Scatteing for hydrogen-like systems in a constant magnetic field ∗∗

We prove asymptotic completeness for a system of two particles with opposite charges interacting via a long-range potential decaying faster than |r|^-1/2 at infinity in a homogeneous magnetic field. In particular, our result implies asymptotic completeness for the hydrogen atom with finite nuclear mass in a uniform magnetic field, which has not been proven previously.     

Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case

We establish global wellposedness and scattering for the -critical defocusing NLS in 3D

assuming radial data , . In particular, it proves global existence of classical solutions in the radial case. The same result is obtained in 4D for the equation

     

Geometric Formulation and Multi‐dark Soliton Solution to the Defocusing Complex Short Pulse Equation

In the present paper, we study the defocusing complex short pulse (CSP) equations both geometrically and algebraically. From the geometric point of view, we establish a link of the complex coupled dispersionless (CCD) system with the motion of space curves in Minkowski space , then with the defocusing CSP equation via a hodograph (reciprocal) transformation, the Lax pair is constructed naturally for the defocusing CSP equation. We also show that the CCD system of both the focusing and defocusing types can be derived from the fundamental forms of surfaces such that their curve flows are formulated. In the second part of the paper, we derive the defocusing CSP equation from the single‐component extended Kadomtsev‐Petviashvili (KP) hierarchy by the reduction method. As a by‐product, the N‐dark soliton solution for the defocusing CSP equation in the form of determinants for these equations is provided.     

On the Essential Spectrum of Two-Dimensional Pauli Operators with Repulsive Potentials

We investigate the spectrum of the two-dimensional Pauli operator, describing a spin-\({\frac{1}{2}}\) particle in a magnetic field B, with a negative scalar potential V, such that |V| grows at infinity. In particular, we obtain criteria for discrete and dense pure-point spectrum.     

On the existence of dark solitons of the defocusing cubic nonlinear Schrödinger equation with periodic inhomogeneous nonlinearity

We provide a simple proof of the existence of dark solitons of the defocusing cubic nonlinear Schrödinger equation with periodic inhomogeneous nonlinearity. Moreover, our proof allows for a broader class of inhomogeneities and gives some new properties of the solutions. We also apply our approach to the defocusing cubic–quintic nonlinear Schrödinger equation with a periodic potential.     

Semilinear Schrödinger flows on hyperbolic spaces: scattering in <Emphasis Type="Italic">H</Emphasis><Superscript>1</Superscript>

We prove global well-posedness and scattering in H ¹ for the defocusing nonlinear Schrödinger equations

$\left\{\begin{array}{ll}(i\partial_t+\Delta_g)u=u|u|^{2\sigma};\\u(0)=\phi,\end{array}\right.$

on the hyperbolic spaces \({\mathbb{H}^d}\), d ≥ 2, for exponents \({\sigma \in (0, 2/(d-2))}\). The main unexpected conclusion is scattering to linear solutions in the case of small exponents σ; for comparison, on Euclidean spaces scattering in H ¹ is not known for any exponent \({\sigma \in (1/d, 2/d]}\) and is known to fail for \({\sigma \in (0, 1/d]}\). Our main ingredients are certain noneuclidean global in time Strichartz estimates and noneuclidean Morawetz inequalities.

     

Scattering Operators for Matrix Zakharov-Shabat Systems

In this article the scattering matrix pertaining to the defocusing matrix Zakharov-Shabat system on the line is related to the scattering operator arising from time-dependent scattering theory. Further, the scattering data allowing for a unique retrieval of the potential in the defocusing matrix Zakharov-Shabat system are characterized. The research leading to this article was supported in part by the Italian Ministery of Education and Research (MIUR) under PRIN grant no. 2006017542-003, and by INdAM. One of the authors (CvdM) wishes to express his gratitude for the hospitality of the Department of Mathematics of the University of Texas at Arlington during the Spring Semester of 2008 when part of the paper was written.     

Maximal chordal subgraphs

An algorithm for finding maximal chordal subgraphs is developed that has worst-case time complexity of O(|E|Δ), where |E| is the number of edges in G and Δ is the maximum vertex degree in G. The study of maximal chordal subgraphs is motivated by their usefulness as computationally efficient structures with which to approximate a general graph. Two examples are given that illustrate potential applications of maximal chordal subgraphs. One provides an alternative formulation to the maximum independent set problem on a graph. The other involves a novel splitting scheme for solving large sparse systems of linear equations.     

Scattering in H1 for the intercritical NLS with an inverse-square potential

We study the nonlinear Schrödinger equation with an inverse-square potential in dimensions $3\le d\le 6$. We consider both focusing and defocusing nonlinearities in the mass-supercritical and energy-subcritical regime. In the focusing case, we prove a scattering/blowup dichotomy below the ground state. In the defocusing case, we prove scattering in $H^1$ for arbitrary data.     

R2中非局部椭圆型方程基态解的存在性

本文考虑了一类非局部椭圆型方程-△u+V(x)u=(1/|x|μ*Q(x)F(u)/|x|β)Q(x)f(u)|x|β,x∈R^x,其中V是正的连续位势函数,0<μ<2,0≤β<1/2,2β+μ≤2,F(s)是f(s)的原函数.假设非线性项f(s)满足Trudinger-Moser型次临界指数增长,利用变分方法证明了该方程基态解的存在性.     

The defocusing energy-supercritical NLS in higher dimensions

We consider the defocusing nonlinear Schr?dinger equations iu_t +△u =|u|~(p_u) with p being an even integer in dimensions d≥ 5. We prove that an a priori bound of critical norm implies global well-posedness and scattering for the solution.     

一类非线性发展方程组整体解的存在性和不存在性

本文考虑如下非线性发展方程组的初边值问题u″+ Au -△ u′-α| v|ρ+ 2 | u|ρu =0 ,　 v″+ Av -△ v′-α| u|ρ+ 2 | v|ρv =0 ,其中 A为 p-Laplace算子 ,α∈ R,ρ>-1均为常数 ,证明了整体解的存在性及渐近性 ,得到了当初始能量为正但有上界时解的爆破性条件     

Scattering theory for nonlinear Schrödinger equations with inverse-square potential

We study the long-time behavior of solutions to nonlinear Schrödinger equations with some critical rough potential of a|x|⁻²$a{|x|}^{-2}$ type. The new ingredients are the interaction Morawetz-type inequalities and Sobolev norm property associated with _Pa=−Δ+a|x|⁻²$P_a=-\mathrm{\Delta }+a{|x|}^{-2}$. We use such properties to obtain the scattering theory for the defocusing energy-subcritical nonlinear Schrödinger equation with inverse square potential in energy space H¹(Rⁿ)$H^1({\mathbb{R}}^n)$.     

Quantum Scattering in a Periodically Pulsed Magnetic Field

In this paper, we study the quantum dynamics of a charged particle in the plane in the presence of a periodically pulsed magnetic field perpendicular to the plane. We show that by controlling the cycle when the magnetic field is switched on and off appropriately, the result of the asymptotic completeness of wave operators can be obtained under the assumption that the potential V satisfies the decaying condition \({|V(x)| \le C(1 + |x|)^{-\rho}}\) for some \({\rho > 0}\).     

一类积分方程组正解的对称性和单调性

本文讨论积分方程组（？）解的性质,其中G_α是α阶贝塞尔位势核,0≤β〈α（n-α＋β）/n,1/（q＋1）＋1/（r＋1）〉（n-α＋β）/n,1/（r＋1）＋1/（p＋1）〉（n-α＋β）/n.我们用积分形式的移动平面法证明上述积分方程组的正解是径向对称且单调的.     

Multiple Solutions for a Semilinear Elliptic Equation with Critical Growth and Magnetic Field

In this paper, we are concerned with the multiplicity of nontrivial solutions for the following class of complex problems $$(-i\nabla - A(x))^2{u} = \mu|u|^{q-2}u + |u|^{2^*-2}u\, {\rm in}\, \Omega,\quad u=0\, {\rm on}\, \partial\, \Omega$$ where \({\Omega \subset \mathbb{R}^N(N \geq 4)}\) is a bounded domain with smooth boundary, \({A: \overline{\Omega} \rightarrow \mathbb{R}^N}\) is a continuous magnetic potential and \({2 \leq q < 2^* = \frac{2N}{N-2}}\) . Using the Lusternik-Schnirelman theory, we relate the number of solutions with the topology of Ω.     

Convergence of the Klein-Gordon equation to the wave map equation with magnetic field

This paper is devoted to the proof of the convergence from the modulated cubic nonlinear defocusing Klein-Gordon equation with magnetic field to the wave map equation. More precisely, we discuss the nonrelativistic-semiclassical limit of the modulated cubic nonlinear Klein-Gordon equation with magnetic field where the Planck's constant ?=ε and the speed of light c are related by c=ε−α for some α?1. When α=1 the limit wave function satisfies the wave map with one extra term coming from the magnetic field. However, α>1, the effect of the magnetic filed disappears and the limit is the typical wave map equation only.     

Matter-coupled de Sitter supergravity

The de Sitter supergravity describes the interaction of supergravity with general chiral and vector multiplets and also one nilpotent chiral multiplet. The extra universal positive term in the potential, generated by the nilpotent multiplet and corresponding to the anti-D3 brane in string theory, is responsible for the de Sitter vacuum stability in these supergravity models. In the flat-space limit, these supergravity models include the Volkov–Akulov model with a nonlinearly realized supersymmetry. We generalize the rules for constructing the pure de Sitter supergravity action to the case of models containing other matter multiplets. We describe a method for deriving the closed-form general supergravity action with a given potential K, superpotential W, and vectormatrix fAB interacting with a nilpotent chiral multiplet. It has the potential V = e^K(|F²|+|DW|²-3|W|²), where F is the auxiliary field of the nilpotent multiplet and is necessarily nonzero. The de Sitter vacuums are present under the simple condition that |F²|-3|W|² > 0. We present an explicit form of the complete action in the unitary gauge.     

On the well-posedness for stochastic fourth-order Schrödinger equations

The influence of the random perturbations on the fourth-order nonlinear Schrödinger equations,

$iu_t + \Delta ^2 u + \varepsilon \Delta u + \lambda |u|^{p - 1} u = \dot \xi ,(t,x) \in \mathbb{R}^ + \times \mathbb{R}^n ,n \geqslant 1,\varepsilon \in \{ - 1,0, + 1\} ,$

, is investigated in this paper. The local well-posedness in the energy space H ²(? ⁿ ) are proved for \(p > \tfrac{{n + 4}}{{n + 2}}\), and p ≤ 2^# ? 1 if n ≥ 5. Global existence is also derived for either defocusing or focusing L ²-subcritical nonlinearities.

     

Modified Wave Operators for the Hartree Equation with Data,Image and Convergence in the Same Space,II

We study modified wave operators for the Hartree equation with a long-range potential |x|^-n |x|^{-\nu} , extending the result in [12] to the whole range of the Dollard type 1/2 < n \nu < 1. We construct the modified wave operators in the whole space of (1 + |x|)^-sL² (1 + |x|)^{-s}L^2 . We also have the image, strong continuity and strong asymptotic approximation in the same space. The lower bound $ s > 1 - \nu / 2 $ s > 1 - \nu / 2 of the weight is sharp from the scaling argument. Those maps are homeomorphic onto open subsets, which implies in particular asymptotic completeness for small data.