关于一般的Cwikel-Lieb-Rozenblum和Lieb-Thirring不等式
On general Cwikel-Lieb-Rozenblum and Lieb-Thirring inequalities
摘要：
这些经典的不等式允许人们估计一类广泛的薛定谔算子的负特征值的个数和$S_γ=\sum |λ_i|^γ$。我们提供了一个详细的证明，这些不等式的运营商的功能在度量空间中使用经典的Lieb方法的基础上的卡茨-费曼公式。本文的主要目标是一组新的例子，其中包括扰动的安德森运营商，运营商的自由，幂零和可解群，运营商的量子图形，马尔可夫过程与独立增量。对例子的研究需要对相应的抛物半群在对角线上的核的精确估计。在某些情况下，内核以指数方式衰减为$t\to \infty $。这使我们能够考虑非常缓慢的衰减潜力，并获得一些结果是精确的，在数学尺度。
英文摘要：
These classical inequalities allow one to estimate the number of negative eigenvalues and the sums $S_γ=\sum |λ_i|^γ$ for a wide class of Schrödinger operators. We provide a detailed proof of these inequalities for operators on functions in metric spaces using the classical Lieb approach based on the Kac-Feynman formula. The main goal of the paper is a new set of examples which include perturbations of the Anderson operator, operators on free, nilpotent and solvable groups, operators on quantum graphs, Markov processes with independent increments. The study of the examples requires an exact estimate of the kernel of the corresponding parabolic semigroup on the diagonal. In some cases the kernel decays exponentially as $t\to \infty $. This allows us to consider very slow decaying potentials and obtain some results that are precise in the logarithmical scale.
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