeurs n’étaient pas de simples exécutants des instructions du Quai ...d’Orsay, mais de vrais conseillers des Ministres, bien plus éphémères dans leurs fonctions. Son rôle dans les relations internationales ne fut pas mince puisqu’il contribua à éviter que l’Italie entre en guerre aux côtés de l’Allemagne et de l’Autriche-Hongrie et à faire qu’elle rejoigne la Triple Entente. Sur son action, l’auteur Enrico Serra qui partageait avec Barrère sa passion du journalisme, écrivit en 1950 un livre d’histoire diplomatique à la fois savant et élégant. Utilisant toutes les sources alors disponibles ainsi que plusieurs archives privées, l’auteur arrive à des conclusions qui ont été confirmées par les recherches les plus récentes. Il nous offre une œuvre pleine de perspicacité et de sympathie pour l’hôte du Palais Farnèse, tout en ne cachant pas les foucades de ce Républicain à la fois épris de la grandeur de la France et attaché au rapprochement des sœurs latines. Par amitié pour Serra (avec lequel il avait fondé et animé le Comité franco-italien d’études historiques), et par admiration pour Barrère, ce diplomate exceptionnel qui avait tenu tête à Clemenceau, Jean-Baptiste Duroselle avait entrepris la traduction de ce livre. Plus d’un demi-siècle plus tard, voici donc cette étude enfin disponible en français.
We prove a conjecture by De Giorgi, which states that global weak solutions of nonlinear wave equations such as □w + |w| p-2 w = 0 can be obtained as limits of functions that minimize suitable ...functionals of the calculus of variations. These functionals, which are integrals in space-time of a convex Lagrangian, contain an exponential weight with a parameter ε, and the initial data of the wave equation serve as boundary conditions. As ε tends to zero, the minimizers v ε converge, up to subsequences, to a solution of the nonlinear wave equation. There is no restriction on the nonlinearity exponent, and the method is easily extended to more general equations.
We study the Dirichlet problem in a ball for the Henon equation with critical growth and we establish, under some conditions, the existence of a positive, non radial solution. The solution is ...obtained as a minimizer of the quotient functional associated to the problem restricted to appropriate subspaces of H01 invariant for the action of a subgroup of O(N). Analysis of compactness properties of minimizing sequences and careful level estimates are the main ingredients of the proof. PUBLICATION ABSTRACT
Phenomenological models aiming to join gravity and quantum mechanics often predict effects that are potentially measurable in refined low-energy experiments. For instance, modified commutation ...relations between position and momentum, that account for a minimal scale length, yield a dynamics that can be codified in additional Hamiltonian terms. When applied to the paradigmatic case of a mechanical oscillator, such terms, at the lowest order in the deformation parameter, introduce a weak intrinsic nonlinearity and, consequently, deviations from the classical trajectory. This point of view has stimulated several experimental proposals and realizations, leading to meaningful upper limits to the deformation parameter. All such experiments are based on classical mechanical oscillators, i.e., excited from a thermal state. We remark indeed that decoherence, that plays a major role in distinguishing the classical from the quantum behavior of (macroscopic) systems, is not usually included in phenomenological quantum gravity models. However, it would not be surprising if peculiar features that are predicted by considering the joined roles of gravity and quantum physics should manifest themselves just on purely quantum objects. On the basis of this consideration, we propose experiments aiming to observe possible quantum gravity effects on macroscopic mechanical oscillators that are preliminary prepared in a high purity state, and we report on the status of their realization.
Graphical abstract
In this work, we present an Opto-Electro-Mechanical Modulator (OEMM) for RF-to-optical transduction realized via an ultra-coherent nanomembrane resonator capacitively coupled to an rf injection ...circuit made of a microfabricated read-out able to improve the electro-optomechanical interaction. This device configuration can be embedded in a Fabry-Perot cavity for electromagnetic cooling of the LC circuit in a dilution refrigerator exploiting the opto-electro-mechanical interaction. To this aim, an optically measured steady-state frequency shift of 380 Hz was seen with a polarization voltage of 30 V and a
-factor of the assembled device above 106 at room temperature. The rf-sputtered titanium nitride layer can be made superconductive to develop efficient quantum transducers.
PDF
