Ito diffusions, modified capacity and harmonic measure. Applications to Schrodinger operators

Potential capacity and applications

We introduce a new capacity associated to a non negative function V. We apply this notion to the study of a necessary and sufficient condition to ensure the existence and uniqueness of a Schrodinger type equation with measure data and with an operator whose coefficients are discontinuous . Our method is based among others on a new topology and density results.

离散图上拟线性算子的Hardy权空间

The space of Hardy-weights for quasilinear operators on discrete graphs

本文在一般加权图上研究p-Schrödinger算子的Hardy不等式，通过广义容量给出Hardy权空间的Maz'ya型刻画，并利用临界性理论探讨极小化元存在性及其与Cheeger型估计的联系。

Wavelets, Multiplier spaces and application to Schrödinger type operators with non-smooth potentials

In this paper, we employ Meyer wavelets to characterize multiplier spaces between Sobolev spaces without using capacity. Further, we introduce logarithmic Morrey spaces $M^{t,τ}_{r,p}(\mathbb{R}^{n})$ to establish the inclusion relation between Morrey spaces and multiplier spaces. By wavelet characterization and fractal skills, we construct a counterexample to show that the scope of the index $τ$ of $M^{t,τ}_{r,p}(\mathbb{R}^{n})$ is sharp. As an application, we consider a Schrödinger type operator with potentials in $M^{t,τ}_{r,p}(\mathbb{R}^{n})$.

Dirac方程中长程扰动的初等交换子方法

Elementary commutator method for the Dirac equation with long-range perturbations

本文提出了一种初等方法，用于研究Dirac方程在长程扰动下的广义特征函数不存在性和共振估计，为稳态散射理论提供理论基础。

Strong maximum principle for Schrödinger operators with singular potential

We prove that for every $p > 1$ and for every potential $V \in L^p$, any nonnegative function satisfying $-Δu + V u \ge 0$ in an open connected set of $\mathbb{R}^N$ is either identically zero or its level set $\{u = 0\}$ has zero $W^{2, p}$ capacity. This gives an affirmative answer to an open problem of Bénilan and Brezis concerning a bridge between Serrin-Stampacchia's strong maximum principle for $p > \frac{N}{2}$ and Ancona's strong maximum principle for $p = 1$. The proof is based on the construction of suitable test functions depending on the level set $\{u = 0\}$ and on the existence o…

具有反射的非线性随机扩散-对流方程

A nonlinear stochastic diffusion-convection equation with reflection

本文研究了任意空间维度下具有反射的非线性伪单调随机扩散-对流演化问题，证明了带非负初值的方程解及防止解取负值的反射测度的存在性，并通过研究抛物容度下解的拟处处定义代表元证明了测度的极小性条件。

涉及到边界子流形距离和梯度源非线性项的带Hardy势的半线性Schrödinger方程

Semilinear Schrödinger equations with Hardy potentials involving the distance to a boundary submanifold and gradient source nonlinearities

本文研究了带Hardy势和梯度源非线性项的半线性Schrödinger方程的正解，证明了在边界测度数据总质量较小时两种主要情形下解的存在性，并给出了超临界范围下基于容度的存在性判据。

On the pointwise convergence of the integral kernels in the Feynman-Trotter formula

We study path integrals in the Trotter-type form for the Schrödinger equation, where the Hamiltonian is the Weyl quantization of a real-valued quadratic form perturbed by a potential $V$ in a class encompassing that - considered by Albeverio and Itô in celebrated papers - of Fourier transforms of complex measures. Essentially, $V$ is bounded and has the regularity of a function whose Fourier transform is in $L^1$. Whereas the strong convergence in $L^2$ in the Trotter formula, as well as several related issues at the operator norm level are well understood, the original Feynman's idea concerne…