Reflectionless CMV matrices and scattering theory. (English) Zbl 1343.47014
Authors’ abstract: Reflectionless CMV matrices are studied using scattering theory. By changing a single Verblunsky coefficient, a full-line CMV matrix can be decoupled and written as the sum of two half-line operators. Explicit formulas for the scattering matrix associated to the coupled and decoupled operators are derived. In particular, it is shown that a CMV matrix is reflectionless iff the scattering matrix is off-diagonal which in turn provides a short proof of an important result of J. Breuer et al. [Commun. Math. Phys. 295, No. 2, 531–550 (2010; Zbl 1190.47032)]. These developments parallel those recently obtained for Jacobi matrices by V. Jakšić et al. [Commun. Math. Phys. 332, No. 2, 827–838 (2014; Zbl 1298.47042)].
Reviewer: Vladimir A. Derkach (Donetsk)
MSC:
| 47A40 | Scattering theory of linear operators |
| 47B39 | Linear difference operators |
| 34L40 | Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) |
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