A new proof of the Cwikel-Lieb-Rosenbljum bound
Concerning the operator \(-\Delta +V\) acting on \(L^ 2(R^ 3)\) where V is a potential in \(L^{3/2}(R^ 3)\) and the dimension N(V) of the spectral projection of \(-\Delta +V\) on (-\(\infty,0]\) there is known the inequality \[ (1.1)\quad N(V)\leq C\int_{R^ 3}| V_- (x)|^{3/2}dx. \] \textit{E. Lieb} [Proc. Symp. Pure Math. 36, 241-252 (1980; Zbl 0445.58029)] gives as the best values of \(C=.116\). Here the author presents a new derivation of (1.1) with the constant \(C=.168\). To derive this result the author considers the following problem: Assuming V(x)\(\leq 0\) \(\forall x\in R^ 3\) and \(V(x)=-W^ 2(x),W(x)\geq 0\) \(\forall x\in R^ 3\), the operator A on \(L^ 2(R^ 3)\) generated by the integral kernel \(a(x,y)=W(x)W(y)[4\pi | x-y|)^{-1}\) is a positive Hilbert-Schmidt operator and has a discrete spectrum \(\mu_ 1\mu_ 2...\geq 0\). The author proves that \(\sum^{N}_{i=1}\mu_ i\leq C^{2/3}N^{1/3}\| W\|^ 2_ 3\) whence the inequality {\#}\(\{\) \(\mu\) \({}_ i\geq \lambda^{-1}\); \(i=1,2,...\}\leq C\lambda^{3/2}\| W\|^ 3_ 3\) \(\forall \lambda >0\) and finally the inequality (1.1) with the mentioned value \(C=.168\).
scientific article; zbMATH DE number 3929524