The one-dimensional Levinson’s theorem is derived and used to study zero-energy resonances in a double-potential system. The low energy behavior of time delay is also investigated. In particular, it is shown that the quantum mechanical time delay admits a classical lower bound, in the low energy limit, if the potential has no bound-state solutions.

Sassoli de Bianchi, M. (1994). Levinson’s theorem, zero-energy resonances, and time delay in one-dimensional scattering systems. J. Math. Phys. 35, pp. 2719-2733. Doi: 10.1063/1.530481.