1,720,975 research outputs found
Scattering of a light particle by a system of harmonic oscillators
No full textWe discuss the asymptotic wave function of a quantum system in R^3 composed by heavy and light particles, in the case where the light particles are in scattering states and no interaction
is assumed among particles of the same kind. We first review a recent result concerning the case
of K heavy and N light particles, where the one-particle potential acting on each heavy particle
decays at infinity. Then we consider the case of one light particle interacting with a system of
harmonic oscillators and prove the same kind of result following, with some modification, the
proof of the previous case. A possible application to the analysis of the scattering of a light
particle from condensed matter is also outlined
Nonlinear singular perturbations of the fractional Schrödinger equation in dimension one
Full text linkThe paper discusses nonlinear singular delta-type perturbations of the fractional Schrödinger equation ı∂ψ/∂t = (−∆)^s ψ, with s ∈ (1/2 , 1] , in dimension one. In particular, we investigate local and global well-posedness (in a strong
sense), conservation laws and the existence of blow-up solutions and standing waves
Topology-induced bifurcations for the nonlinear Schrödinger equation on the tadpole graph
No full textIn this paper we give the complete classification of solitons for a cubic nonlinear Schrödinger equation on the simplest network with a nontrivial topology: the tadpole graph, i.e., a ring with a half line attached to it and free boundary conditions at the junction. This is a step toward the modelization of condensate propagation and confinement in quasi-one-dimensional traps. The model, although simple, exhibits a surprisingly rich behavior and in particular we show that it admits: (i) a denumerable family of continuous branches of embedded solitons vanishing on the half line and bifurcating from linear eigenstates and threshold resonances of the system; (ii) a continuous branch of edge solitons bifurcating from the previous families at the threshold of the continuous spectrum with a pitchfork bifurcation; and (iii) a finite family of continuous branches of solitons without linear analog. 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. In this sense the magnetic field acts as a control parameter. Finally we highlight the role of resonances in the linearization as a signature of the occurrence of bifurcations of solitons from the continuous spectrum
Ground state and orbital stability for the NLS equation on a general starlike graph with potentials
No full textWe consider a nonlinear Schrödinger equation (NLS)osed on a graph (or network) composed of a generic compact part to which a finite number of half-lines are attached. We call this structure a starlike graph. At the vertices of the graph interactions of Î ́-type can be present and an overall external potential is admitted. Under general assumptions on the potential, we prove that the NLS is globally well-posed in the energy domain. We are interested in minimizing the energy of the system on the manifold of constant mass (L 2-norm). When existing, the minimizer is called ground state and it is the profile of an orbitally stable standing wave for the NLS evolution. We prove that a ground state exists for sufficiently small masses whenever the quadratic part of the energy admits a simple isolated eigenvalue at the bottom of the spectrum (the linear ground state). This is a wide generalization of a result previously obtained for a star-graph with a single vertex. The main part of the proof is devoted to prove the concentration compactness principle for starlike structures; this is non trivial due to the lack of translation invariance of the domain. Then we show that a minimizing, bounded, H 1 sequence for the constrained NLS energy with external linear potentials is in fact convergent if its mass is small enough. Moreover we show that the ground state bifurcates from the vanishing solution at the bottom of the linear spectrum. Examples are provided with a discussion of the hypotheses on the linear part
The point-like limit for a NLS equation with concentrated nonlinearity in dimension three
Full text linkWe consider a scaling limit of a nonlinear Schrödinger 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∂∂tψε(t)=-δψε(t)+g(ε,μ,|(ρε,ψε(t))|2μ)(ρε,ψε(t))ρε where ρε→δ0 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
The three-body problem in dimension one: From short-range to contact interactions
Full text linkWe consider a Hamiltonian describing three quantum particles in dimension one interacting through two-body short-range potentials. We prove that, as a suitable scale parameter in the potential terms goes to zero, such a Hamiltonian converges to one with zero-range (also called delta or point) interactions. The convergence is understood in the norm resolvent sense. The two-body rescaled potentials are of the form vσε(xσ)=ε-1vσ(ε-1xσ), where σ = 23, 12, 31 is an index that runs over all the possible pairings of the three particles, xσis the relative coordinate between two particles, and ε is the scale parameter. The limiting Hamiltonian is the one formally obtained by replacing the potentials vσ with ασδσ, where δσis the Dirac delta-distribution centered on the coincidence hyperplane xσ= 0 and ασ= Rvσdxσ. To prove the convergence of the resolvents, we make use of Faddeev's equations
Stable standing waves for a NLS on star graphs as local minimizers of the constrained energy
No full textOn a star graph made of N≥3 halflines (edges) we consider a Schrödinger equation with a subcritical power-type nonlinearity and an attractive delta interaction located at the vertex. From previous works it is known that there exists a family of standing waves, symmetric with respect to the exchange of edges, that can be parametrized by the mass (or L2-norm) of its elements. Furthermore, if the mass is small enough, then the corresponding symmetric standing wave is a ground state and, consequently, it is orbitally stable. On the other hand, if the mass is above a threshold value, then the system has no ground state.Here we prove that orbital stability holds for every value of the mass, even if the corresponding symmetric standing wave is not a ground state, since it is anyway a local minimizer of the energy among functions with the same mass.The proof is based on a new technique that allows to restrict the analysis to functions made of pieces of soliton, reducing the problem to a finite-dimensional one. In such a way, we do not need to use direct methods of Calculus of Variations, nor linearization procedures
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