An estimate on the number of bound states of Schrödinger operators
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.
scientific article; zbMATH DE number 902528