Point interaction controls for the energy transfer in 3-D quantum systems

Abstract

We consider the problem of energy-mass transfer from scattering to bound states for a one body quantum system subject to the action of a time dependent point interaction in 3-D. Under suitable assumptions on the initial state of the particle, we prove a result of local controllability of this process. Our proof exploits the finite time asymptotic analysis of fractional integral equations and the rank theorem for maps denned on Banach spaces.

Acknowledgments

I would like to thank Prof. J.-M. Coron and Prof. R. Figari for useful discussions. The paper was supported through a European Community Marie-Curie Fellowship and in the framework of CTS.

Notes

†From the physical point of view, the parameter α is linked to the inverse scattering length of H _α (Albeverio et al. 1988).

‡For the definitions and the properties concerning point interactions operators, we refer to the book (Albeverio et al. 1988).