Domenico Finco

FAST SOLITONS ON STAR GRAPHS

We define the Schrodinger 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 δ and δ′ 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 [17] 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.

Coupling in the singular limit of thin quantum waveguides

We analyze the problem of approximating a smooth quantum waveguide with a quantum graph. We consider a planar curve with compactly supported curvature and a strip of constant width around the curve. We rescale the curvature and the width in such a way that the strip can be approximated by a singular limit curve, consisting of one vertex and two infinite, straight edges, i.e., a broken line. We discuss the convergence of the Laplacian, with Dirichlet boundary conditions on the strip, in a suitable sense and we obtain two possible limits: the Laplacian on the line with Dirichlet boundary conditions in the origin and a nontrivial family of point perturbations of the Laplacian on the line. The first case generically occurs and corresponds to the decoupling of the two components of the limit curve, while in the second case a coupling takes place. We present also two families of curves which give rise to coupling.

On the structure of critical energy levels for the cubic focusing NLS on star graphs

We provide information on a non-trivial structure of phase space of the cubic nonlinear Schrodinger (NLS) on a three-edge star graph. We prove that, in contrast to the case of the standard NLS on the line, the energy associated with the cubic focusing Schrodinger equation on the three-edge star graph with a free (Kirchhoff) vertex does not attain a minimum value on any sphere of constant L2-norm. We moreover show that the only stationary state with prescribed L2-norm is indeed a saddle point.

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∗ ≃ (13.607)−1 a self-adjoint and lower bounded Hamiltonian H0 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 ∈ (m∗,m∗∗), where m∗∗ ≃ (8.62)−1, there is a further family of self-adjoint and lower bounded Hamiltonians H0,β, β ∈ 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.

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 ≥ 2 identical fermions, with unit mass, interacting with a different particle, with mass m, via a zero-range interaction of strength α ∈ ℝ. We construct the corresponding renormalized quadratic (or energy) form and the so-called Skornyakov–Ter–Martirosyan symmetric extension Hα, 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 is closed and bounded from below. As a consequence, defines a unique self-adjoint and bounded from below extension of Hα and therefore the system is stable. On the other hand, we also show that the 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.

On the asymptotic behaviour of a quantum two-body system in the small mass ratio limit

We consider a quantum system of two particles in dimension three interacting via a smooth potential. We characterize the asymptotic dynamics in the limit of small mass ratio for an initial state given in product form, with an explicit control of the error. An application to the decoherence effect produced on the heavy particle is also discussed.

Quadratic Forms for the Fermionic Unitary Gas Model

We consider a quantum system in dimension three composed by a group of N identical fermions, with mass 1/2, interacting via zero-range interaction with a group of M identical fermions of a different type, with mass m/2. Exploiting a renormalization procedure, we construct the corresponding quadratic form and define the so-called Skornyakov-Ter-Martirosyan extension Hα, which is the natural candidate as a possible Hamiltonian of the system. It is shown that if the form is unbounded from below then Hα is not a self-adjoint and bounded from below operator, and this in particular suggests that the so-called Thomas effect could occur. In the special case N = 2, M = 1 we prove that this is in fact the case when a suitable condition on the parameter m is satisfied.

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.

The NLS Equation in Dimension One with Spatially Concentrated Nonlinearities: the Pointlike Limit

The nonlinear Schrödinger equation with spatially dependent nonlinearities has been considered in many contexts, both for its mathematical interest and for its relevance in several physical models. They represent situations in which there is a spatial inhomogeneity of the response of the medium to the wavefunction propagation. Here we are interested in the special case in which the nonlinearity is strongly concentrated in space around a finite number of points. More precisely, we investigate the limit where the nonlinearity becomes strictly pointlike. Such nonlinearities appear in several physical applications, for example: nonlinear diffraction of electrons from a thin layer (see [6]), analysis of nonlinear resonant tunneling (see [11]), models of pattern formation and bifurcation of solitons in nonlinear media (see [9, 12, 15, 16, 19] and references therein). Rigorous analysis of asymptotic stability of standing waves for some of the above models can be found in [7, 14]. To make precise the considered problem, we will consider the following nonlinear Schrödinger type equation

The point-like limit for a NLS equation with concentrated nonlinearity in dimension three

Abstract We consider a scaling limit of a nonlinear Schrodinger equation (NLS) with a nonlocal nonlinearity showing that it reproduces in the limit of cutoff removal a NLS equation with nonlinearity concentrated at a point. The regularized dynamics is described by the equation i ∂ ∂ t ψ e ( t ) = − Δ ψ e ( t ) + g ( e , μ , | ( ρ e , ψ e ( t ) ) | 2 μ ) ( ρ e , ψ e ( t ) ) ρ e where ρ e → δ 0 weakly and the function g embodies the nonlinearity and the scaling and has to be fine tuned in order to have a nontrivial limit dynamics. The limit dynamics is a nonlinear version of point interaction in dimension three and it has been previously studied in several papers as regards the well-posedness, blow-up and asymptotic properties of solutions. Our result is the first justification of the model as the point limit of a regularized dynamics.