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Controllability of the Schrödinger Equation via Intersection of Eigenvalues

We introduce two models of controlled infinite dimensional quantum system whose Hamiltonian operator has a purely discrete spectrum. For any couple of eigenstates we construct a path in the space of controls that approximately steers the system from one eigenstate to the other. To this purpose we use the adiabatic theory for quantum systems, and therefore the strategy requires large times. keywords: Quantum Control, Controllability of PDEs, Adiabatic Theory, δ-like Interactions