On the non-propagation theorem and applications (Q2862224)

One way to understand the spectrum of selfadjoint operators is show that these operators are affiliated to \(C^*\)-algebras. If one considers the quotient of these \(C^*\)-algebras by the ideal of compact operators, then one obtains some information about the essential spectrum of these operators. If one quotients these \(C^*\)-algebra by other ideals, then one may deduce some localization results that might have some consequences for the propagation properties of the unitary groups generated by these selfadjoint operators, or actually non-propagation properties. The paper under review treats a class of generalized Schrödinger operators in the setting of trees. After a description of the \(C^*\)-algebra associated to a tree, the class of Schrödinger operators is introduced and the statements following the outlined program are derived in this setting. The paper closes with some examples, where the Schrödinger operator has a bounded and periodic potential, or a potential with asymptotic vanishing oscillation.