  In this paper, we treat quantum walks in a two-dimensional lattice with
cutting edges along a straight boundary introduced by Asboth and Edge (2015
Phys.Rev. A 91 022324) in order to study one-dimensional edge states
originating from topological phases of matter and to obtain collateral evidence
of how a quantum walker reacts to the boundary. Firstly, we connect this model
to the CMV matrix, which provides a 5-term recursion relation of the Laurent
polynomial associated with spectral measure on the unit circle. Secondly,we
explicitly derive the spectra of bulk and edge states of the quantum walk with
the boundary using spectral analysis of the CMV matrix. Thirdly, while
topological numbers of the model studied so far are well-defined only when gaps
in the bulk spectrum exist, we find a new topological number defined only when
there are no gaps in the bulk spectrum. We confirm that the existence of the
spectrum for edge states derived from the CMV matrix is consistent with the
prediction from a bulk-edge correspondence using topological numbers calculated
in the cases where gaps in the bulk spectrum do or do not exist. Finally, we
show how the edge states contribute to the asymptotic behavior of the quantum
walk through limit theorems of the finding probability. Conversely, we also
propose a differential equation using this limit distribution whose solution is
the underlying edge state.
