Levinson's theorem for the Klein-Gordon equation in one dimension

Abstract

Levinson's theorem for the one-dimensional Schrödinger equation with a symmetric potential, which decays at infinity faster than x −2 , is established by the Sturm-Liouville theorem. The critical case, where the Schrödinger equation has a finite zero-energy solution, is also analyzed. It is demonstrated that the number of bound states with even (odd) parity n+ (n−) is related to the phase shift η+(0)[η−(0)] of the scattering states with the same parity at zero momentum asfor the non − critical case,

Spectral density, Levinson's theorem, and the extra term in the second virial coefficient for the one-dimensional δ -function potential

“…In the case of delta-function potential which does not support any half bound state, I 1 = −1/2, and δ(0) = π/2, giving N B = 1, which is exactly the number of bound states supported by the potential. One can show Levinson's theorem in 1D by using a number of methods, including Sturm-Liouville method, and the S-matrix method [12,13,26,27]. From those calculations one ends up with the following form of Levinson's theorem (in the even parity case),…”