Cwikel的边界重新加载
Cwikel's bound reloaded
摘要：
著名的Cwikel-Lieb-Rozenblum（Cwikel-Lieb-Rozenblum）界，这是薛定谔算符束缚态个数的半经典界，在20世纪70年代得到了证明。在Cwikel，Lieb和Rozenblum的相当不同的证明中，Lieb给出了最好的常数，Rozenblum似乎没有给出任何合理的常数估计，而Cwikel的证明据说给出了一个常数，它至少是大约2个数量级的真理。在过去的40多年里，这种情况没有太大变化。事实证明，这种共同的信念，即Cwikel的方法产生坏的常数，是不是一成不变的：我们给一个激烈的简化Cwikel的原始方法，导致一个令人惊讶的好界的常数在Einstein不等式。我们的证明也是相当灵活的，并导致相当精确的范围为一大类薛定谔型算子广义动能。此外，它突出了一个自然的，但被忽视的连接的最大傅立叶乘子从调和分析的边界的最大的傅立叶界。
英文摘要：
There are a couple of proofs by now for the famous Cwikel--Lieb--Rozenblum (CLR) bound, which is a semiclassical bound on the number of bound states for a Schrödinger operator, proven in the 1970s. Of the rather distinct proofs by Cwikel, Lieb, and Rozenblum, the one by Lieb gives the best constant, the one by Rozenblum does not seem to yield any reasonable estimate for the constants, and Cwikel's proof is said to give a constant which is at least about 2 orders of magnitude off the truth. This situation did not change much during the last 40+ years. It turns out that this common belief, i.e, Cwikel's approach yields bad constants, is not set in stone: We give a drastic simplification of Cwikel's original approach which leads to an astonishingly good bound for the constant in the CLR inequality. Our proof is also quite flexible and leads to rather precise bounds for a large class of Schrödinger-type operators with generalized kinetic energies. Moreover, it highlights a natural but overlooked connection of the CLR bound with bounds for maximal Fourier multipliers from harmonic analysis.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
