Spectral inequalities for operators on \(H\)-type groups (Q414727)

Let \(\mathbb{G}\) be an \(H\)-type group and consider the operator \(\mathcal{L}=-\Delta_{\mathbb{G}}+\nabla_\mathbb{G}U\cdot\nabla_\mathbb{G}\) where \(\Delta_\mathbb{G}\) is the sub-Laplacian, \(\nabla_\mathbb{G}\) is the sub-gradient and \(U\) is some potential. The author proves some conditions on the growth and smoothness of \(U\) that ensure that the operator has empty essential spectrum. He then turns to proving functional inequalities, first when \(U\) is defined in terms of the Carnot-Carathéodory distance, and secondly when the Carnot-Carathéodory distance is replaced by the Kaplan distance. It is shown that these cases result in operators with significantly different spectra.