2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/186006In the light of the Sturm-Liouville theorem, the Levinson theorem for the Schrödinger equation with both local and non-local cylindrically symmetric potentials is studied. It is proved that the two-dimensional Levinson theorem holds for the case with both local and non-local cylindrically symmetric cutoff potentials, which is not necessarily separable. In addition, the problems related to the positive-energy bound states and the physically redundant state are also discussed in this paper.Latex 11 pages, no figure, submitted to J. Phys. A Email: dongsh@bepc4.ihep.ac.cn, mazq@bepc3.ihep.ac.cnQuantum Physicstext