Some generalizations of Cwikel-Lieb-Rozenblum and Lieb-Thirring inequalities

二维 Pauli 和磁 Schrödinger 算子的 Cwikel-Lieb-Rozenblum 型估计

Cwikel-Lieb-Rozenblum type estimates for the Pauli and magnetic Schrödinger operator in dimension two

本文证明了二维 Pauli 算子负特征值数量的 Cwikel-Lieb-Rozenblum 型不等式，所得上界在弱和强耦合极限下均尖锐，并导出了磁 Schrödinger 算子的不同上界，其性质取决于是否考虑自旋轨道耦合。

Hardy-Schrödinger算子的Cwikel-Lieb-Rozenblum型不等式

Cwikel-Lieb-Rozenblum type inequalities for Hardy-Schrödinger operator

本研究证明了Hardy-Schrödinger算子负特征值个数的Cwikel-Lieb-Rozenblum型不等式，给出W的加权Lᵈ/²范数形式的最优界，并推广到分数阶Laplacian情形。

Cwikel's bound reloaded

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, …

Lieb-Thirring and Cwickel-Lieb-Rozenblum inequalities for perturbed graphene with a Coulomb impurity

We study the two dimensional massless Coulomb-Dirac operator restricted to its positive spectral subspace and prove estimates on the negative eigenvalues created by electromagnetic perturbations.

On general Cwikel-Lieb-Rozenblum and Lieb-Thirring inequalities

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 es…

Cwikel's theorem and the CLR inequality

We give a short proof of the Cwikel-Lieb-Rozenblum (CLR) bound on the number of negative eigenvalues of Schrödinger operators. The argument, which is based on work of Rumin, leads to remarkably good constants and applies to the case of operator-valued potentials as well. Moreover, we obtain the general form of Cwikel's estimate about the singular values of operators of the form $f(X) g(-i\nabla)$.