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arXiv · 2512.23286

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

Stefan Le Coz·Boris Shakarov
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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.

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Stefan Le Coz, Boris Shakarov. 2025-12-29. Ground States for the Nonlinear Schr{\"o}dinger Equation on Open Books and Dimensional Reduction to Metric Graphs. https://arxiv.org/abs/2512.23286

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