锥中与稳态的薛定谔算子相关的广义Martin函数无穷远处的控制（英文）

本文研究了稳态的薛定谔算子的Dirichlet问题和Martin函数的边界行为.利用广义Martin表示和稳态的薛定谔算子对应的常微分方程基本解,在具有光滑边界的锥形区域中获得了与稳态的薛定谔算子相关的广义Martin函数无穷远处广义调和控制的一些刻画,推广了拉普拉斯算子情形的结果.     

锥中与稳态的薛定谔算子相关的容度与广义Green能量（英文）

本文首先引入了与稳态的薛定谔算子相关的广义容度和Green能量,然后得到它们和概率测度间的相互关系.进而,利用容度工具刻画了锥中与稳态的薛定谔算子相关的极细集和稀疏集.     

薛定谔型算子在广义Morrey空间上的有界性

研究薛定谔型算子V~(β_1)▽(-△+V)~(-β_2).当位势函数V属于逆Holder函数类RH_s(sn/2)时,利用函数分解技巧,得到了这个算子在广义Morrey空间上的有界性质.这个结果丰富和改进了一些已有的一些结论.     

平面负位势相关的Schrdinger方程的非负广义解

设β是复平面上圆盘 内的一个零容紧致集.考虑 上的定常Schrodinger方程(-△+μ)u=0,其中位势μ≤0是Kato类Radon测度,将方程在广义函数意义下的在 ,上取极限值0的非负连续解族记为μH+.对Ωαβ的Kerekjato-Stoilow意义下的理想边界β的任一点ζ,本文通过定义μH+→μH+的线性算子πζ,引入Martin函数Kζ,证明了μH+= ,其中     

与薛定谔算子相关的g_λ~*-函数

令L=-△+V为一个薛定谔算子,其中△是欧式空间R~d上的拉普拉斯算子,V是属于逆Hlder类B_(d/2)的非负位势.该文将研究与薛定谔算子L相关的g_λ~*-函数的有界性.     

锥中无穷远处与稳态的薛定谔算子相关的稀疏集的覆盖性质

在锥形区域里关于任意正测度得到了与稳态的薛定谔算子有关的Green位势和Poisson积分在无穷远处的值分布,然后提出了a-稀疏集的一个覆盖性质.     

白噪声分析中广义算子值函数的Bochner-Wick积分

白噪声广义算子在白噪声分析理论及其应用中起着十分重要的作用. 本文主要讨论了白噪声广义算子值函数的积分及相关问题. 主要工作有: 引入了广义算子值测度的概念, 分别讨论了这种测度在象征和算子p-范数意义下的变差及相互关系; 借助于广义算子的Wick积运算, 引入了广义算子值函数关于广义算子值测度的一种积分---Bochner-Wick积分, 讨论了这种积分的性质, 建立了相应的收敛定理并且展示了其在量子白噪声理论中的应用; 探讨了Bochner-Wick积分的Fubini定理及相关问题.     

广义函数法边界积分方程的建立

从广义函数论出发,本文引入一特殊广义函数δθ_P,通过它以及它的各阶导数建立了任一足够光滑函数的各阶导数的边界积分方程。对于由线性偏微分算子定义的问题,只要存在着相应的基本解,问题的偏微分方程总可转换成边界积分方程。     

基于广义Greiner算子的Ostrowski型不等式和带有边界项的Hardv不等式（英文）

本文基于广义Greiner算子建立了一类Ostrowski型不等式和带有边界项的Hardy不等式.采用的技巧是先建立函数表示公式及水平梯度的L~∞范数,再进一步获得球域及一般有界域上的Ostrowski型不等式.利用同样的技巧,获得了带有边界项的Hardy不等式.     

满足H(m)条件的奇异积分算子的交换子的有界性

主要讨论了满足H(m)条件的奇异积分算子与Lipschitz函数的交换子在L~p和Hardy空间的有界性,并把这个结果应用于与薛定谔算子相关的Riesz变换.     

A new type of minimal thinness with respect to the stationary Schrödinger operator and its applications

In this paper we define a new type of minimal thinness with respect to the stationary Schrödinger operator at a fixed Martin–Sch boundary point of a cylinder. We also discuss some criteria for it.     

Operator kernel estimates for functions of generalized Schrödinger operators

We study the decay at large distances of operator kernels of functions of generalized Schrödinger operators, a class of semibounded second order partial differential operators of mathematical physics, which includes the Schrödinger operator, the magnetic Schrödinger operator, and the classical wave operators (i.e., acoustic operator, Maxwell operator, and other second order partial differential operators associated with classical wave equations). We derive an improved Combes-Thomas estimate, obtaining an explicit lower bound on the rate of exponential decay of the operator kernel of the resolvent. We prove that for slowly decreasing smooth functions the operator kernels decay faster than any polynomial.

     

On decay of solutions to nonlinear Schrödinger equations

We present general results on exponential decay of finite energy solutions to stationary nonlinear Schrödinger equations. Under certain natural assumptions we show that any such solution is continuous and vanishes at infinity. This allows us to interpret the solution as a finite multiplicity eigenfunction of a certain linear Schrödinger operator and, hence, apply well-known results on the decay of eigenfunctions.

     

On the singular spectrum of Schrödinger operators with decaying potential

The relation between Hausdorff dimension of the singular spectrum of a Schrödinger operator and the decay of its potential has been extensively studied in many papers. In this work, we address similar questions from a different point of view. Our approach relies on the study of the so-called Krein systems. For Schrödinger operators, we show that some bounds on the singular spectrum, obtained recently by Remling and Christ-Kiselev, are optimal.

     

Sub-exponential decay of operator kernels for functions of generalized Schrödinger operators

We study the decay at large distances of operator kernels of functions of generalized Schrödinger operators. We prove sub-exponential decay for functions in Gevrey classes and exponential decay for real analytic functions.

     

The Schrödinger operator

In this paper we study the maximum dissipative extension of the Schrödinger operator, introduce the generalized indefinite metric space, obtain the representation of the maximum dissipative extension of the Schrödinger operator in the natural boundary space and make preparation for the further study of the longtime chaotic behavior of the infinite-dimensional dynamics system in the Schrödinger equation.

     

The Schrödinger equation with a moving point interaction in three dimensions

In the case of a single point interaction we improve, by using different methods, the existence theorem for the unitary evolution generated by a Schrödinger operator with moving point interactions obtained by Dell'Antonio, Figari and Teta.

     

Beurling-Nevanlinna inequality for subfunctions of the stationary Schrödinger operator

The classical Beurling-Nevanlinna upper bound for subharmonic functions is extended to subsolutions of the stationary Schrödinger equation.

     

Resolvent Estimates and Matrix-Valued Schrödinger Operator with Eigenvalue Crossings; Application to Strichartz Estimates

We consider a semi-classical Schrödinger operator with a matrix-valued potential presenting eigenvalue crossings on isolated points. We obtain estimates for the boundary values of the resolvent under a generalized non-trapping assumption. As a consequence, we prove the smoothing effect of this operator, derive Strichartz type estimate for the propagator and get an existence theorem for a system of non-linear Schrödinger equations.     

Asymptotic Dirichlet problem for the Schrödinger operator via rough isometry

We pose and solve the asymptotic Dirichlet problem for the Schrödinger operator via rough isometries on a certain class of Riemannian manifolds. With suitable potentials, we give the solvability of the problem for a naturally defined class of data functions.