Resonant states in momentum representation

Abstract

Gamow states in momentum representation are defined as solutions of a homogeneous Lippmann-Schwinger equation for purely outgoing particles. We study their properties when the potential, local or nonlocal, is such that the trace of the kernel of the Lippmann-Schwinger equation exists. It is found that, contrary to what happens in position representation, Gamow states in momentum representation are square integrable functions. A norm is defined and expressions for matrix elements of operators between arbitrary states and properly normalized resonant states are given, free of divergence difficulties. It is also shown that bound and resonant states form a biorthonormal set of functions with their adjoints and that a square integrable function may be expanded in terms of a set containing bound states, resonant states, and a continuum of scattering functions. Resonant states may be transformed from momentum to position representation modifying, in a suitable way, the usual rule.