Which notion of energy for bilinear quantum systems?*

In this note we investigate what is the best Lp-norm in order to describe the relation between the evolution of the state of a bilinear quantum system with the Lp-norm of the external field. Although L2 has a structure more easy to handle, the L1 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 Lp norm. Explicit optimal costs for the L1 norm are computed on an example.