Levinson theorem for the Dirac equation in one dimension

The Levinson theorem for the (1+1)-dimensional Dirac equation with a symmetric potential is proved with the Sturm-Liouville theorem. The half-bound states at the energies E=±M, whose wave function is finite but does not decay at infinity fast enough to be square integrable, are discussed. The number n_± of bound states is equal to the sum of the phase shifts at the energies E=±M:_±(M)+_±(–M)=(n_±+a), where the subscript ± denotes the parity and the constant a is equal to –1/2 when no half-bound state occurs, to 0 when one half-bound state occurs at E=M or at E=–M, and to 1/2 when two half-bound states occur at both E=±M.

©2006 The American Physical Society