Seminar Analysis and Mathematical Physics: Simone Dovetta (CNR-IMATI, Pavia)

Variational problems for nonlinear Schroedinger equations on metric graphs

The talk overviews some recent developments about nonlinear Schroedinger equations (NLS) on metric graphs. Precisely, we concentrate on variational problems for the NLS energy functional subject to the mass constraint. After a brief recap of the well-known behaviour of such model on the real line, we address the existence of NLS ground states on noncompact metric graphs, with a specific focus on periodic graphs and infinite trees. The emergence of threshold phenomena rooted in the nature of these graphs is discussed. Finally, we provide some insights on the uniqueness of ground states at fixed mass. On the one hand, uniqueness is shown to hold for two classes of graphs with halflines. On the other hand, a counterexample to uniqueness in full generality is exhibited. The matter we discuss is part of a wider research line, developed in collaboration with several authors. The results explicitly covered by the talk refer to a series of papers, some of which are joint works with Riccardo Adami, Enrico Serra and Paolo Tilli.

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