\(C^*\)-algebras of anisotropic Schrödinger operators on trees
Let \(\Gamma\) be a \(\nu\)-fold tree and \(\widehat\Gamma= \Gamma\cup\partial\Gamma\) be its the hyperbolic compactification. On \(\ell^2(\Gamma)\) defined is the differential operator given by \((\partial f)(x)=\sum_{y'=x}f(y)\). Let \(\mathcal{D}\) be the \(C^*\)-algebra generated by \(\partial\). By \(C(\widehat\Gamma)\) is denoted the set of complex-valued continuous functions on \(\widehat\Gamma\). Denote by \(\mathcal{C}(\widehat\Gamma)\) the \(C^*\)-algebra generated by \(\mathcal{D}\) and \(C(\widehat\Gamma)\). This algebra contains the set \(\mathbb{K}(\Gamma)\) of compact operators on \(\ell^2(\Gamma)\). In this paper the algebra \(\mathcal{C}(\widehat\Gamma)\) is studied. Following [\textit{V. Georgescu, A. Iftimovici}, Commun. Math. Phys. 228, 519--560 (2002; Zbl 1005.81026)] the quotient \(\mathcal{C}(\widehat\Gamma)/\mathbb{K}(\Gamma)\) is considered. Namely, it is proved the following Theorem. Let \(\nu>1\). There is a unique morphism \(\Phi:\mathcal{C}(\widehat\Gamma)\to \mathcal{D}\otimes C(\partial\Gamma)\) such that \(\Phi(D)=D\otimes 1\), for all \(D\in\mathcal{D}\) and \(\Phi(\varphi(Q))=1\otimes(\varphi| _{\partial\Gamma})\). This morphism is surjective and its kernel is \(\mathbb{K}(\Gamma)\). This result allowed to compute the essential spectrum of self-adjoint operators affiliated to \(\mathcal{C}(\widehat\Gamma)\). The results cover Schrödinger operators with highly anisotropic potential.
scientific article; zbMATH DE number 2133173
scientific article