Propagation estimates in the one-communicator theory

Sylvain Golenia (1) , Marc-Adrien Mandich (1)

1 IMB - Institut de Mathématiques de Bordeaux

Working in the abstract framework of Mourre theory, we derive a pair of propagation estimates for scattering states at certain energies of a Hamiltonian H. The propagation of these states is understood in terms of a conjugate operator A. A similar estimate has long been known for Hamiltonians having a good regularity with respect to A thanks to the limiting absorption principle (LAP). We show that in general some propagation estimates still hold when H has less regularity with respect to A, even in situations where the LAP has not yet been established. The estimates obtained are further discussed in relation to the RAGE and Riemann-Lebesgue formulae. Based on several examples, including continuous and discrete Schrödinger operators, it appears that the derived propagation estimates are a new result for multi-dimensional Hamiltonians.

Mots clés: Propagation estimate; Mourre theory; Mourre estimate; Commutator; RAGE Theorem; Schr\"odinger operators

Domaines: Théorie spectrale [math.SP]; Physique mathématique [math-ph]

Sylvain Golenia, Marc-Adrien Mandich. Propagation estimates in the one-communicator theory. 2017. ⟨hal-01494284v1⟩