The spectral bound of Schrödinger operators (Q1917791)

The authors study the spectral bound \(s(\lambda)\) of the Schrödinger operator \((1/2)\Delta - \lambda V(x)\), \(x\in \mathbb{R}^N\) as \(\lambda\to\infty\). They prove that \(s(\infty)=\inf\{\sigma < 0 : \inf_{E\in F_{\sigma}}\int_{E}V > 0\}\). Here \(F_{\sigma}\) is a class of all Borel sets \(E\) such that the spectral bound of the Laplace operator \((1/2)\Delta\) is greater than \(\sigma\). The authors investigate also the question whether this limit value is attained for finite \(\lambda\).