ISOMORPHISMS OF VARIABLE HARDY SPACES ASSOCIATED WITH SCHRÖDINGER OPERATORS

Let L:=-△+V be the Schrodinger operator on Rⁿwith n≥3,where V is a non-negative potential satisfying△^-1(V)∈L^∞(Rⁿ).Let w be an L-harmonic function,determined by V,satisfying that there exists a positive constantδsuch that,for any x∈Rn,0<δ≤w(x)≤1.Assume that p(·):Rⁿ→(0,1]is a variable exponent satisfying the globally log-H?lder continuous condition.In this article,the authors show that the mappings HL^p(·))(Rⁿ)■f■wf∈H^p(·)(Rⁿ)and HL^p(·)(Rⁿ)■f■(-△)^1/2L^-1/2(f)∈H^p(·)(Rⁿ)are isomorphisms between the variable Hardy spaces HL^p(·)(Rⁿ),associated with L,and the variable Hardy spaces H^p(·)(Rⁿ).     

STRICHARTZ ESTIMATES FOR A SCHRÖDINGER OPERATOR WITH NONSMOOTH COEFFICIENTS

ABSTRACT

We prove Strichartz type estimates for the Schrödinger equation corresponding to a second order elliptic operator with variable coefficients. We assume that the coefficients are a C ² compactly supported perturbation of the identity, satisfying a nontrapping condition.     

CONVERGENCE OF THE SCHRÖDINGER–POISSON SYSTEM TO THE INCOMPRESSIBLE EULER EQUATIONS

ABSTRACT

We study here an asymptotic limit of the Schrödinger–Poisson system. We prove that the current converges toward a dissipative solution of the Euler equations when we consider a semi-classical quasi-neutral limit. The proof involves the modulated energy introduced by Brenier in Ref. [2] Brenier, Y. 2000. Convergence of the Vlasov-Poisson System to the Incompressible Euler Equations. Comm. Partial Differential Equations, 25(3–4): 737–754. [Taylor & Francis Online], [Web of Science ®] , [Google Scholar].     

LITTLEWOOD-PALEY THEOREM FOR SCHR DINGER OPERATORS

Let H be a Schrodinger operator on Rn. Under a polynomial decay condition for the kernel of its spectral operator, we show that the Besov spaces and Triebel-Lizorkin spaces associated with H are well defined. We further give a Littlewood-Paley characterization of Lp spaces in terms of dyadic functions of H. This generalizes and strengthens the previous result when the heat kernel of H satisfies certain upper Gaussian bound.     

A STRUCTURE-PRESERVING DISCRETIZATION OFNONLINEAR SCHRDINGER EQUATION

1.IntroductionManyimportantdifferentialequationsofevolutiontypeinphysicsandmechanicshavespecificgeometricstructure.Forinstance,theHamiltoniansystemsinclassicalmechanics,theSchrodingerequationinquantum,theKorteweg-deVriesandKleinGordonequationsofnonli...     

ORBITAL INSTABILITY OF STANDING WAVES FOR THE GENERALIZED 3D NONLOCAL NONLINEAR SCHR?DINGER EQUATIONS

The existence and orbital instability of standing waves for the generalized threedimensional nonlocal nonlinear Schr¨odinger equations is studied. By defining some suitable functionals and a constrained variational problem, we first establish the existence of standing waves, which relys on the inner structure of the equations under consideration to overcome the drawback that nonlocal terms violate the space-scale invariance. We then show the orbital instability of standing waves. The arguments depend upon the conservation laws of the mass and of the energy.     

THE INVARIANT MANIFOLDS FOR A PERTURBED QUINTIC-CUBIC SCHRDINGER EQUATION

In this paper, the authors apply the analytical method to establish the persistence of invariant manifolds for certain perturbation of the quintic-cubic Schrodinger equation under even periodic boundary conditions.     

ON INITIAL BOUNDARY VALUE PROBLEMS FOR NONLINEAR SCHRDINGER EQUATIONS

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HARNACK’S INEQUALITY FOR GENERALIZED SUBELLIPTIC SCHRDINGER OPERATORS

We prove a uniform Harnack inequality for nonnegative solutions of ΔGu Gμu = 0, where ΔG is a sublaplacian, μ is a non-negative Radon measure and satisfying scale-invariant Kato condition.     

AREA FUNCTIONS ON HARDY SPACES ASSOCIATED TO SCHRDINGER OPERATORS

In this paper, the author gives a characterization of atomic Hardy spaces associated to Schrodinger operators by using area functions, and hence gets the dual spaces of atomic Hardy spaces.     

EXPLICIT DISSIPATIVE SCHEMES FOR BOUNDARY PROBLEMS OF GENERALIZED SCHRDINGER SYSTEMS

In this paper, the author constructs a class of explicit schemes, spanning two time levels, forthe initial--boundary-value problems of generalized nonlinear Schrodinger systems, and proves theconvergence of these schemes with a series of prior estimates. For a single Schrodinger equation, theschemes are identical with those of the article [1].     

SOME TRACE FORMULAS FOR THE SCHRDINGER EQUATION WITH ENERGY-DEPENDENT POTENTIAL

The importance of the inverse scattering problem for the Schrdinger equation with energy-dependent potential:     

NODAL BOUND STATES WITH CLUSTERED SPIKES FOR NONLINEAR SCHRDINGER EQUATIONS

We consider the following nonlinear Schr¨odinger equations -ε2△u + u = Q(x)|u|p-2u in RN, u ∈ H1(RN),where ε is a small positive parameter, N ≥ 2, 2 p ∞ for N = 2 and 2 p 2N/N-2 for N ≥ 3. We prove that this problem has sign-changing(nodal) semi-classical bound states with clustered spikes for sufficiently small ε under some additional conditions on Q(x).Moreover, the number of this type of solutions will go to infinity as ε→ 0+.     

HOMOGENIZATION OF SOME LINEAR AND SEMILINEAR SCHRDINGER EQUATIONS WITHREAL POTENTIAL

1 IntroductionIn this paPer, we collsider the initial and boundaxy value probIem of linear and sentilillearScl1r5dinger equation as fOllows:l ie(x):5:i".,'<">"'>.'<",,'>,"=' 1:!\:?0t:)t>, (l1)wllere E > 0, fl C RN is a bounded open dolllain and T is an ar…     

SOME TRACE FORMULAS FOR THE SCHRDINGER EQUATION WITH ENERGY-DEPENDENT POTENTIAL

The importance of the inverse scattering problem for the Schrdinger equation with energy-dependent potential: has been noticed by quite a few authors. Jaulent and Miodek have used it to solve some nonlinear evolution equations with isospectral condition ξ_t=0. Li and Zhuang have further made use of (1) to expand the solvable class of nonlinear evolution equations. Several physical problems in absorption media can be reduced to the inverse scattoring problem of(1) (see [4]).