ABSTRACT. We analyze the energy transfer from continuous to point spectrum of a time dependent Point Interaction operator. Under suitable regularity assumptions for the initial state of the associated quantum system, we prove a result of local controllability of this process.
KEYWORDS: Quantum Control Systems; Schrödinger Equation
Since their appearance in Theoretical Physics , point interaction operators have become a natural tool in order to describe short range forces or "small" obstacles for the scattering of waves and particles.
In this work we build up a point interaction model of a time dependent Schrödinger operator; this interaction will be used as a control for the energy transfer between continuous and discrete spectrum of a one body quantum system. We will investigate the possibility of finding a time dependence profile such that part of the energy of the particle, initially placed in a scattering state, moves on a bound state in finite time.
The subject we treat has its natural collocation in the framework of quantum systems control theory. Our analysis may find applications in those contexts where short range quantum potentials will be used as control tools.
Consider a quantum particle subject to the action of a point ...