Study of the resolvent of the perturbed Schrödinger operator

by Viviana GRASSELLI (Institut de Mathématiques de Toulouse)

Salle Katherine Johnson, bâtiment 1R3 (Institut de Mathématiques de Toulouse)

Salle Katherine Johnson, bâtiment 1R3

Institut de Mathématiques de Toulouse

118 route de Narbonne 31062 Toulouse Cedex 9

In this thesis we study spectral properties of the Schrödinger operator on non compact domains equipped with Riemannian metrics. More precisely, we are interested in the asymptotic behavior of the resolvent on the boundary of the resolvent set, in different frequency regimes.

The first part of our analysis it is aimed at extending to the Riemannian framework results which are known in the Euclidean case. We start by treating the low frequency regime in the case of asymptotically conical manifolds. We prove the existence of the limiting resolvents and of its powers for a Schrödinger operator with potential. Notably, this result allows to recover the local energy decay of the low frequency part of solutions to the Schrödinger, wave and Klein-Gordon equations. From the point of view of technique, we employ Mourre theory to prove the limiting absorption principle, which requires an adapted pseudodifferential calculus in order to treat operators depending on the spectral parameter. Next, we treat the high frequency regime, in a more general framework with respect to the previous one. On the one hand we consider a class of manifolds including not only the asymptotically conical case, but also the asymptotically hyperbolic one. On the other hand, we treat order one perturbations of the Schrödinger operator with potential. Under these assumptions, we obtain an optimal estimate for the resolvent on the non compact part of the manifold: it is bounded by the inverse square root of the spectral parameter. Moreover, these estimates are obtained using Besov type norms, making it possible to consider stronger topologies than what is often used in the literature. To conclude, we treat the compact region via Carleman estimates.

In the last chapter, we consider the Schrödinger operator in the Euclidean space in dimensions three and four and with a potential lying in a Lorentz space. More precisely, we study the nature of the frequency zero as well as the decaying properties of the associated states. We prove that any resonant state belonging to the homogeneous Sobolev space of order one is also in a weak Lebesgue space. Under classical assumptions of orthogonality between the resonant state and the potential, we obtain $L^2$, $L^{1,\infty}$ and finally $L^1$ integrability of the resonant state.