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CLR-estimate revisited: Lieb’s approach with no path integrals. (English) Zbl 1027.47020

Journées “Équations aux dérivées partielles”, Saint-Jean-de-Monts, 2 Juin au 6 Juin 1997. Exposés Nos. I-XVIII. Palaiseau: École Polytechnique, Centre de Mathématiques. Exp. No. 16, 10 p. (1997).
Let \(B\) be a positive selfadjoint operator in a Hilbert space which is perturbed by the operator \(-V\leq 0\). In many mathematical and physical applications arises the question of estimating the number \(N_-(H)\) of negative eigenvalues of \(H= B-V\). For the Schrödinger operator \(H= -\Delta-V\) in \(L_2(\mathbb{R}^d)\), where \(V\geq 0\) is a measurable function, the following Cwikel-Lieb-Rozenblum estimation (CLR) holds: \[ N_-(-\Delta- V)\leq C(d) \int_{\mathbb{R}^d} V(x)^{{d\over 2}}dx,\quad d\geq 3. \] The proof of CLR given by Lieb used the path integration formalism. The aim of the paper under review is to translate this proof into a pure operator-theoretical language and so to generalize this estimation.
For the entire collection see [Zbl 0990.00048].

MSC:

47B25 Linear symmetric and selfadjoint operators (unbounded)
35J10 Schrödinger operator, Schrödinger equation
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