An estimate on the number of bound states of Schrödinger operators (Q1916811)

The author investigates the eigenvalue problem \[ (- \Delta+ V(x)) u= \lambda u\quad \text{on } \Omega,\quad u|_{\partial\Omega}= 0,\tag{1} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^2\) and \(V\) is a measurable function defined on \(\Omega\). The main result of the paper is that the number of non-positive eigenvalues of problem (1) \(N(V, \Omega)\) is bounded by \(N(V, \Omega)\leq C \int_\Omega V_-(1+ \log^+ V_-) dx\), where \(C\) is a positive constant depending only on \(\Omega\) and \(V_-\) is the negative part of \(V\). The author uses some ideas of the technique of K. Tanaka as well as the extrapolation theorem of O. N. Capri and N. A. Fava.