Schrödinger semigroups and the Hörmander hypoellipticity condition

2024/06/06 by Nicola Garofalo, Garofalo, Nicola, Alessandra Lunardi +1 · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2406.04441

openalex publication_date 2024/06/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

We introduce a class of (possibly) degenerate dispersive equations with a drift. We prove that, under the Hörmander hypoellipticity condition, the relevant Cauchy problem can be uniquely solved in the Schwartz class, and the solution operator can be uniquely extended to a strongly continuous semigroup \\mathcal T(t)\_{t≥ 0} in L²(\Rm). Finally, we prove that for t>0 the operator \mathcal T(t) satisfies a sharp form of dispersive estimate in L^p, for any 1≤ p≤ 2, and an uncertainty principle.

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• Dispersive equations with invariant measures

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