引力下的测不准原理引理与薛定谔算子的离散谱

Boundedness of Schr{ö}dinger operator in energy space

On a complete weighted Riemannian manifold $(M^n,g,μ)$ satisfying the doubling condition and the Poincar{é} inequalities, we characterize the class of function $V$ such that the Schr{ö}dinger operator $Δ-V$ maps the homogeneous Sobolev space $W_o^{1,2} (M)$ to its dual space. On Euclidean space, this result is due to Maz'ya and Verbitsky. In the proof of our result, we investigate the weighted $L^2$-boundedness of the Hodge projector.

Lower bound of Schr{ö}dinger operators on Riemannian manifolds

We show that a weighted manifold which admits a relative Faber Krahn inequality admits the Fefferman Phong inequality V $ψ$, $ψ$ $\le$ CV $ψ$ 2 , with the constant depending on a Morrey norm of V , and we deduce from it a condition for a L 2 Hardy inequality to holds, as well as conditions for Schr{ö}dinger operators to be positive. We also obtain an estimate on the bottom of the spectrum for Schr{ö}dinger operators.

标准N-球面和双曲N-空间上非线性Schr $\bf{\ddot{O}}$ dinger方程解的爆破

Blow-up of solutions to the nonlinear Schr$\bf{\ddot{O}}$dinger equations on standard N-sphere and hyperbolic N-space

本文部分解决了标准2-球面$S^2$和双曲2-空间$H^{2}(-1)$上的非线性Schr $\ddot{o}$ dinger方程有限时间爆破性质的长期未解决问题.利用类似的思想，我们在高维标准球面和双曲$n$ -空间上建立了这样的爆破结果。给出了具有$n\geq 2$的$n$维黎曼曲积流形的推广.