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In this paper, we consider an extended coined Szegedy model and discuss theexistence of the point spectrum of induced quantum walks in terms of recurrenceproperties of the underlying birth and death process. We obtain that if theunderlying random walk is not null recurrent, then the point spectrum exists inthe induced quantum walks. As an application, we provide a simple computationalway of the dispersion relation of the edge state part for the topological phasemodel driven by quantum walk using the recurrence properties of underlyingbirth and death process.

Eigenvalues of quantum walk induced by recurrence properties of the underlying birth and death process: application to computation of an edge state