Cwikel's theorem and the CLR inequality

Abstract

We give a short proof of the Cwikel-Lieb-Rozenblum (CLR) bound on the number of negative eigenvalues of Schr\"odinger 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 $f(X) g(-i\nabla)$.