Normalized nonlinear Schrödinger ground states on defected grids with unbounded defects
Abstract
We study normalized ground states for the nonlinear Schrödinger energy on two-dimensional square grids with unbounded defects. We first show that unbounded defects do not necessarily destroy the dimensional crossover, proving its persistence on the half-grid and the quarter-grid. We then consider a prototypical defected grid where a macroscopically two-dimensional region coexists with an unbounded one-dimensional channel. On this graph, normalized ground states do not exist for sufficiently small or sufficiently large masses, showing that the one-dimensional component can dominate the variational problem and induce loss of compactness. Finally, we show that this mechanism is sensitive to the local geometry of the graph: suitable additional defects preserve small-mass nonexistence but restore the existence of ground states for large masses. Our analysis combines small-mass energy asymptotics with a large-mass blow-up argument.
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BibTeX
@misc{borrelli2026normalized,
title = {{Normalized nonlinear Schrödinger ground states on defected grids with unbounded defects}},
author = {William Borrelli and Nicolò Cangiotti and Simone Dovetta and Lorenzo Tentarelli},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09908},
eprint = {2610.09908},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09908},
}
RIS
TY - UNPB
TI - Normalized nonlinear Schrödinger ground states on defected grids with unbounded defects
AU - William Borrelli
AU - Nicolò Cangiotti
AU - Simone Dovetta
AU - Lorenzo Tentarelli
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.09908
UR - https://arxiv.org/abs/2610.09908
AB - We study normalized ground states for the nonlinear Schrödinger energy on two-dimensional square grids with unbounded defects. We first show that unbounded defects do not necessarily destroy the dimensional crossover, proving its persistence on the half-grid and the quarter-grid. We then consider a prototypical defected grid where a macroscopically two-dimensional region coexists with an unbounded one-dimensional channel. On this graph, normalized ground states do not exist for sufficiently small or sufficiently large masses, showing that the one-dimensional component can dominate the variational problem and induce loss of compactness. Finally, we show that this mechanism is sensitive to the local geometry of the graph: suitable additional defects preserve small-mass nonexistence but restore the existence of ground states for large masses. Our analysis combines small-mass energy asymptotics with a large-mass blow-up argument.
ER -