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Register a new userAsymptotic properties of generalized eigenfunctions for multi-dimensional quantum walks
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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 definition of the shift operator for the free QW. Then we define an anisotropic Banach space as a modified Agmon-H{o}rmanders $mathcal{B}^*$ 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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We consider the time-independent scattering theory for time evolution operators of one-dimensional two-state quantum walks. The scattering matrix associated with the position-dependent quantum walk naturally appears in the asymptotic behavior at spatial infinity of generalized eigenfunctions. The asymptotic behavior of generalized eigenfunctions is a consequence of an explicit expression of the Green function associated with the free quantum walk. When the position-dependent quantum walk is a finite rank perturbation of the free quantum walk, we derive a kind of combinatorial constructions of the scattering matrix by counting paths of quantum walkers. We also mention some remarks on the tunneling effect.
We clarify that coined quantum walk is determined by only the choice of local quantum coins. To do so, we characterize coined quantum walks on graph by disjoint Euler circles with respect to symmetric arcs. In this paper, we introduce a new class of coined quantum walk by a special choice of quantum coins determined by corresponding quantum graph, called quantum graph walk. We show that a stationary state of quantum graph walk describes the eigenfunction of the quantum graph.
We demonstrate how the Moutard transformation of two-dimensional Schrodinger operators acts on the Faddeev eigenfunctions on the zero energy level and present some explicitly computed examples of such eigenfunctions for smooth fast decaying potentials of operators with non-trivial kernel and for deformed potentials which correspond to blowing up solutions of the Novikov-Veselov equation.
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 ${bf Z}$. 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.
In a recent paper we proposed a non-Markovian random walk model with memory of the maximum distance ever reached from the starting point (home). The behavior of the walker is at variance with respect to the simple symmetric random walk (SSRW) only when she is at this maximum distance, where, having the choice to move either farther or closer, she decides with different probabilities. If the probability of a forward step is higher then the probability of a backward step, the walker is bold and her behavior turns out to be super-diffusive, otherwise she is timorous and her behavior turns out to be sub-diffusive. The scaling behavior vary continuously from sub-diffusive (timorous) to super-diffusive (bold) according to a single parameter $gamma in R$. We investigate here the asymptotic properties of the bold case in the non ballistic region $gamma in [0,1/2]$, a problem which was left partially unsolved in cite{S}. The exact results proved in this paper require new probabilistic tools which rely on the construction of appropriate martingales of the random walk and its hitting times.
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