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.

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).

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.

Inverse problems for radial Schrodinger operators with the missing part of eigenvalues

We study inverse spectral problems for radial Schrodinger operators in L^(2)(0,1).It is well known that for a radial Schrodinger operator,two spectra for the different boundary conditions can uniquely determine the potential.However,if the spectra corresponding to the radial Schrodinger operators with the two potential functions miss a finite number of eigenvalues,what is the relationship between the two potential functions?Inspired by Hochstadt(1973)'s work,which handled the Sturm-Liouville operator with the potential q∈L^(1)(0,1),we give a corresponding result for radial Schrodinger operators with a larger class of potentials than L^(1)(0,1).When q∈L^(1)(0,1),we also consider the case where the spectra corresponding to the radial Schrodinger operators with the two potential functions miss an infinite number of eigenvalues and the eigenvalues are close in some sense.

A Marcinkiewicz criterion for L^(p)-multipliers related to Schrodinger operators with constant magnetic fields

In this paper,we follow Dappa’s work to establish the Marcinkiewicz criterion for the spectral multipliers related to the Schrdinger operator with a constant magnetic field.We prove that if m and m′are locally absolutely continuous on(0,∞)and ‖m‖∞+sup j∈Z2j 2i+1 r|m′′(r)|dr<∞,then the multiplier defined by m(t)is bounded on Lpfor 2n/(n+3)<p<2n/(n-3)with n 3.Our approach is based on the estimates for the generalized Littlewood-Paley functions of the spectral representation of the Schrdinger operator with a constant magnetic field.

Generalized Weighted Morrey Estimates for Marcinkiewicz Integrals with Rough Kernel Associated with Schrodinger Operator and Their Commutators

Let L =-△+V(x) be a Schrodinger operator, where △ is the Laplacian on R^n,while nonnegative potential V(x) belonging to the reverse Holder class. The aim of this paper is to give generalized weighted Morrey estimates for the boundedness of Marcinkiewicz integrals with rough kernel associated with Schrodinger operator and their commutators.Moreover, the boundedness of the commutator operators formed by BMO functions and Marcinkiewicz integrals with rough kernel associated with Schrodinger operators is discussed on the generalized weighted Morrey spaces. As its special cases, the corresponding results of Marcinkiewicz integrals with rough kernel associated with Schrodinger operator and their commutators have been deduced, respectively. Also, Marcinkiewicz integral operators, rough Hardy-Littlewood(H-L for short) maximal operators, Bochner-Riesz means and parametric Marcinkiewicz integral operators which satisfy the conditions of our main results can be considered as some examples.

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.

On 4-order Schrodinger operator and Beam operator

Wc show that there is no localization for the 4-order Schrodinger operator Jt,Af and Bearn operator 38%more precisely,on the one hand,we show that the 4-order Schrodinger operator Atf, does not converge pointwise to zero as t→0 provided f∈H^s(R)with compact support and 0<s<1/4 by constructing a counterexample in R.On the other hand,we show that the Beam operator Btf also has the same property with the 4-order Schrodinger operator Jt,4f.Hence,we find that the Hausdorff dimension of the divergence set for Jt,4f and Btf is a1,J4(s)=a1,B(s)=1 as 0<s<1/4.

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.

A Note on the Convergence of the Schrodinger Operator along Curve

In this paper we set up a convergence property for the fractional Schodinger operator for 0<α<1.Moreover,we extend the known results to non-tangent convergence and the convergence along Lipschitz curves.

Boundedness of Area Functions Related to Schrodinger Operators and Their Commutators in Weighted Hardy Spaces

In this paper,we consider the area function SQ related to the Schrodinger operator ■ and its commutator SQ,b´establish the boundedness of SQ from Hp(w)to L^(p)_(ρ)(w)to L^(p)(w)or WL^(p)(w),as well as the boundedness of SQ,b´from H^(1)_(ρ)(w)to WL^(1)(w).

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.

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.

Hardy Spaces H_L^p(R^n) Associated with Higher-Order Schrdinger Type Operators

Let L = L0 ＋ V be the higher order Schrodiger type operator where L0 is a homogeneous elliptic operator of order 2m in divergence form with bounded coeffi- cients and V is a real measurable function as multiplication operator （e.g., including （-△）m＋v （m∈N） as special examples）. In this paper, assume that V satisfies a strongly subcritical form condition associated with L0, the authors attempt to establish a the- HL （R） （0 〈 p ≤ 1） associated with the higher order Schrodinger ory of Hardy space P n type operator L. Specifically, we first define the molecular Hardy space Hp （JRn） by the so-called （p, q,ε, M） molecule associated to L and then establish its characterizations by the area integral defined by the heat semigroup e-tL.

The Boundedness of the Commutator for Riesz Potential Associated with Schr dinger Operator on Morrey Spaces

Let L=-△＋V be the Schrodinger operator on Rd, where？is the Laplacian on Rd and V ≠=0 is a nonnegative function satisfying the reverse Holder’s inequality. The authors prove that Riesz potential Iβand its commutator [b,Iβ] associated with L map from Mp,qα,v into Mp1,q1α,v .