Wave equation with concentrated nonlinearities

In this paper we address the problem of wave dynamics in presence of concentrated nonlinearities. Given a vector field $V$ on an open subset of \CO^n and a discrete set Y\subset\RE^3 with $n$ elements, we define a nonlinear operator $\Delta_{V,Y}$ on L^2(\RE^3) which coincides with the free Laplacian when restricted to regular functions vanishing at $Y$, and which reduces to the usual Laplacian with point interactions placed at $Y$ when $V$ is linear and is represented by an Hermitean matrix. We then consider the nonlinear wave equation $\ddot \phi=\Delta_{V,Y}\phi$ and study the corresponding Cauchy problem, giving an existence and uniqueness result in the case $V$ is Lipschitz. The solution of such a problem is explicitly expressed in terms of the solutions of two Cauchy problem: one relative to a free wave equation and the other relative to an inhomogeneous ordinary differential equation with delay and principal part $\dot\zeta+V(\zeta)$. Main properties of the solution are given and, when $Y$ is a singleton, the mechanism and details of blow-up are studied.Comment: Revised version. To appear in Journal of Physics A: Mathematical and General, special issue on Singular Interactions in Quantum Mechanics: Solvable Model

Rigorous Dynamics and Radiation Theory for a Pauli-Fierz Model in the Ultraviolet Limit

The present paper is devoted to the detailed study of quantization and evolution of the point limit of the Pauli-Fierz model for a charged oscillator interacting with the electromagnetic field in dipole approximation. In particular, a well defined dynamics is constructed for the classical model, which is subsequently quantized according to the Segal scheme. To this end, the classical model in the point limit is reformulated as a second order abstract wave equation, and a consistent quantum evolution is given. This allows a study of the behaviour of the survival and transition amplitudes for the process of decay of the excited states of the charged particle, and the emission of photons in the decay process. In particular, for the survival amplitude the exact time behaviour is found. This is completely determined by the resonances of the systems plus a tail term prevailing in the asymptotic, long time regime. Moreover, the survival amplitude exhibites in a fairly clear way the Lamb shift correction to the unperturbed frequencies of the oscillator.Comment: Shortened version. To appear in J. Math. Phy

Point interactions in acoustics: one dimensional models

A one dimensional system made up of a compressible fluid and several mechanical oscillators, coupled to the acoustic field in the fluid, is analyzed for different settings of the oscillators array. The dynamical models are formulated in terms of singular perturbations of the decoupled dynamics of the acoustic field and the mechanical oscillators. Detailed spectral properties of the generators of the dynamics are given for each model we consider. In the case of a periodic array of mechanical oscillators it is shown that the energy spectrum presents a band structure.Comment: revised version, 30 pages, 2 figure

TOPICS IN ALGEBRAIC SUPERGEOMETRY OVER PROJECTIVE SPACES

In questa tesi vengono studiati alcuni argomenti in supergeometria algebrica, con particolare attenzione al caso in cui le variet\ue0 ridotte delle supervariet\ue0 in esame siano spazi proiettivi complessi $\mathbb{P}^n$. Dopo aver introdotto le definizioni di base e alcune nozioni generali della supergeometria, viene studiata in dettaglio la geometria dei superspazi proiettivi $\mathbb{P}^{n|m}$. In questo contesto, vengono dati risultati sulla struttura e la coomologia dei fasci invertibili, sugli automorfismi e le deformazioni infinitesime. Attenzione speciale \ue8 riservata al caso della supercurva di Calabi-Yau $\mathbb{P}^{1|2}$. In seguito, vengono studiate le variet\ue0 non-projected su $\mathbb{P}^n$ e se ne fornisce una classificazione nel caso la dimensione dispari sia $2$, mostrando che esistono supervariet\ue0 non-projected solamente sulla linea proiettiva $\mathbb{P}^1$ e sul piano proiettivo $\mathbb{P}^2$. In particolare, si dimostra che tutte le supervariet\ue0 non-projected su $\mathbb{P}^2$ sono Calabi-Yau, cio\ue8 hanno fascio Bereziniano banale, ed inoltre sono non proiettive: non possono cio\ue8 essere immerse in un superspazio proiettivo $\mathbb{P}^{n|m}$. Si dimostra, invece, che esse possono sempre essere immerse in super Grassmanniane. In questo contesto, alcune immersioni di supervariet\ue0 non-projected significative vengono realizzate esplicitamente. Infine, \ue8 data una nuova costruzione dei $\Pi$-spazi proiettivi come supervariet\ue0 non-projected connesse al fascio cotangente su $\mathbb{P}^n$.The aim of this thesis is to study some topics in algebraic supergeometry, in particular in the case the supermanifolds have their reduced manifolds given by complex projective spaces $\mathbb{P}^n$. After the main definitions and notions in supergeometry are introduced, the geometry of complex projective superspaces $\mathbb{P}^{n|m}$ is studied in detail. Invertible sheaves and their cohomology, infinitesimal automorphisms and deformations are studied for $\mathbb{P}^{n|m}$. Special attention is paid to the case of the Calabi-Yau supercurve $\mathbb{P}^{1|2}$. The focus is then moved to non-projected supermanifolds over $\mathbb{P}^n$. A complete classification is given in the case the odd dimension is $2$, showing that there exist non-projected supermanifolds only over the projective line $\mathbb{P}^1$ and projective plane $\mathbb{P}^2$. In particular, it is shown that all of the non-projected supermanifolds over $\mathbb{P}^2$ are Calabi-Yau's, i.e.\ they have trivial Berezinian sheaf, and they are all non-projective, i.e.\ they cannot be embedded into any ordinary projective superspace $\mathbb{P}^{n|m}$. Instead, it is shown that there always exist an embedding of these supermanifolds in super Grassmannians, and some meaningful examples are realised explicitly. Finally, a new construction of $\Pi$-projective spaces as non-projected supermanifolds related to the cotangent sheaf over $\mathbb{P}^n$ is given

A Dirac field interacting with point nuclear dynamics

The system describing a single Dirac electron field coupled with classically moving point nuclei is presented and studied. The model is a semi-relativistic extension of corresponding time-dependent one-body Hartree-Fock equation coupled with classical nuclear dynamics, already known and studied both in quantum chemistry and in rigorous mathematical literature. We prove local existence of solutions for data in H\u3c3 with \u3c3 08[1,32[. In the course of the analysis a second new result of independent interest is discussed and proved, namely the construction of the propagator for the Dirac operator with several moving Coulomb singularities

Fast solitons on star graphs

We define the Schr\"odinger equation with focusing, cubic nonlinearity on one-vertex graphs. We prove global well-posedness in the energy domain and conservation laws for some self-adjoint boundary conditions at the vertex, i.e. Kirchhoff boundary condition and the so called $\delta$ and $\delta'$ boundary conditions. Moreover, in the same setting we study the collision of a fast solitary wave with the vertex and we show that it splits in reflected and transmitted components. The outgoing waves preserve a soliton character over a time which depends on the logarithm of the velocity of the ingoing solitary wave. Over the same timescale the reflection and transmission coefficients of the outgoing waves coincide with the corresponding coefficients of the linear problem. In the analysis of the problem we follow ideas borrowed from the seminal paper \cite{[HMZ07]} about scattering of fast solitons by a delta interaction on the line, by Holmer, Marzuola and Zworski; the present paper represents an extension of their work to the case of graphs and, as a byproduct, it shows how to extend the analysis of soliton scattering by other point interactions on the line, interpreted as a degenerate graph.Comment: Sec. 2 revised; several misprints corrected; added references; 32 page

Orbital and asymptotic stability for standing waves of a NLS equation with concentrated nonlinearity in dimension three

We begin to study in this paper orbital and asymptotic stability of standing waves for a model of Schr\"odinger equation with concentrated nonlinearity in dimension three. The nonlinearity is obtained considering a {point} (or contact) interaction with strength $\alpha$, which consists of a singular perturbation of the laplacian described by a selfadjoint operator $H_{\alpha}$, where the strength $\alpha$ depends on the wavefunction: $i\dot u= H_\alpha u$, $\alpha=\alpha(u)$. If $q$ is the so-called charge of the domain element $u$, i.e. the coefficient of its singular part, we let the strength $\alpha$ depend on $u$ according to the law $\alpha=-\nu|q|^\sigma$, with $\nu > 0$. This characterizes the model as a focusing NLS with concentrated nonlinearity of power type. For such a model we prove the existence of standing waves of the form $u (t)=e^{i\omega t}\Phi_{\omega}$, which are orbitally stable in the range $\sigma \in (0,1)$, and orbitally unstable for $\sigma \geq 1.$ Moreover, we show that for $\sigma \in (0,\frac{1}{\sqrt 2})$ every standing wave is asymptotically stable in the following sense. Choosing initial data close to the stationary state in the energy norm, and belonging to a natural weighted $L^p$ space which allows dispersive estimates, the following resolution holds: $u(t) = e^{i\omega_{\infty} t} \Phi_{\omega_{\infty}} +U_t*\psi_{\infty} +r_{\infty}, \quad \textrm{as} \;\; t \rightarrow +\infty$, where $U$ is the free Schr\"odinger propagator, $\omega_{\infty} > 0$ and $\psi_{\infty}$, $r_{\infty} \in L^2(\R^3)$ with $\| r_{\infty} \|_{L^2} = O(t^{-5/4}) \quad \textrm{as} \;\; t \rightarrow +\infty$. Notice that in the present model the admitted nonlinearity for which asymptotic stability of solitons is proved is subcritical.Comment: Comments and clarifications added; several misprints correcte

Transfer of Axial Chirality to the Nanoscale Endows Carbon Nanodots with Circularly Polarized Luminescence

We report the synthesis, purification and characterization of chiral carbon nanodots starting from atropoisomeric precursors. The obtained atropoisomeric carbon nanodots are soluble in organic solvents and have good thermal stability, which are desirable features for technological applications. The synthetic protocol is robust, as it supports a number of variations in terms of molecular doping agents. Remarkably, the combination of axially chiral precursors and 1,4-benzoquinone as doping agent results in green-emissive carbon dots displaying circularly polarized luminescence. Dissymmetry factors of |3.5|×10−4 are obtained in solution, without the need of any additional element of chirality. Introducing axial chirality expands the strategies available to tailor the properties of carbon nanodots, paving the way for carbon nanoparticles that combine good processability in organic solvents with engineered advanced chiroptical properties

Dynamics and Lax-Phillips scattering for generalized Lamb models

This paper treats the dynamics and scattering of a model of coupled oscillating systems, a finite dimensional one and a wave field on the half line. The coupling is realized producing the family of selfadjoint extensions of the suitably restricted self-adjoint operator describing the uncoupled dynamics. The spectral theory of the family is studied and the associated quadratic forms constructed. The dynamics turns out to be Hamiltonian and the Hamiltonian is described, including the case in which the finite dimensional systems comprises nonlinear oscillators; in this case the dynamics is shown to exist as well. In the linear case the system is equivalent, on a dense subspace, to a wave equation on the half line with higher order boundary conditions, described by a differential polynomial $p(\partial_x)$ explicitely related to the model parameters. In terms of such structure the Lax-Phillips scattering of the system is studied. In particular we determine the incoming and outgoing translation representations, the scattering operator, which turns out to be unitarily equivalent to the multiplication operator given by the rational function $-p(i\kappa)^*/p(i\kappa)$, and the Lax-Phillips semigroup, which describes the evolution of the states which are neither incoming in the past nor outgoing in the future