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\(C^*\)-algebras of anisotropic Schrödinger operators on trees. (English) Zbl 1068.81022

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 [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.

MSC:

81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
46L08 \(C^*\)-modules
81R15 Operator algebra methods applied to problems in quantum theory

Citations:

Zbl 1005.81026
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