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Ground States for the Nonlinear Schr{ö}dinger Equation on Open Books and Dimensional Reduction to Metric Graphs

Stefan Le Coz, Boris Shakarov

TL;DR

This work develops a variational framework for the nonlinear Schrödinger equation on open books, a class of two-dimensional network-like domains, and studies their dimensional reduction to metric graphs in the shrinking limit. By formulating action minimization on the Nehari manifold and proving exponential decay of ground states, it establishes existence results for finite, periodic, and graph-based books. A key contribution is the sharp transverse-thickness threshold: for graph-based books $\mathcal{G}\times[0,L]$, ground states remain graph-extended for small $L$ (independent of the transverse variable) and become genuinely two-dimensional once $L$ exceeds a critical value $L_{min}$, with $L_{min}>0$ when $s_{\omega,\mathcal{G}}<s_{\omega,\mathcal{G}}^{\infty}$. The analysis employs a rescaled formulation $v(x,y)=u(x,Ly)$ to compare the shrinking problem with line and strip models, proving continuity and monotonicity properties of $s_{\omega,L}$ and a minimizer rigidity result. Overall, the paper rigorously justifies quantum-graph as an effective one-dimensional model for thin nonlinear networks and characterizes the precise transition to higher-dimensional ground states as the transverse width grows.

Abstract

In this work, we study the dimensional reduction of stationary states in the shrinking limit for a broad class of two-dimensional domains, called open books, to their counterparts on metric graphs. An open book is a two-dimensional structure formed by rectangular domains sharing common boundaries. We first develop a functional-analytic framework suited to variational problems on open books and establish the existence of solutions as constrained action minimizers. For graph-based open books (i.e., those isomorphic to the product of a graph with an interval) we prove the existence of a sharp transition in the dimensionality of ground states. Specifically, there exists a critical transverse width: below this threshold, all ground states coincide with the ground states on the underlying graph trivially extended in the transverse direction; above it, ground states become genuinely two-dimensional.

Ground States for the Nonlinear Schr{ö}dinger Equation on Open Books and Dimensional Reduction to Metric Graphs

TL;DR

This work develops a variational framework for the nonlinear Schrödinger equation on open books, a class of two-dimensional network-like domains, and studies their dimensional reduction to metric graphs in the shrinking limit. By formulating action minimization on the Nehari manifold and proving exponential decay of ground states, it establishes existence results for finite, periodic, and graph-based books. A key contribution is the sharp transverse-thickness threshold: for graph-based books , ground states remain graph-extended for small (independent of the transverse variable) and become genuinely two-dimensional once exceeds a critical value , with when . The analysis employs a rescaled formulation to compare the shrinking problem with line and strip models, proving continuity and monotonicity properties of and a minimizer rigidity result. Overall, the paper rigorously justifies quantum-graph as an effective one-dimensional model for thin nonlinear networks and characterizes the precise transition to higher-dimensional ground states as the transverse width grows.

Abstract

In this work, we study the dimensional reduction of stationary states in the shrinking limit for a broad class of two-dimensional domains, called open books, to their counterparts on metric graphs. An open book is a two-dimensional structure formed by rectangular domains sharing common boundaries. We first develop a functional-analytic framework suited to variational problems on open books and establish the existence of solutions as constrained action minimizers. For graph-based open books (i.e., those isomorphic to the product of a graph with an interval) we prove the existence of a sharp transition in the dimensionality of ground states. Specifically, there exists a critical transverse width: below this threshold, all ground states coincide with the ground states on the underlying graph trivially extended in the transverse direction; above it, ground states become genuinely two-dimensional.
Paper Structure (15 sections, 13 theorems, 167 equations, 3 figures)

This paper contains 15 sections, 13 theorems, 167 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Functional setting
  4. Notation
  5. Metric structure
  6. Exponential decay
  7. Examples
  8. Existence of an action minimizer
  9. The problem at infinity
  10. Finite books
  11. Periodic Books
  12. Shrinking limit for graph-based books
  13. The rescaled problem
  14. The action level function
  15. Minimizers rigidity

Key Result

Theorem 1.3

Let $\mathcal{L}$ be a connected book, either finite or periodic, and let $\omega > -\omega_\mathcal{L}$. If $\mathcal{L}$ is finite, assume additionally that $s_{\omega}<s_{\omega}^\infty$, where the action level at infinity $s_{\omega}^\infty$ is defined in eqSOmInfinit. Then $s_\omega>0$ and ther

Figures (3)

  • Figure 1: An open-book with seven pages
  • Figure 2: Open book versions of classical graphs: star-graph, tadpole and dumbbell (from left to right)
  • Figure 3: The open-book versions of the 2-d grid and the torus

Theorems & Definitions (30)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3
  • Definition 1.4
  • Theorem 1.5
  • Proposition 2.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 20 more