## A new proof of the Cwikel-Lieb-Rosenbljum bound.(English)Zbl 0581.35014

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 $(1.1)\quad N(V)\leq C\int_{R^ 3}| V_- (x)|^{3/2}dx.$ 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 $$C=.168$$.
Reviewer: P.T.Craciunaş

### MSC:

 35J10 Schrödinger operator, Schrödinger equation 35P15 Estimates of eigenvalues in context of PDEs 47A10 Spectrum, resolvent

### Keywords:

Cwikel-Lieb-Rosenbljum bound

Zbl 0445.58029
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