Schrödinger equation on noncompact symmetric spaces

Équation de Schrödinger sur les espaces symétriques non-compacts

Jean-Philippe Anker (1) , Stefano Meda (2) , Vittoria Pierfelice (1) , Maria Vallarino (3) , Hong-Wei Zhang (1, 4)

1 IDP - Institut Denis Poisson
2 Dipartimento di Matematica e Applicazioni [Milano]
3 Dipartimento di Scienze Matematiche
4 Department of Mathematics [Gent/Ghent]

We establish sharp-in-time pointwise kernel estimates for the Schrödinger equation on noncompact symmetric spaces of general rank. A well-known difficulty in higher rank analysis, namely the fact that the Plancherel density is not a differential symbol in general, is overcome by using a spectral decomposition introduced recently by two of the authors in the study of the wave equation. We deduce the dispersive property of the Schrödinger propagator and prove global-in-time Strichartz inequalities for a large family of admissible pairs. As consequences, we extend the global well-posedness and small data scattering results previously obtained on real hyperbolic spaces to general Riemannian symmetric spaces of noncompact type.

Mots clés: semilinear Schrödinger equation; pointwise kernel estimates; dispersive property; Strichartz inequality; global well-posedness; noncompact symmetric space

Domaines: Equations aux dérivées partielles [math.AP]

Jean-Philippe Anker, Stefano Meda, Vittoria Pierfelice, Maria Vallarino, Hong-Wei Zhang. Schrödinger equation on noncompact symmetric spaces. 2021. ⟨hal-03187413v1⟩