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, an edge defect is a set \( \{ y-1 ,y\} \) for \(y\in \mathbf{Z}\), in 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. In view of applications, by checking the spectrum, we can detect the existence of disconnecting edge (in the sense of edge defects above) on the line without directly watching it.