Irreversible behaviour and collapse of wave packets in a quantum system with point interactions

A system of a particle and a harmonic oscillator, which have pure point spectra if uncoupled, is known to acquire absolutely continuous spectrum when they are coupled by a sufficiently strong point interaction. Here the dynamical mechanism underlying this spectral phenomenon is exposed. The energy of the oscillator is proven to exponentially diverge in time, while the spatial probability distribution of the particle collapses into a δ-function in the interaction point. On account of this result, a generalized model with many oscillators which interact with the particle at different points is argued to provide a formal model for approximate measurement of position, and collapse of wave packets. PACS numbers: 03.65.Yz; 02.30.Tb I. INTRODUCTION. A.