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

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 η₊ (±M) [η_- (±M)] of the scattering states with the same parity at zero momentum as η₊(M) - η₊(-M) = {(N+ - 1/2) π for the non-critical case N₊π for the critical case E = ±M and η_-(M) - η_- (-M) = { N_-π for the non-critical case (N_- + 1/2) π for the critical case E = ±M. The solution of the one-dimensional Klein-Gordon equation with the energy M or -M is called as a half bound state if it is finite but does not decay fast enough at infinity to be square integrable.