https://doi.org/10.1007/s100530070079
Levinson's theorem for the Klein-Gordon equation in one dimension
Physical and Theoretical Chemistry Laboratory, University of
Oxford, Oxford OX1 3QZ, UK
and Department of Physics, Cardwell Hall,
Kansas State University, Manhattan, Kansas 66506, USA
Received:
22
December
1999
Published online: 15 August 2000
In terms of the modified Sturm-Liouville theorem, the Levinson theorem for the one-dimensional Klein-Gordon equation with a symmetric potential V(x) is established.
It is shown that the number N+ (N-) of bound states with even (odd) parity is related to the phase shift ![$\eta_{+}(\pm M)[\eta_{-}(\pm M)]$](/articles/epjd/abs/2000/08/d9332/d9332_tex_eq1.png)
of the scattering states with the same parity at zero momentum as
and
The solution of the one-dimensional Klein-Gordon equation with the energy
M or 
is called as a
half bound state if it is finite but does not decay fast enough at infinity to
be square integrable.
PACS: 03.65.Ge – Solutions of wave equations: bound states / 11.80.-m – Relativistic scattering theory / 73.50.Bk – General theory, scattering mechanisms
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 2000
