Levinson's theorems in classical scattering

Abstract

This paper states and proves Levinson's theorems for the classical scattering of two particles. The derivation is carried out by using time-delay theory. In both classical and quantum mechanics Levinson's theorems are shown to be the consequence of a common principle. This principle is the spectral property of time delay. Consider an arbitrary space region v. In quantum and classical mechanics a time-dependent scattering state traverses v in a definite time. The spectral property states that the sum of transit times for all orbits with total energy ε is proportional to the state density in region v. In classical mechanics it is established that this property can be derived from simple features of the time-evolution group.