Ground states for the NLS on non-compact graphs with an attractive potential

Riccardo Adami; Ivan Gallo; David Spitzkopf; Riccardo Adami; Ivan Gallo; David Spitzkopf

doi:10.3934/nhm.2025015

Networks and Heterogeneous Media

2025, Volume 20, Issue 1: 324-344. doi: 10.3934/nhm.2025015

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Ground states for the NLS on non-compact graphs with an attractive potential

1.
Dipartimento di Scienze Matematiche "G.L. Lagrange", Politecnico di Torino Corso Duca degli Abruzzi, 24, Torino 10129, Italy
2.
Nuclear Physics Institute, Czech Academy of Sciences, Hlavní 130, Řež near Prague 25068, Czech Republic
3.
Faculty of Mathematics and Physics, Charles University, V Holešovičkách 2, Prague 18000, Czech Republic

Received: 18 January 2025 Revised: 28 March 2025 Accepted: 08 April 2025 Published: 23 April 2025

We consider the subcritical nonlinear Schrödinger equation on non-compact quantum graphs with an attractive potential supported in the compact core, and investigate the existence and the nonexistence of ground states, defined as minimizers of the energy at fixed $ L^2 $-norm, or mass. We finally reach the following picture: for small and large mass there are ground states. Moreover, according to the metric features of the compact core of the graph and to the strength of the potential, there may be an interval of intermediate masses for which there are no ground states. The study was inspired by the research on quantum waveguides, in which the curvature of a thin tube induces an effective attractive potential.
- nonlinear Schrödinger equation,
- quantum graphs,
- ground states,
- concentration compactness techniques
Citation: Riccardo Adami, Ivan Gallo, David Spitzkopf. Ground states for the NLS on non-compact graphs with an attractive potential[J]. Networks and Heterogeneous Media, 2025, 20(1): 324-344. doi: 10.3934/nhm.2025015

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Abstract

We consider the subcritical nonlinear Schrödinger equation on non-compact quantum graphs with an attractive potential supported in the compact core, and investigate the existence and the nonexistence of ground states, defined as minimizers of the energy at fixed $ L^2 $-norm, or mass. We finally reach the following picture: for small and large mass there are ground states. Moreover, according to the metric features of the compact core of the graph and to the strength of the potential, there may be an interval of intermediate masses for which there are no ground states. The study was inspired by the research on quantum waveguides, in which the curvature of a thin tube induces an effective attractive potential.

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