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

Riccardo Adami, Ivan Gallo, David Spitzkopf

TL;DR

This work analyzes the subcritical nonlinear Schrödinger equation on noncompact metric graphs with an attractive potential supported on the compact core, characterizing ground states under a mass constraint. By extending an existing energy-criterion to include the potential term, the authors prove ground states exist for both small and large masses and identify possible nonexistence in a middle mass range depending on graph geometry and potential strength. The approach combines variational methods, concavity/ subadditivity of the constrained infimum, and concrete test-function constructions localized to the potential core. The results illuminate how graph topology and localized attraction shape nonlinear bound states and connect to curvature-induced potentials in quantum waveguides, providing a framework for understanding ground-state existence and nonexistence on networks.

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.

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

TL;DR

This work analyzes the subcritical nonlinear Schrödinger equation on noncompact metric graphs with an attractive potential supported on the compact core, characterizing ground states under a mass constraint. By extending an existing energy-criterion to include the potential term, the authors prove ground states exist for both small and large masses and identify possible nonexistence in a middle mass range depending on graph geometry and potential strength. The approach combines variational methods, concavity/ subadditivity of the constrained infimum, and concrete test-function constructions localized to the potential core. The results illuminate how graph topology and localized attraction shape nonlinear bound states and connect to curvature-induced potentials in quantum waveguides, providing a framework for understanding ground-state existence and nonexistence on networks.

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 -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.
Paper Structure (8 sections, 11 theorems, 96 equations, 4 figures)

This paper contains 8 sections, 11 theorems, 96 equations, 4 figures.

Table of Contents

  1. Introduction
  2. General framework
  3. General Properties
  4. Existence criterion for $E(\cdot, \mathcal{G})$
  5. Existence
  6. Existence for large mass
  7. Existence for small mass
  8. Nonexistence

Key Result

Proposition 2.3

There exists $M_p > 0$ such that Moreover for every $u \in H^1({\mathcal{G}})$ and every metric graph $\mathcal{G}$ with $n$ infinite edges, $n \geq 1$.

Figures (4)

  • Figure 1: The 2-bridge graph satisfies the assumption (H) that prevents the existence of ground states for $E_{\rm{NLS}}$ at every mass.
  • Figure 2: A pictorial representation of the function $\widetilde{f}_\mu$ on the 2-bridge graph.
  • Figure 3: A representation of the function $u_\mu$ on the two-bridge graph.
  • Figure 4: A n-fork graph consisting of one halfline and $n$ edges of length $l$.

Theorems & Definitions (25)

  • Remark 2.1
  • Remark 2.2
  • Proposition 2.3: Gagliardo-Nirenberg inequalities
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Remark 2.6
  • Remark 2.7
  • Proposition 2.8: Monotone rearrangement
  • ...and 15 more