The LpL^p boundedness of wave operators for Schr\"odinger operators with threshold singularities II. Even dimensional case

In this paper we consider the wave operators $W_{\pm}$ for a Schr\"odinger operator $H$ in ${\bf{R}}^n$ with $n\geq 4$ even and we discuss the $L^p$ boundedness of $W_{\pm}$ assuming a suitable decay at infinity of the potential $V$. The analysis heavily depends on the singularities of the resolvent for small energy, that is if 0-energy eigenstates exist. If such eigenstates do not exist $W_{\pm}: L^p \to L^p$ are bounded for $1 \leq p \leq \infty$ otherwise this is true for $\frac{n}{n-2} < p < \frac{n}{2}$. The extension to Sobolev space is discussed.Comment: 59 page

Energy lower bound for the unitary N+1 fermionic model

We consider the stability problem for a unitary N+1 fermionic model, i.e., a system of $N$ identical fermions interacting via zero-range interactions with a different particle, in the case of infinite two-body scattering length. We present a slightly more direct and simplified proof of a recent result obtained in \cite{CDFMT}, where a sufficient stability condition is proved under a suitable assumption on the mass ratio.Comment: 7 page

Nonlinear singular perturbations of the fractional Schr\"odinger equation in dimension one

The paper discusses nonlinear singular perturbations of delta type of the fractional Schr\"odinger equation $\imath\partial_t\psi=\left(-\triangle\right)^s\psi$, with $s\in(\frac{1}{2},1]$, in dimension one. Precisely, we investigate local and global well posedness (in a strong sense), conservations laws and existence of blow-up solutions and standing waves.Comment: 28 pages. Some minor revisions have been made with respect to the previous versio

Stationary States of NLS on Star Graphs

We consider a generalized nonlinear Schr\"odinger equation (NLS) with a power nonlinearity |\psi|^2\mu\psi, of focusing type, describing propagation on the ramified structure given by N edges connected at a vertex (a star graph). To model the interaction at the junction, it is there imposed a boundary condition analogous to the \delta potential of strength \alpha on the line, including as a special case (\alpha=0) the free propagation. We show that nonlinear stationary states describing solitons sitting at the vertex exist both for attractive (\alpha0, a potential barrier) interaction. In the case of sufficiently strong attractive interaction at the vertex and power nonlinearity \mu<2, including the standard cubic case, we characterize the ground state as minimizer of a constrained action and we discuss its orbital stability. Finally we show that in the free case, for even N only, the stationary states can be used to construct traveling waves on the graph.Comment: Revised version, 5 pages, 2 figure

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 NLS on a three-edge star graph. We prove that, contrarily to the case of the standard NLS on the line, the energy associated to the cubic focusing Schr\"odinger equation on the three-edge star graph with a free (Kirchhoff) vertex does not attain a minimum value on any sphere of constant $L^2$-norm. We moreover show that the only stationary state with prescribed L^2-norm is indeed a saddle point

Stable standing waves for a NLS on star graphs as local minimizers of the constrained energy

On a star graph made of $N \geq 3$ halflines (edges) we consider a Schr\"odinger 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 $L^2$-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 {\em 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.Comment: 18 pages, 2 figure

Asymptotic Expansion for the Wave Function in a one-dimensional Model of Inelastic Interaction

We consider a two-body quantum system in dimension one composed by a test particle interacting with an harmonic oscillator placed at the position $a>0$. At time zero the test particle is concentrated around the position $R_0$ with average velocity $\pm v_0$ while the oscillator is in its ground state. In a suitable scaling limit, corresponding for the test particle to a semi-classical regime with small energy exchange with the oscillator, we give a complete asymptotic expansion of the wave function of the system in both cases $R_0 <a$ and $R_0 >a$.Comment: 23 page

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

On the Asymptotic Dynamics of a Quantum System Composed by Heavy and Light Particles

We consider a non relativistic quantum system consisting of $K$ heavy and $N$ light particles in dimension three, where each heavy particle interacts with the light ones via a two-body potential $\alpha V$. No interaction is assumed among particles of the same kind. Choosing an initial state in a product form and assuming $\alpha$ sufficiently small we characterize the asymptotic dynamics of the system in the limit of small mass ratio, with an explicit control of the error. In the case K=1 the result is extended to arbitrary $\alpha$. The proof relies on a perturbative analysis and exploits a generalized version of the standard dispersive estimates for the Schr\"{o}dinger group. Exploiting the asymptotic formula, it is also outlined an application to the problem of the decoherence effect produced on a heavy particle by the interaction with the light ones.Comment: 38 page