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Dispersive estimates for quantum walks on 1D lattice. (English) Zbl 07470024

The paper focuses on the study in the long time of quantum walks (QWs) within coins that converge to some constant coin \(C_0\) as \(|x|\rightarrow \infty\). The core tool used to tackle the \(\ell^\infty-\ell^1-\) decay estimate provided by eq. (1.1) – see Theorem 1.9. – are centered on the notion of Jost solutions (Section 5).
To this end, the authors take into account the so-called Cantero-Moral-Velázquez (CMV) representation of QWs (see [M. J. Cantero et al., Commun. Pure Appl. Math. 63, No. 4, 464–507 (2010; Zbl 1186.81036)]) to describe the pair \((\lambda,u)\) of solutions of \((U-e^{i\lambda})u=0\) through the the transfer matrix representation of the eigenvalue problem \((U-\lambda)u=0\) (see Section 3).
The paper under consideration is situated on the crossroads of orthogonal polynomials and spectral analysis. And it involves also the framework of Pauli matrices, an essential tool to describe the symmetries of the transfer matrix of the aforementioned eigenvalue problem. For the interested reader, this paper shall be considered as a starting paper on the scope of QWs. Indeed, the proofs enclosed are structured, self-contained and easy to reach.

MSC:

35Q41 Time-dependent Schrödinger equations and Dirac equations
81U30 Dispersion theory, dispersion relations arising in quantum theory
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics

Citations:

Zbl 1186.81036
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References:

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