微分算子的迹恒等式及其对本征值分布的应用
Trace identities for commutators, with applications to the distribution of eigenvalues
摘要：
我们证明了迹恒等式的算子，这是用来获得和规则和尖锐的普遍界的特征值的周期Schroedinger算子和Schroedinger算子浸入流形上。特别是，我们证明了本征值lambda_{N+1}的下界谱，本征值的平均值的比率上的界，和普遍单调性的特征值矩，这意味着尖锐版本的Lieb-Thirring不等式。在几何的背景下，我们推导出一个版本的赖利不等式，通过一个普适常数乘以最大中曲率的平方乘以N^{2/d}，将Laplace-Beltrami算子的本征值lambda_{N+1}限定在d维浸入流形上。
英文摘要：
We prove trace identities for commutators of operators, which are used to derive sum rules and sharp universal bounds for the eigenvalues of periodic Schroedinger operators and Schroedinger operators on immersed manifolds. In particular, we prove bounds on the eigenvalue lambda_{N+1} in terms of the lower spectrum, bounds on ratios of means of eigenvalues, and universal monotonicity properties of eigenvalue moments, which imply sharp versions of Lieb-Thirring inequalities. In the geometric context we derive a version of Reilly's inequality, bounding the eigenvalue lambda_{N+1} of the Laplace-Beltrami operator on an immersed manifold of dimension d by a universal constant times the square of the maximal mean curvature times N^{2/d}.
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