The authors consider Schrödinger operators with inverse square potentials on the half-line, and depending on some parameters. Such operators can have either a finite or an infinite number of complex eigenvalues. The Schrödinger operator of the form \[ -\partial_r^2 + \left(m^2-\frac{1}{4}\right)\frac{1}{r^2} \] defined on the half-line \(\mathbb{R}_+\) is considered. The parameter \(m\in\mathbb{C}\) with \(\mathfrak{R}(m) > -1\) is used for describing the coupling constant for the potential. For \(m\neq 0\) an additional parameter \(\kappa\in\mathbb{C}\) is used for defining the boundary condition at \(r = 0\), while for \(m = 0\) another family of operators indexed by a boundary parameter \(\nu\) is defined. The study of the corresponding families of closed operators \(H_{m,\kappa}\) and \(H_0^\nu\) in \(L^2(\mathbb{R}_+)\) has been initiated and developed by J. Dereziński and S. Richard [Ann. Henri Poincaré 18, No. 3, 869–928 (2017; Zbl 1370.81070)].
The main statement discussed here is Levinson’s theorem. It gives a certain relation between the number of bound states of a quantum mechanical system and an expression related to the scattering part of that system. Spectral singularities embedded in the continuous spectrum which the authors call exceptional situations are considered. The spectral and the scattering theory for the above stated operators is discussed. Some new results for the exceptional cases are provided. The known index theorems in scattering theory are also considered. The question why some results cannot be extended to the exceptional cases is discussed as well.