Detection of edge defects by embedded eigenvalues of quantum walks

Abstract

We consider a position-dependent quantum walk on $\mathbf{Z}$. In particular, we derive a detection method for edge defects by embedded eigenvalues of its time evolution operator. In the present paper, the set of edge defects is that of points in $\mathbf{Z}$ on which the coin operator is an anti-diagonal matrix. In fact, under some suitable assumptions, the existence of a finite number of edge defects is equivalent to the existence of embedded eigenvalues of the time evolution operator.

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• Asymptotic properties of generalized eigenfunctions for multi-dimensional quantum walks
• Explicit expression of scattering operator of some quantum walks on impurities
• An eigenfunction expansion formula for one-dimensional two-state quantum walks
• A discontinuity of the energy of quantum walk in impurities
• A weak limit theorem for a class of long range type quantum walks in 1d
• Complex translation methods and its application to resonances for quantum walks