Theorem of Levinson via the Spectral Density

We deduce Levinson’s theorem in non-relativistic quantum mechanics in one dimension as a sum rule for the spectral density constructed from asymptotic data. We assume a self-adjoint Hamiltonian which guarantees completeness; the potential needs not to be isotropic and a zero-energy resonance is automatically taken into account. Peculiarities of this one-dimension case are explained because of the “critical” character of the free case u(x) = 0, in the sense that any attractive potential forms at least a bound state. We believe this method is more general and direct than the usual one in which one proves the theorem first for single wave modes and performs analytical continuation.