Global controllability of 1D Schrödinger-Poisson equation

MARIANO DE LEO, CONSTANZA SÁNCHEZ FERNÁNDEZ DE LA VEGA, DIEGO RIAL

REVISTA DE LA UNIóN MATEMáTICA ARGENTINA

UNION MATEMATICA ARGENTINA

Lugar: Bahia Blanca; Año: 2012

0041-6932

This paper is concerned with both the local and global internal controllability of the 1D Schr¨dinger?oPoisson equation i ut (x, t) = −uxx + V (u) u, which arises in quantum semiconductor models. Here V (u)is a Hartree-type nonlinearity stemming from the coupling with the 1D Poisson equation, which includesthe so-called doping profile or impurities. More precisely, it is shown that for both attractive and repulsiveself-consistent potentials ?depending on the balance between the total charge and the impurities? thisproblem is globally internal controllable in a suitable Sobolev space.