Positivity criteria for a hyperbolic Schrödinger operator (Q385941)

Let \(\widetilde{\mathbb{H}^d}\) be an open hyperbolic set, \(\Delta_{\mathrm{hyp}}\) the hyperbolic-Laplacian and \(V\in L^1_{\mathrm{loc}}(\widetilde{\mathbb{H}^d},d\nu_{\mathrm{hyp}})\). This paper concerns a hyperbolic Schrödinger operator \(H_{V,\widetilde{\mathbb{H}^d}}:=-\Delta_{\mathrm{hyp}}+V\) on \(L^2(\widetilde{\mathbb{H}^d},d\nu_{\mathrm{hyp}})\) with Dirichlet boundary condition. A precise lower bound of the hyperbolic Schrödinger operator is given when the potential is positive and not identically zero with a compact support.