Generalized eigenfunctions and scattering matrices for position-dependent quantum walks

Abstract

We study the spectral analysis and the scattering theory for time evolution operators of position-dependent quantum walks. Our main purpose of this paper is construction of generalized eigenfunctions of the time evolution operator. Roughly speaking, the generalized eigenfunctions are not square summable but belong to ${\ell }^{\infty }$-space on $\mathbf{Z}$. Moreover, we derive a characterization of the set of generalized eigenfunctions in view of the time-harmonic scattering theory. Thus we show that the S-matrix associated with the quantum walk appears in the singularity expansion of generalized eigenfunctions.

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• Design for implementation of discrete-time quantum walk with circulant matrix on graph by optical polarizing elements
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• Asymptotic properties of generalized eigenfunctions for multi-dimensional quantum walks
• Index Theorems for One-dimensional Chirally Symmetric Quantum Walks with Asymptotically Periodic Parameters
• Feynman Integral in Quantum Walk, Barrier-top Scattering and Hadamard Walk
• An eigenfunction expansion formula for one-dimensional two-state quantum walks
• Complex translation methods and its application to resonances for quantum walks
• The Witten Index for One-dimensional Non-unitary Quantum Walks with Gapless Time-evolution
• A discontinuity of the energy of quantum walk in impurities
• Resonances in finitely perturbed quantum walks, resonance expansion and generic simplicity
• Stationary measure for three-state quantum walk