In this study, we review some topics on spectral and scattering theory for Schrödinger operators or time-independent wave equations and inverse scattering problems. First, we consider an example of an inverse problem for a one-dimensional wave equation with a piecewise constant coefficient. Nonscattering energy naturally appears in the process of reconstruction of the coefficient. A similar problem is known in multidimensional cases. This problem can be reduced to an interior transmission eigenvalue problem. Furthermore, herein, we refer to the shape resonance model for the Schrödinger equation as a related topic.

The graph zeta function began from the Ihara zeta function. We state the history and the definition of the Ihara zeta function and explain its properties. Next, we introduce the second weighted zeta function of a graph as a generalization of the Ihara zeta function and provide its determinant expression. As an application of the second weighted zeta function, we explain an explicit formula, i.e., the Konno–Sato Theorem for the time evolution matrix (the Grover matrix) of the Grover walk as a discrete-time quantum walk on a graph. Finally, as an application of the Konno–Sato Theorem, we briefly discuss Grover/Zeta Correspondence, which makes an initial move of a series of Zeta Correspondence.