Quantitative Green's function estimates for lattice quasi-periodic Schrodinger operators

In this paper,we establish quantitative Green’s function estimates for some higher-dimensional lattice quasi-periodic(QP)Schrodinger operators.The resonances in the estimates can be described via a pair of symmetric zeros of certain functions,and the estimates apply to the sub-exponential-type non-resonance conditions.As the application of quantitative Green’s function estimates,we prove both the arithmetic version of Anderson localization and the finite volume version of(1/2-)-Holder continuity of the integrated density of states(IDS)for such QP Schrodinger operators.This gives an affirmative answer to Bourgain’s problem in Bourgain(2000).

Holder Continuity of Spectral Measures for the Finitely Differentiable Quasi-Periodic Schrodinger Operators

In the present paper,we prove the 1/2-Holder continuity of spectral measures for the C^k Schrodinger operators.This result is based on the quantitative almost reducibility and an estimate for the growth of the Schrodinger cocycles in[5].

LOCAL ESTIMATE ABOUT SCHRODINGER MAXIMAL OPERATOR ON H-TYPE GROUPS

Let △ be full Laplacian on H-type group G. Then for every compact set D Ga local estimate of the Schrodinger maximal operator holds, that is,∫D^sup0〈t〈1｜e^it△f（x）｜^2dx≤｜｜f｜｜^2H^s,s〉1/2We also show that the above inequality fails when s 〈 1/4.

BOUNDEDNESS OF VARIATION OPERATORS ASSOCIATED WITH THE HEAT SEMIGROUP GENERATED BY HIGH ORDER SCHRODINGER TYPE OPERATORS

In this article,we derive the Lp-boundedness of the variation operators associated with the heat semigroup which is generated by the high order Schrodinger type operator(—Δ)^2+V%2 in R%n(n≥5)with V being a nonnegative potential satisfying the reverse Holder inequality.Furt her more,we prove the boundedness of the variation operators on associated Morrey spaces.In the proof of the main results,we always make use of the variation inequalities associated with the hea t semigroup genera ted by the biharmonic operator(-Δ)2.

AN UPBOUND OF HAUSDORFF’S DIMENSION OF THE DIVERGENCE SET OF THE FRACTIONAL SCHRODINGER OPERATOR ON H^(s)(R^(n))

Given n≥2 and α≥1/2,we obtained an improved upbound of Hausdorff's dimension of the fractional Schrodinger operator;that is,supf∈H^(s)(R^(n)) dim_(H){x∈R^(n):limt→0 e^(it)(-△)^(α) f(x)≠f(x)}≤n+1-2(n+1)s/n for n/2(n+1)<s≤n/2.

An Optimization Problem of Boundary Type for Cooperative Hyperbolic Systems Involving Schrodinger Operator

In this paper, we consider cooperative hyperbolic systems involving Schr?dinger operator defined on ?Rn. First we prove the existence and uniqueness of the state for these systems. Then we find the necessary and sufficient conditions of optimal control for such systems of the boundary type. We also find the necessary and sufficient conditions of optimal control for same systems when the observation is on the boundary.

Characterizations of product Hardy space associated to Schrodinger operators

Let L1 and L2 be the Schrodinger operators on Rn and Rm, respectively. By using different maximal functions and Littlewood-Paley g function on distinct variables, in this paper,some characterizations for functions in the product Hardy space HL1,L21(R^n×R^m) associated to operators L1 and L2 are obtained.

Schrodinger Operators on Graphs and Branched Manifolds

We consider the Schrodinger operators on graphs with a finite or countable number of edges and Schr?dinger operators on branched manifolds of variable dimension. In particular, a description of self-adjoint extensions of symmetric Schr?dinger operator, initially defined on a smooth function, whose support does not contain the branch points of the graph and branch points of the manifold. These results are obtained for graphs with a single vertex, graphs with multiple vertices and graphs with a single vertex and countable set of rays.

Carleson Measure Associated with the Fractional Heat Semigroup of Schrodinger Operator

Let L=-△+V be a Schrodinger operator,where△is the Laplacian on Rd and the nonnegative potential V belongs to the reverse Holder class Ba/2.In this paper,we define a new version of Carleson measure associated with the fractional heat semigroup of Schrodinger operator L.We will characterize the Campanato spaces and the predual spaces of the Hardy spaces by the new Carleson measure.

Reducibility for Schrodinger Operator with Finite Smooth and Time-Quasi-periodic Potential

In this paper, the author establishes a reduction theorem for linear Schr?dinger equation with finite smooth and time-quasi-periodic potential subject to Dirichlet boundary condition by means of KAM(Kolmogorov-Arnold-Moser) technique. Moreover, it is proved that the corresponding Schr?dinger operator possesses the property of pure point spectra and zero Lyapunov exponent.

一类Schrodinger型算子在关于非负位势的变指数Morrey空间上的有界性

本文考虑了一类Schr?dinger型算子和其交换子的有界性问题.利用其在L^p空有界性间上的,获得了一类Schr?dinger型算子和其交换子在变指数Morrey空间上的有界性.

关于Schrodinger算子的任意相邻两特征值之差的上界估计

设Ω是Ｒ＂中有界光滑区域．用λ记Ω上Ｓｃｈｒｏｄｉｎｇｅｒ算子的具Ｄｉｒｉｃｈｌｅｔ边界条件的特征值问题的第ｉ个特征值。本文利用极大极小原理研究任意相邻两特征值之差λｋ＋１－λｋ的上界，推广了Ｐａｙｎｅ和Ｓ．Ｔ．Ｙａｎ等人的有关结果。

与高阶Schrodinger型算子相关的变分算子的Lipschitz交换子

设L=(-Δ)^(2)+V^( 2)是ℝn(n≥5)上的高阶Schrodinger型算子,其中非负位势V属于反向Holder类RH q(q>n/2).记Vρ(e-t L)为与高阶Schr dinger型算子L相关的变分算子.基于Herz型Hardy空间的原子分解理论,利用Schrodinger型算子的性质,证明该类变分算子与Lipschitz函数构成的交换子的L q有界性,并进一步证明该类变分算子的交换子从Herz型Hardy空间到Herz空间是有界的,在Morrey-Herz空间上也是有界的.

加权模不等式和Schrodinger算子的特征值估计

本文应用二进极大算子的加权模不等式和I_1算子的性质。估计了广义Schrodinger算子H=-A-U+V的最小特征值,并与龙瑞麟先生对某些H=-A-U型算子的特征值估计进行了比较。

与高阶Schrodinger-型算子相关的变分算子在Herz-型空间的有界性

设L=(-Δ)^(2)+V^(2)是R^(n)(n≥5)上的高阶Schrodinger-型算子,其中非负位势V属于反向H lder类RH q q>n 2.记V_(ρ)(e^(-t L))为与高阶Schrodinger-型算子L生成的热半群相关的变分算子,基于此类变分算子在L^(q)空间上有界的结果,应用Herz-型Hardy空间的原子分解理论,以及Schrodinger-型算子的性质,进行不等式的估计,证明了V_(ρ)(e^(-t L))是从Herz-型Hardy空间到Herz空间的有界算子,也是Morrey-Herz空间上的有界算子.

SchrOdinger算子的极大耗散扩张

本文研究Ｓｃｈｒｏｄｉｎｇｅｒ算子的极大耗散扩张，引入广义不定度规空间，得到在自然边界空间中，Ｓｃｈｒｏｄｉｎｇｅｒ算子极大耗散扩张表示形式，为进一步研究非线性Ｓｃｈｒｏｄｉｎｇｅｒ方程无穷维动力系统的长期混沌行为作准备。

BOUNDEDNESS OF FRACTIONAL MAXIMAL OPERATOR AND THEIR HIGHER ORDER COMMUTATORS IN GENERALIZED MORREY SPACES ON CARNOT GROUPS

In the article we consider the fractional maximal operator Mα, 0 ≤α 〈 Q on any Carnot group G （i.e., nilpotent stratified Lie group） in the generalized Morrey spaces Mp,φ（G）, where Q is the homogeneous dimension of G. We find the conditions on the pair （φ1, φ2） which ensures the boundedness of the operator Ms from one generalized Morrey space Mp,φ1 （G） to another Mq,φ2 （G）, 1. 〈 p ≤q 〈 ∞. 1/p - 1/q = α/Q, and from the space M1,φ1 （G） to the weak space Wq,φ2 （G）, 1 〈 q 〈 ∞, 1 - 1/q = α/Q. Also find conditions on the φ which ensure the Adams type boundedness of the Ms from M α （G） from Mp,φ^1/p（G）to Mq,φ^1/q（G） for 1 〈p〈q〈∞ and fromM1,φ（G） toWMq,φ^1/q（G）for 1〈q〈∞. In the case b ∈ BMO（G） and 1 〈 p 〈 q 〈 ∞, find the sufficient conditions on the pair （φ1, φ2） which ensures the boundedness of the kth-order commutator operator Mb,α,k from Mp,φ1 （G） to Mq,φ2（G） with 1/p - 1/q = α/Q. Also find the sufficient conditions on the φ which ensures the boundedness of the operator Mb,α,k from Mp,φ^1/p（G） tom Mp,φ^1/p （G） for 1 〈p〈q〈∞. In all the cases the conditions for the boundedness of Mα are given it terms of supremaltype inequalities on （φ1, φ2） and φ , which do not assume any assumption on monotonicity of （φ1, φ2） and φ in r. As applications we consider the SchrSdinger operator -△G ＋ V on G, where the nonnegative potential V belongs to the reverse Holder class B∞（G）. The MB,φ1 - Mq,φ2 estimates for the operators V^γ（-△G ＋ V）^-β and V^γ△↓G（-△G ＋ V）^-β are obtained.

AREA FUNCTIONS ON HARDY SPACES ASSOCIATED TO SCHRDINGER 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.

The Fractional Maximal Operator and Marcinkiewicz Integrals Associated with Schr?dinger Operators on Morrey Spaces with Variable Exponent

In this paper, we prove the boundedness of the fractional maximal operator, Hardy-Littlewood maximal operator and marcinkiewicz integrals associated with Schrodinger operator on Morrey spaces with variable exponent.

GROUND STATES FOR FRACTIONAL SCHRODINGER EQUATIONS WITH ELECTROMAGNETIC FIELDS AND CRITICAL GROWTH

In this article,we study the following fractional Schrodinger equation with electromagnetic fields and critical growth(-Δ)^sAu+V(x)u=|u|^2^*s-2)u+λf(x,|u|^2)u,x∈R^n,where(-Δ)^sA is the fractional magnetic operator with 0<s<1,N>2s,λ>0,2^*s=2N/(N-2s),f is a continuous function,V∈C(R^n,R)and A∈C(R^n,R^n)are the electric and magnetic potentials,respectively.When V and f are asymptotically periodic in x,we prove that the equation has a ground state solution for largeλby Nehari method.