Stationary scattering theory for unitary operators with an application to quantum walks
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Publication:6330859
DOI10.1016/J.JFA.2020.108704zbMath1518.47019arXiv1912.04960MaRDI QIDQ6330859
Publication date: 10 December 2019
Abstract: We present a general account on the stationary scattering theory for unitary operators in a two-Hilbert spaces setting. For unitary operators $U_0,U$ in Hilbert spaces ${cal H}_0,{cal H}$ and for an identification operator $J:{cal H}_0 o{cal H}$, we give the definitions and collect properties of the stationary wave operators, the strong wave operators, the scattering operator and the scattering matrix for the triple $(U,U_0,J)$. In particular, we exhibit conditions under which the stationary wave operators and the strong wave operators exist and coincide, and we derive representation formulas for the stationary wave operators and the scattering matrix. As an application, we show that these representation formulas are satisfied for a class of anisotropic quantum walks recently introduced in the literature.
Selfadjoint operator theory in quantum theory, including spectral analysis (81Q10) Scattering theory of linear operators (47A40) Nonselfadjoint operator theory in quantum theory including creation and destruction operators (81Q12)
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