Schrödinger operators with \(\delta\)- and \(\delta'\)-interactions on Lipschitz surfaces and chromatic numbers of associated partitions (Q6486832)

This paper deals with Schrödinger operators, \(-\Delta_{\delta,\alpha}\) and \(-\Delta_{\delta',\beta}\), with \(\delta\)-interaction of strength \(\alpha\) and \(\delta'\)-interaction of strength \(\beta\) supported on the Euclidean space \(\mathbb R^d\) divided into a finite number of Lipschitz domains \(\Omega_k\), \(k=1, \dots, n\).NEWLINENEWLINEThe main result in the paper is an inequality between the Schrödinger operators in terms of the chromatic number \(\chi\) of the Lipschitz partition, which is the minimal number of colors with which one can color all domains \(\Omega_k\) in such a way that any two neighboring domains have distinct colors. To be specific, it asserts that if \(0 \leq \beta \leq \frac{4}{\alpha} \sin^2(\pi/\chi)\), then there exists an unitary operator \(U\) in \(L^2(\mathbb R^d)\) such that \(U^{-1}(-\Delta_{\delta',\beta})U\leq-\Delta_{\delta,\alpha}\) holds. In the end, this result is used to discuss the essential spectra and bound states of Schrödinger operators with \(\delta\) and \(\delta'\)-interactions on Lipschitz partitions.