2025/12/29 by Stefan Le Coz, Coz, Stefan Le, Boris Shakarov +1
Mathematics · Physics and Astronomy · #Analysis of PDEs (math.AP) #Class (philosophy) #FOS: Mathematics #Graph #Ground state #Limit (mathematics) #Metric (unit) #Nonlinear Partial Differential Equations #Nonlinear Photonic Systems #Nonlinear system #Open set #Reduction (mathematics) #Spectral Theory in Mathematical Physics
paper · open access · doi:10.48550/arxiv.2512.23286
published in arXiv (Cornell University) (Cornell University)
openalex publication_date 2025/12/29 · openalex created_date 2025/12/31 · openalex updated_date 2026/08/06
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.