The Levinson theorem

The Levinson theorem is a fundamental theorem in quantum scattering theory, which shows the relation between the number of bound states and the phase shift at zero momentum for the Schrödinger equation. The Levinson theorem was established and developed mainly with the Jost function, with the Green function and with the Sturm–Liouville theorem. In this review, we compare three methods of proof, study the conditions of the potential for the Levinson theorem and generalize it to the Dirac equation. The method with the Sturm–Liouville theorem is explained in some detail. References to development and application of the Levinson theorem are introduced.