Theorem of Levinson Via The Spectral Density

Abstract

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 atractive 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.