arXiv subjects

D. Finco

Publications and source records attributed to D. Finco.

11 recordsLinked to original sources

arXiv2022

Failure of scattering for the NLSE with a point interaction in dimension two and three

In this paper we consider the NLS equation with power nonlinearity and a point interaction (a "$\delta$-potential" in the physical literature) in dimension two and three. We will show that for low power nonlinearities there is failure of scattering to the free dynamics or to standing waves. In the recent paper Murphy and Nakanishi consider the NLS equation with potentials and measures, singular enough to include the $\delta$-potential in dimension one and they show analogous properties. In this respect our contribution is an extension to higher dimension and it needs a different treatment of the linear part of the interaction, due the qualitatively different and stronger character of the singularity involved.

math-ph↗

arXiv2022

Blow-up and instability of standing waves for the NLS with a point interaction in dimension two

In the present note we study the NLS equation in dimension two with a point interaction and in the supercritical regime, showing two results. After obtaining the (nonstandard) virial formula, we exhibit a set of initial data that blow-up. Moreover we show the standing waves $e^{i\omega t} \varphi_\omega$ corresponding to ground states $\varphi_\omega$ of the action are strongly unstable, at least for sufficiently high $\omega$.

math.AP↗

arXiv2019

A Quantum Model of Feshbach Resonances

We consider a quantum model of two-channel scattering to describe the mechanism of a Feshbach resonance. We perform a rigorous analysis in order to count and localize the energy resonances in the perturbative regime, i.e., for small inter-channel coupling, and in the non-perturbative one. We provide an expansion of the effective scattering length near the resonances, via a detailed study of an effective Lippmann-Schwinger equation with energy-dependent potential.

math-ph↗

arXiv2015

A Class of Hamiltonians for a Three-Particle Fermionic System at Unitarity

We consider a quantum mechanical three-particle system made of two identical fermions of mass one and a different particle of mass $ m $, where each fermion interacts via a zero-range force with the different particle. In particular we study the unitary regime, i.e., the case of infinite two-body scattering length. The Hamiltonians describing the system are, by definition, self-adjoint extensions of the free Hamiltonian restricted on smooth functions vanishing at the two-body coincidence planes, i.e., where the positions of two interacting particles coincide. It is known that for $ m $ larger than a critical value $ m^* \simeq (13.607)^{-1} $ a self-adjoint and lower bounded Hamiltonian $ H_0 $ can be constructed, whose domain is characterized in terms of the standard point-interaction boundary condition at each coincidence plane. Here we prove that for $ m \in( m^*,m^{**}) $, where $ m^{**} \simeq (8.62)^{-1} $, there is a further family of self-adjoint and lower bounded Hamiltonians $ H_{0,\beta} $, $ \beta \in \mathbb{R} $, describing the system. Using a quadratic form method, we give a rigorous construction of such Hamiltonians and we show that the elements of their domains satisfy a further boundary condition, characterizing the singular behavior when the positions of all the three particles coincide.

math-ph↗

arXiv2014

Variational properties and orbital stability of standing waves for NLS equation on a star graph

We study standing waves for a nonlinear Schrödinger equation on a star graph {$\mathcal{G}$} i.e. $N$ half-lines joined at a vertex. At the vertex an interaction occurs described by a boundary condition of delta type with strength $α\leqslant 0$. The nonlinearity is of focusing power type. The dynamics is given by an equation of the form $ i \frac{d}{dt}Ψ_t = H Ψ_t - | Ψ_t |^{2μ} Ψ_t $, where $H$ is the Hamiltonian operator which generates the linear Schrödinger dynamics. We show the existence of several families of standing waves for every sign of the coupling at the vertex for every $ω> \frac{α^2}{N^2}$. Furthermore, we determine the ground states, as minimizers of the action on the Nehari manifold, and order the various families. Finally, we show that the ground states are orbitally stable for every allowed $ω$ if the nonlinearity is subcritical or critical, and for $ω<ω^\ast$ otherwise.

math-ph↗

arXiv2014

Topology induced bifurcations for the NLS on the tadpole graph

In this paper we give the complete classification of solitons for a cubic NLS equation on the simplest network with a non-trivial topology: the tadpole graph, i.e. a ring with a half-line attached to it and free boundary conditions at the junction. The model, although simple, exhibits a surprisingly rich behavior and in particular we show that it admits: 1) a denumerable family of continuous branches of embedded solitons bifurcating from linear eigenstates and threshold resonances of the system; 2) a continuous branch of edge solitons displaying a pitchfork symmetry breaking bifurcation at the threshold of the continuous spectrum; 3) a finite family of continuous branches of solitons without linear analogue. All the solutions are explicitly constructed in terms of elliptic Jacobian functions. Moreover we show that families of nonlinear bound states of the above kind continue to exist in the presence of a uniform magnetic field orthogonal to the plane of the ring when a well definite flux quantization condition holds true. Finally we highlight the role of resonances in the linearization as a signature of the occurrence of bifurcations of solitons from the continuous spectrum.

math-ph↗

arXiv2014

The NLS equation in dimension one with spatially concentrated nonlinearities: the pointlike limit

In the present paper we study the following scaled nonlinear Schr\"odinger equation (NLS) in one space dimension: \[ i\frac{d}{dt} \psi^{\varepsilon}(t) =-\Delta\psi^{\varepsilon}(t) + \frac{1}{\epsilon}V\left(\frac{x}{\epsilon}\right)|\psi^{\varepsilon}(t)|^{2\mu}\psi^{\varepsilon}(t) \quad \quad \epsilon>0\ ,\quad V\in L^1(\mathbb{R},(1+|x|)dx) \cap L^\infty(\mathbb{R}) \ . \] This equation represents a nonlinear Schr\"odinger equation with a spatially concentrated nonlinearity. We show that in the limit $\epsilon\to 0$, the weak (integral) dynamics converges in $H^1(\mathbb{R})$ to the weak dynamics of the NLS with point-concentrated nonlinearity: \[ i\frac{d}{dt} \psi(t) =H_{\alpha}\psi(t) . \] where $H_{\alpha}$ is the laplacian with the nonlinear boundary condition at the origin $\psi'(t,0+)-\psi'(t,0-)=\alpha|\psi(t,0)|^{2\mu}\psi(t,0)$ and $\alpha=\int_{\mathbb{R}}Vdx$. The convergence occurs for every $\mu\in \mathbb{R}^+$ if $V \geq 0$ and for every $\mu\in (0,1)$ otherwise. The same result holds true for a nonlinearity with an arbitrary number $N$ of concentration points

math-ph↗

arXiv2012

Constrained energy minimization and orbital stability for the NLS equation on a star graph

We consider a nonlinear Schrödinger equation with focusing nonlinearity of power type on a star graph ${\mathcal G}$, written as $ i \partial_t Ψ(t) = H Ψ(t) - |Ψ(t)|^{2μ}Ψ(t)$, where $H$ is the selfadjoint operator which defines the linear dynamics on the graph with an attractive $δ$ interaction, with strength $α< 0$, at the vertex. The mass and energy functionals are conserved by the flow. We show that for $0<μ<2$ the energy at fixed mass is bounded from below and that for every mass $m$ below a critical mass $m^*$ it attains its minimum value at a certain $\hat Ψ_m \in H^1(\GG) $, while for $m>m^*$ there is no minimum. Moreover, the set of minimizers has the structure ${\mathcal M}={e^{iθ}\hat Ψ_m, θ\in \erre}$. Correspondingly, for every $m<m^*$ there exists a unique $ω=ω(m)$ such that the standing wave $\hatΨ_ωe^{iωt} $ is orbitally stable. To prove the above results we adapt the concentration-compactness method to the case of a star graph. This is non trivial due to the lack of translational symmetry of the set supporting the dynamics, i.e. the graph. This affects in an essential way the proof and the statement of concentration-compactness lemma and its application to minimization of constrained energy. The existence of a mass threshold comes from the instability of the system in the free (or Kirchhoff's) case, that in our setting corresponds to $\al=0$.

math-ph↗

arXiv2012

Stability for a System of N Fermions Plus a Different Particle with Zero-Range Interactions

We study the stability problem for a non-relativistic quantum system in dimension three composed by $ N \geq 2 $ identical fermions, with unit mass, interacting with a different particle, with mass $ m $, via a zero-range interaction of strength $ α\in \R $. We construct the corresponding renormalised quadratic (or energy) form $ \form $ and the so-called Skornyakov-Ter-Martirosyan symmetric extension $ H_α $, which is the natural candidate as Hamiltonian of the system. We find a value of the mass $ m^*(N) $ such that for $ m > m^*(N)$ the form $ \form $ is closed and bounded from below. As a consequence, $ \form $ defines a unique self-adjoint and bounded from below extension of $ H_α$ and therefore the system is stable. On the other hand, we also show that the form $ \form $ is unbounded from below for $ m < m^*(2)$. In analogy with the well-known bosonic case, this suggests that the system is unstable for $ m < m^*(2)$ and the so-called Thomas effect occurs.

math-ph↗

arXiv2008

Graph-like models for thin waveguides with Robin boundary conditions

We discuss the limit of small width for the Laplacian defined on a waveguide with Robin boundary conditions. Under suitable hypothesis on the scaling of the curvature, we prove the convergence of the Robin Laplacian to the Laplacian on the corresponding graph. We show that the projections on each transverse mode generically give rise to decoupling conditions between the edges of the graph while exceptionally a coupling can occur. The non decoupling conditions are related to the existence of resonances at the thresholds of the continuum spectrum.

math-ph↗

arXiv2007

Spectral Analysis of a Two Body Problem with Zero Range Perturbation

We consider a class of singular, zero-range perturbations of the Hamiltonian of a quantum system composed by a test particle and a harmonic oscillators in dimension one, two and three and we study its spectrum. In facts we give a detailed characterization of point spectrum and its asymptotic behavior with respect to the parameters entering the Hamiltonian. We also partially describe the positive spectrum and scattering properties of the Hamiltonian.

math-ph↗