Section: New Results

New results: quantum control

New results have been obtained for the control of the bilinear Schrödinger equation, with two different approaches.

• In [2] we proved an approximate controllability result by finite-dimensional methods, considering the Galerkin approximations. The approach improves the technique that we developed in [40] . The result requires less restrictive non-resonance hypotheses on the spectrum of the uncontrolled Schrödinger operator than those already known. The control operator is not required to be bounded and we are able to extend the controllability result to the density matrices. The proof is based on fine controllability properties of the finite-dimensional Galerkin approximations and allows to get estimates for the $L^1$ norm of the control. The general controllability result is applied to the problem of controlling the rotation of a bipolar rigid molecule confined on a plane by means of two orthogonal external fields.

• In [4] we presented a constructive method to control the bilinear Schrödinger equation via two controls. The method is based on adiabatic theory and works if the spectrum of the Hamiltonian admits conical eigenvalue intersections. We provided sharp estimates of the relation between the error and the controllability time. We also showed that for a Hamiltonian of the kind $-\Delta +V_0\left(x\right)+u_1{V}_1\left(x\right)+u_2{V}_2\left(x\right)$ on a domain of ${ℝ}^n$ the eigenvalue intersections are conical generically with respect to $V_0,V_1,V_2$.