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

TL;DR

This paper investigates how stationary states of the nonlinear Schrödinger equation on open book domains reduce to solutions on metric graphs as the domain shrinks, revealing a critical width where the nature of ground states changes.

Contribution

It develops a functional-analytic framework for variational problems on open books and proves a sharp transition in ground state dimensionality based on transverse width.

Findings

Existence of solutions as constrained action minimizers.

Identification of a critical transverse width for ground state transition.

Genuine two-dimensional ground states above the critical width.

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.