In this paper, we introduce a multidimensional generalization of Kitagawa's splitstep discrete-time quantum walk, study the spectrum of its evolution operator for the case of one defect coins, and prove localization of the walk. Using a spectral mapping theorem, we can reduce the spectral analysis of the evolution operator to that of a discrete Schrödinger operator with variable coefficients, which is analyzed using the Feshbach map.
For given two unitary and self-adjoint operators on a Hilbert space, a spectral mapping theorem was proved in [14]. In this paper, as an application of the spectral mapping theorem, we investigate the spectrum of a one-dimensional split-step quantum walk. We give a criterion for when there is no eigenvalues around ±1 in terms of a discriminant operator. We also provide a criterion for when eigenvalues ±1 exist in terms of birth eigenspaces. Moreover, we prove that eigenvectors from the birth eigenspaces decay exponentially at spatial infinity and that the birth eigenspaces are robust against perturbations.
Mathematical investigations on quantum Zeno effect (QZE) are presented, including the following aspects: (i) QZE by frequent measurements made by an arbitrary partition of a time interval [0, t] (t > 0); (ii) non-occurrence of QZE for vector states which are not in the domain of the Hamiltonian of the quantum system under consideration; (iii) asymptotic behavior of the survival probability characterizing QZE in the number N of divisions of [0, t]; (iv) QZE along a curve in the Hilbert space of state vectors.Mathematics Subject Classification (2010). 81Q10, 81P15, 47B25.
This paper proves a weak limit theorem for a one-dimensional split-step quantum walk and investigates the limit density function. In the density function, the difference between two Konno's functions appears.
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