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 consider the Schrödinger equation on Riemannian symmetric spaces of noncompact type. Previous studies in rank one included sharp-in-time pointwise estimates for the Schrödinger kernel, dispersive properties, Strichartz inequalities for a large family of admissible pairs, and global well-posedness and scattering, both for small initial data. In this paper we establish analogous results in the higher rank case. The kernel estimates, which is our main result, are obtained by combining a subordination formula, an improved Hadamard parametrix for the wave equation, and a barycentric decomposition initially developed for the wave equation, which allows us to overcome a well-known problem, namely the fact that the Plancherel density is not always a differential symbol.

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. Journal of Differential Equations, 2023, 356, pp.163-187. ⟨10.1016/j.jde.2023.02.003⟩. ⟨hal-03187413v3⟩