A REPRESENTATION FORMULA RELATED TO SCHRDINGER OPERATORS

Schrdinger operator is a central subject in the mathematical study of quanturn mechanics.Consider the Schrodinger operator H=-△+Ⅴ on R,where △=d2/dx2 and the potentialfunction Ⅴ is real valued.In Fourier analysis.it is well—known that a square integrable functionadmits an expansion with exponentials as eigenfunctions of—△.A natural Conjecture is that anL2 function admits a similar expansion in terms of“eigenfunctions”of H,a perturbation of theLaplacian(see [7],Ch.Ⅺ and the notes),under certain condition on Ⅴ.     

A CLASS OF BOUNDARY PROBLEMS FOR SOME HYPERBOLIC EQUATIONS

A boundary problem for the Klein-Gordon equation in the strip O≤t≤T is considered with the boundary condition:the initial state at t=O and the final state at t=T.It is proven that the problem admits of an infinite number of solutions.The same result holds for a generic 2nd order hyperbolic equation in 2-variables.Using the result for the wave operator in 3-space dimensions we give a method to reconstruct functions whose integral on all unit spheres in R~3 is a given function.     

Projective Dirichlet boundary condition with applications to a geometric problem

Given a domain Ω ? R~n, let λ 0 be an eigenvalue of the elliptic operator L :=Σ!(i,j)~n =1?/?xi(a~(ij0 ?/?xj) on Ω for Dirichlet condition. For a function f ∈ L~2(Ω), it is known that the linear resonance equation Lu + λu = f in Ω with Dirichlet boundary condition is not always solvable.We give a new boundary condition P_λ(u|? Ω) = g, called to be pro jective Dirichlet condition, such that the linear resonance equation always admits a unique solution u being orthogonal to all of the eigenfunctions corresponding to λ which satisfies ||u||2,2 ≤ C(||f ||_2 +|| g||_(2,2)) under suitable regularity assumptions on ?Ω and L, where C is a constant depends only on n, Ω, and L. More a priori estimates,such as W~(2,p)-estimates and the C~(2,α)-estimates etc., are given also. This boundary condition can be viewed as a generalization of the Dirichlet condition to resonance equations and shows its advantage when applying to nonlinear resonance equations. In particular, this enables us to find the new indicatrices with vanishing mean(Cartan) torsion in Minkowski geometry. It is known that the geometry of indicatries is the foundation of Finsler geometry.     

Inverse Scattering for a Schroedinger Operator with a Repulsive Potential

We consider a pair of Hamiltonians （H, H0） on L2（R^n）, where H0=p^2 -x^2 is a SchrSdinger operator with a repulsive potential, and H = H0＋V（x）. We show that, under suitable assumptions on the decay of the electric potential, V is uniquely determined by the high energy limit of the scattering operator.     

A REGULARIZATION OF THE POINTWISE SUMMATION OF SINGULAR STURM-LIOUVILLE EXPANSIONS

Suppose that the information which we are given about anL~2[0,∞)function comes from a“harmonic analyzer”,which givesonly the coefficients of the expansion of f in eigenfunctions ofa Sturm-Liouville problem on[0,∞).It is well known that suchan expansion may not sum to the value of f,even at its continuitypoints.Let us further suppose that the analyzer is imperfect,sothat the coefficients it gives deviate somewhat from the correctones.     

ON CONVERGENCE OF WAVELET PACKET EXPANSIONS

It is well known that the Walsh-Fourier expansion of a function from the-block space (?)([0,1)), 1< q≤∞, converges pointwise a. e. We prove that the same result is true for the expansion of a function from (?) in certain periodized smooth periodic non-stationary wavelet packets bases based on the Haar filters. We also consider wavelet packets based on the Shannon filters and show that the expansion of Lp-functions, 1< p<∞. converges in norm and pointwise almost everywhere.     

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

Let L=??+V be the Schr ¨odinger operator on Rd, where?is the Laplacian on Rd and V 6=0 is a nonnegative function satisfying the reverse H¨older’s inequality. The authors prove that Riesz potential Iβa...     

On convergence of wavelet packet expansions

It is well known that the-Walsh-Fourier expansion of a function from the block space ([0, 1 ) ), 1 ＜q≤∞, converges pointwise a.e. We prove that the same result is true for the expansion of a function from in certain periodixed smooth periodic non-stationary wavelet packets bases based on the Haar filters. We also consider wavelet packets based on the Shannon filters and show that the expansion of Lp-functions, 1＜p＜∞, converges in norm and pointwise almost everywhere.     

TIME-DELAY AND SPECTRAL DENSITY FOR STARK HAMILTONIANS(Ⅱ)——ASYMPTOTICS OF TRACE FORMULAE

This paper studies the Schrodinger operator with a homogeneous electric field of the form -△+x_1+V(x), where x= (x_1,…, x_n)∈R~n. It is proved that in the specctral representation of the free Stark Hamiltonian, the time-delay operator in scattering theory can be expressed in trems of scattering matrix and under reasonable assumptions on the decay of the potential V, the on-shell time-delay operator is of trace class and its trace is related to the local spectral density via an explicit integral formula. Some asymptotics for the trace are estabhshed when the energy tends to infinity.     

Segal-Bargmann空间上带无界符号的Toeplitz算子

In this paper,we construct a function φ in L2(Cn,d Vα) which is unbounded on any neighborhood of each point in Cnsuch that Tφ is a trace class operator on the SegalBargmann space H2(Cn,d Vα).In addition,we also characterize the Schatten p-class Toeplitz operators with positive measure symbols on H2(Cn,d Vα).