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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. [ABSTRACT FROM AUTHOR]