Cwikel's theorem and the CLR inequality (Q2450825)

Let \( {\mathcal{S}}_{q,w}(\mathcal{H})\) denote the space of compact operators \(K\) on a separable Hilbert space \(\mathcal{H} \) satisfying \[ \|K\|^q_{q,w} = \sup_{k>0} k^q n(k,(K^*K)^{1/2}) < \infty. \] Here \( n(k,(K^*K)^{1/2}) \) denotes the number of eigenvalues of \( (K^*K)^{1/2} \) larger than \(k\), counting multiplicities. Cwikel's theorem says that \( f \in L_q(\mathbb{R}^d) \) and \( g \in L_{q,w}(\mathbb{R}^d)\) implies \(f(X)g(-i\Delta) \in {\mathcal{S}}_{q,w}(L_2(\mathbb{R}^d)) \) for \(q > 2.\) For the number \( N(0,-\Delta + V) \) of negative eigenvalues of the Schrödinger operator \(-\Delta + V\) in \( L_2(\mathbb{R}^d), d \geq 3, \) it follows with Cwikel's theorem \[ N(0,-\Delta + V) \leq const \int_{\mathbb{R}^d} V(x)^{d/2}_{-} dx, \] where \(V(x)_{-} = \max \{-V(x),0 \}.\) The aim of the paper is a new short proof of Cwikel's theorem and the inequality above. Generalizations of Cwikel's theorem are also given.