We consider the Grover walk on the infinite graph in which an internal finite
subgraph receives the inflow from the outside with some frequency and also
radiates the outflow to the outside. To characterize the stationary state of
this system, which is represented by a function on the arcs of the graph, we
introduce a kind of discrete gradient operator twisted by the frequency. Then
we obtain a circuit equation which shows that (i) the stationary state is
described by the twisted gradient of a potential function which is a function
on the vertices; (ii) the potential function satisfies the Poisson equation
with respect to a generalized Laplacian matrix. Consequently, we characterize
the scattering on the surface of the internal graph and the energy penetrating
inside it. Moreover, for the complete graph as the internal graph, we
illustrate the relationship of the scattering and the internal energy to the
frequency and the number of tails.
