Asymptotically correct lower bound for the sum of negative eigenvalues of Schrödinger operators through a decomposition of unity given by Macke

Abstract

We prove that the semi-classical expression for the sum of the negative eigenvalues of a one-dimensional Schrödinger operator with negative potential −φ is correct up to order O(h log 1/h) in the “Planck” constant h provided φ^1/4 is in the Sobolev space H¹(R).