Abstract
For unitary operators \(U_0,U\) in Hilbert spaces \({\mathcal {H}}_0,{\mathcal {H}}\) and identification operator \(J:{\mathcal {H}}_0\rightarrow {\mathcal {H}}\), we present results on the derivation of a Mourre estimate for U starting from a Mourre estimate for \(U_0\) and on the existence and completeness of the wave operators for the triple \((U,U_0,J)\). As an application, we determine spectral and scattering properties of a class of anisotropic quantum walks on homogenous trees of odd degree with evolution operator U. In particular, we establish a Mourre estimate for U, obtain a class of locally U-smooth operators and prove that the spectrum of U covers the whole unit circle and is purely absolutely continuous, outside possibly a finite set where U may have eigenvalues of finite multiplicity. We also show that (at least) three different choices of free evolution operators \(U_0\) are possible for the proof of the existence and completeness of the wave operators.
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The author thanks the anonymous referees for their suggestions which helped improve the redaction of various definitions and motivated the extension of the results to homogenous trees of odd degree \(>3\).
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Communicated by Alain Joye.
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Partially supported by the Chilean Fondecyt Grant 1210003.
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Tiedra de Aldecoa, R. Spectral and Scattering Properties of Quantum Walks on Homogenous Trees of Odd Degree. Ann. Henri Poincaré 22, 2563–2593 (2021). https://doi.org/10.1007/s00023-021-01066-9
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DOI: https://doi.org/10.1007/s00023-021-01066-9