5,059 research outputs found
Small time reachable set of bilinear quantum systems
This note presents an example of bilinear conservative system in an infinite
dimensional Hilbert space for which approximate controllability in the Hilbert
unit sphere holds for arbitrary small times. This situation is in contrast with
the finite dimensional case and is due to the unboundedness of the drift
operator
Controllability of the bilinear Schr\"odinger equation with several controls and application to a 3D molecule
We show the approximate rotational controllability of a polar linear molecule
by means of three nonresonant linear polarized laser fields. The result is
based on a general approximate controllability result for the bilinear
Schr\"odinger equation, with wavefunction varying in the unit sphere of an
infinite-dimensional Hilbert space and with several control potentials, under
the assumption that the internal Hamiltonian has discrete spectrum
Controllability of the discrete-spectrum Schrodinger equation driven by an external field
We prove approximate controllability of the bilinear Schr\"odinger equation
in the case in which the uncontrolled Hamiltonian has discrete non-resonant
spectrum. The results that are obtained apply both to bounded or unbounded
domains and to the case in which the control potential is bounded or unbounded.
The method relies on finite-dimensional techniques applied to the Galerkin
approximations and permits, in addition, to get some controllability properties
for the density matrix. Two examples are presented: the harmonic oscillator and
the 3D well of potential, both controlled by suitable potentials
Which notion of energy for bilinear quantum systems?
In this note we investigate what is the best L^p-norm in order to describe
the relation between the evolution of the state of a bilinear quantum system
with the L^p-norm of the external field. Although L^2 has a structure more easy
to handle, the L^1 norm is more suitable for this purpose. Indeed for every
p>1, it is possible to steer, with arbitrary precision, a generic bilinear
quantum system from any eigenstate of the free Hamiltonian to any other with a
control of arbitrary small L^p norm. Explicit optimal costs for the L^1 norm
are computed on an example
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