Scattering from local deformations of a semitransparent plane

We study scattering for the couple $(A_{F},A_{0})$ of Schr\"odinger operators in $L^2(\mathbb{R}^3)$ formally defined as $A_0 = -\Delta + \alpha\, \delta_{\pi_0}$ and $A_F = -\Delta + \alpha\, \delta_{\pi_F}$, $\alpha >0$, where $\delta_{\pi_F}$ is the Dirac $\delta$-distribution supported on the deformed plane given by the graph of the compactly supported, Lipschitz continuous function $F:\mathbb{R}^{2}\to\mathbb{R}$ and $\pi_{0}$ is the undeformed plane corresponding to the choice $F\equiv 0$. We provide a Limiting Absorption Principle, show asymptotic completeness of the wave operators and give a representation formula for the corresponding Scattering Matrix $S_{F}(\lambda)$. Moreover we show that, as $F\to 0$, $\|S_{F}(\lambda)-\mathsf 1\|^{2}_{\mathfrak{B}(L^{2}({\mathbb S}^{2}))}={\mathcal O}\!\left(\int_{\mathbb{R}^{2}}d\textbf{x}|F(\textbf{x})|^{\gamma}\right)$, $0<\gamma<1$. We correct a minor mistake in the computation of the scattering matrix, occurring in the published version of this paper (see J. Math. Anal. Appl. 473(1) (2019), pp. 215-257). The mistake was in Section 7, and affected the statement of Corollary 7.2, specifically, Eq. (7.8). Regrettably the formula for $S_F$ in the Corrigendum J. Math. Anal. Appl. 482(1) (2020), 123554, still contains a misprint, the correct expression is the one given here.Comment: We corrected a minor mistake in the computation of the scattering matri

The semi-classical limit with a delta-prime potential

We consider the quantum evolution $e^{-i\frac{t}{\hbar}H_{\beta}} \psi_{\xi}^{\hbar}$ of a Gaussian coherent state $\psi_{\xi}^{\hbar}\in L^{2}(\mathbb{R})$ localized close to the classical state $\xi \equiv (q,p) \in \mathbb{R}^{2}$, where $H_{\beta}$ denotes a self-adjoint realization of the formal Hamiltonian $-\frac{\hbar^{2}}{2m}\,\frac{d^{2}\,}{dx^{2}} + \beta\,\delta'_{0}$, with $\delta'_{0}$ the derivative of Dirac's delta distribution at $x = 0$ and $\beta$ a real parameter. We show that in the semi-classical limit such a quantum evolution can be approximated (w.r.t. the $L^{2}(\mathbb{R})$-norm, uniformly for any $t \in \mathbb{R}$ away from the collision time) by $e^{\frac{i}{\hbar} A_{t}} e^{it L_{B}} \phi^{\hbar}_{x}$, where $A_{t} = \frac{p^{2}t}{2m}$, $\phi_{x}^{\hbar}(\xi) := \psi^{\hbar}_{\xi}(x)$ and $L_{B}$ is a suitable self-adjoint extension of the restriction to $\mathcal{C}^{\infty}_{c}({\mathscr M}_{0})$, ${\mathscr M}_{0} := \{(q,p) \in \mathbb{R}^{2}\,|\,q \neq 0\}$, of ($-i$ times) the generator of the free classical dynamics. While the operator $L_{B}$ here utilized is similar to the one appearing in our previous work [C. Cacciapuoti, D. Fermi, A. Posilicano, The semi-classical limit with a delta potential, Annali di Matematica Pura e Applicata (2020)] regarding the semi-classical limit with a delta potential, in the present case the approximation gives a smaller error: it is of order $\hbar^{7/2-\lambda}$, $0 < \lambda < 1/2$, whereas it turns out to be of order $\hbar^{3/2-\lambda}$, $0 < \lambda < 3/2$, for the delta potential. We also provide similar approximation results for both the wave and scattering operators.Comment: 24 pages. arXiv admin note: text overlap with arXiv:1907.0580

Spin dependent point potentials in one and three dimensions

We consider a system realized with one spinless quantum particle and an array of $N$ spins 1/2 in dimension one and three. We characterize all the Hamiltonians obtained as point perturbations of an assigned free dynamics in terms of some ``generalized boundary conditions''. For every boundary condition we give the explicit formula for the resolvent of the corresponding Hamiltonian. We discuss the problem of locality and give two examples of spin dependent point potentials that could be of interest as multi-component solvable models.Comment: 15 pages, some misprints corrected, one example added, some references modified or adde

Strong electric fields induced on a sharp stellar boundary

Due to a first order phase transition, a compact star may have a discontinuous distribution of baryon as well as electric charge densities, as e.g. at the surface of a strange quark star. The induced separation of positive and negative charges may lead to generation of supercritical electric fields in the vicinity of such a discontinuity. We study this effect within a relativistic Thomas-Fermi approximation and demonstrate that the strength of the electric field depends strongly on the degree of sharpness of the surface. The influence of strong electric fields on the stability of compact stars is discussed. It is demonstrated that stable configurations appear only when the counter-pressure of degenerate fermions is taken into consideration.Comment: 13 pages, 2 figure

Dissipation Function: Nonequilibrium Physics and Dynamical Systems

An exact response theory has recently been developed within the field of Nonequilibrium Molecular Dynamics. Its main ingredient is known as the Dissipation Function, &Omega;. This quantity determines nonequilbrium properties like thermodynamic potentials do with equilibrium states. In particular, &Omega; can be used to determine the exact response of particle systems obeying classical mechanical laws, subjected to perturbations of arbitrary size. Under certain conditions, it can also be used to express the response of a single system, in contrast to the standard response theory, which concerns ensembles of identical systems. The dimensions of &Omega; are those of a rate, hence &Omega; can be associated with the entropy production rate, provided local thermodynamic equilibrium holds. When this is not the case for a particle system, or generic dynamical systems are considered, &Omega; can equally be defined, and it yields formal, thermodynamic-like, relations. While such relations may have no physical content, they may still constitute interesting characterizations of the relevant dynamics. Moreover, such a formal approach turns physically relevant, because it allows a deeper analysis of &Omega; and of response theory than possible in case of fully fledged physical models. Here, we investigate the relation between linear and exact response, pointing out conditions for the validity of the response theory, as well as difficulties and opportunities for the physical interpretation of certain formal results

Data monitoring roadmap. The experience of the Italian Multiple Sclerosis and Related Disorders Register

Introduction Over the years, disease registers have been increasingly considered a source of reliable and valuable population studies. However, the validity and reliability of data from registers may be limited by missing data, selection bias or data quality not adequately evaluated or checked.This study reports the analysis of the consistency and completeness of the data in the Italian Multiple Sclerosis and Related Disorders Register.MethodsThe Register collects, through a standardized Web-based Application, unique patients.Data are exported bimonthly and evaluated to assess the updating and completeness, and to check the quality and consistency. Eight clinical indicators are evaluated.ResultsThe Register counts 77,628 patients registered by 126 centres. The number of centres has increased over time, as their capacity to collect patients.The percentages of updated patients (with at least one visit in the last 24 months) have increased from 33% (enrolment period 2000-2015) to 60% (enrolment period 2016-2022). In the cohort of patients registered after 2016, there were &gt;= 75% updated patients in 30% of the small centres (33), in 9% of the medium centres (11), and in all the large centres (2).Clinical indicators show significant improvement for the active patients, expanded disability status scale every 6 months or once every 12 months, visits every 6 months, first visit within 1 year and MRI every 12 months.ConclusionsData from disease registers provide guidance for evidence-based health policies and research, so methods and strategies ensuring their quality and reliability are crucial and have several potential applications

Supplement: "Localization and broadband follow-up of the gravitational-wave transient GW150914" (2016, ApJL, 826, L13)

This Supplement provides supporting material for Abbott et al. (2016a). We briefly summarize past electromagnetic (EM) follow-up efforts as well as the organization and policy of the current EM follow-up program. We compare the four probability sky maps produced for the gravitational-wave transient GW150914, and provide additional details of the EM follow-up observations that were performed in the different bands

Le recrutement des maîtres de l'enseignement supérieur

Fermi Claudio. Le recrutement des maîtres de l'enseignement supérieur. In: Revue internationale de l'enseignement, tome 49, Janvier-Juin 1905. pp. 251-256

Le recrutement des maîtres de l'enseignement supérieur

Fermi Claudio. Le recrutement des maîtres de l'enseignement supérieur. In: Revue internationale de l'enseignement, tome 49, Janvier-Juin 1905. pp. 251-256