Levinson's theorem in one dimension: heuristics

For partial waves in three dimensions, Levinson's theorem asserts that delta (0)= pi n_b, where delta (p) is the phase shift at wavenumber p, and n_b the number of bound states ( delta ( infinity )=0 by convention). The corresponding theorem in one dimension calls first for a systematic parametrisation of the transmission amplitude T(p)=cos theta e^{i tau} , and of the left (right)-incidence reflection amplitudes R_L,R(p)=i sin theta exp(i tau +or-i rho ), where the phase angles tau , theta , rho are functions of p. If the potential is not everywhere zero, and excluding throughout the exceptional case where it has a zero-energy bound state, heuristic arguments show that R_L,R(0)=-1, that theta (0)=¹/₂ pi (-¹/₂ pi ) when n_b is odd (even), and that tau (0)= pi (n_b-¹/₂); (by convention, tau ( infinity )=0= theta ( infinity )). Thus mod tau (0) mod cannot be less than ¹/₂ pi , no matter how weak the potential; the transition to the limit of zero potential is non-uniform. In the special case of reflection-symmetric potentials, rho =0, one can subdivide n_b=n_b^(e)+n_b^(o), and define even- and odd-parity phase shifts E=¹/₂( tau + theta ) and Delta =¹/₂( tau - theta ); then E(0)= pi (n_b^(e)-¹/₂), Delta (0)= pi n_b^(o). The appendix shows how E(0) is obtainable by suitably adapting the familiar s-wave argument which exploits the analyticity properties of the Jost solutions.