Cwikel's theorem and the CLR inequality

TL;DR

This paper presents a concise proof of the CLR inequality for negative eigenvalues of Schrödinger operators, achieving improved constants and extending to operator-valued potentials, based on Rumin's work.

Contribution

It provides a simplified proof of the CLR bound with better constants and generalizes to operator-valued potentials and singular value estimates.

Findings

Improved constants in the CLR inequality

Extension to operator-valued potentials

General form of Cwikel's estimate for singular values

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 $f(X) g(-i\nabla)$.