Generalized eigenfunctions and scattering matrices for position-dependent quantum walks

“…The quantum walks are certain unitary operators, defined below, in a discrete setting, and they are sometimes regarded as a quantum counterpart of the classical random walks. The homogeneous two-state quantum walks (in one dimension with constant coin matrix) is well understood (see, for example, [8], [5], [24]), and recently the scattering-theoretical aspect, as a perturbation of homogeneous walks, are intensively investigated (see [14], [15], [19], [20], [16]). The Schrödinger operators in one dimension are often called the Sturm-Liouville operators and they are well-studied.…”

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confidence: 99%

An eigenfunction expansion formula for one-dimensional two-state quantum walks

Tate¹

2021

Preprint

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The purpose of this paper is to give a direct proof of an eigenfunction expansion formula for one-dimensional 2-state quantum walks, which is an analog of that for Sturm-Liouville operators due to Weyl, Stone, Titchmarsh and Kodaira. In the context of the theory of CMV matrix it had been already established by Gesztesy-Zinchenko. Our approach is restricted to the class of quantum walks mentioned above whereas it is direct and it gives some important properties of Green functions. The properties given here enable us to give a concrete formula for a positive-matrix-valued measure, which gives directly the spectral measure, in a simplest case of the so-called two-phase model. * The 2 × 2 matrix-valued function T λ (x) is introduced in [10]. There is another matrix called transfer matrix used in [12]. See Appendix in this paper.

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“…For details of the time-dependent scattering theory for QWs, see Suzuki [20] or Morioka [14]. In these previous works, the authors consider 1D QWs.…”

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confidence: 99%

“…Note that the authors adopted commutator method for unitary operators (see [2], [7], [19]). Morioka [14], Morioka-Segawa [15], Maeda et al [13] and Komatsu et al [11] considered the time-independent scattering theory and the absence of eigenvalues embedded in the continuous spectrum. Tiedra de Aldecoa [22] studied an abstract theory of time-independent scattering for unitary operators and its applications to QWs.…”

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confidence: 99%

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Asymptotic properties of generalized eigenfunctions for multi-dimensional quantum walks

Komatsu,

Konno,

Morioka

et al. 2021

Preprint

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We construct a distorted Fourier transformation associated with the multi-dimensional quantum walk. In order to avoid the complication of notations, almost all of our arguments are restricted to two dimensional quantum walks (2DQWs) without loss of generality. The distorted Fourier transformation characterizes generalized eigenfunctions of the time evolution operator of the QW. The 2DQW which will be considered in this paper has an anisotropy due to the bias of the shift of the free QW. Then we define an anisotropic Banach space as a modified Agmon-Hörmander's B * space and we derive the asymptotic behavior at infinity of generalized eigenfunctions in these spaces. The scattering matrix appears in the asymptotic expansion of generalized eigenfunctions.

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Generalized eigenfunctions and scattering matrices for position-dependent quantum walks

Cited by 26 publications

References 40 publications

A walk on max-plus algebra

A walk on max-plus algebra

An eigenfunction expansion formula for one-dimensional two-state quantum walks

Asymptotic properties of generalized eigenfunctions for multi-dimensional quantum walks

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