具有无界缺陷的网格上归一化非线性薛定谔基态
Normalized nonlinear Schrödinger ground states on defected grids with unbounded defects
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中文总结 AI 辅助
本文研究无界缺陷网格上归一化非线性薛定谔基态，证明缺陷不破坏维度交叉，但一维通道导致小/大质量下基态缺失，额外缺陷可恢复大质量存在性。
中文摘要 AI 辅助
我们研究了在具有无界缺陷的二维方形网格上的非线性薛定谔能量的归一化基态。首先，我们证明无界缺陷不一定破坏维度交叉，并证明其在半网格和四分之一网格上的持续性。然后，我们考虑一个典型的缺陷网格，其中宏观二维区域与无界一维通道共存。在此图上，对于足够小或足够大的质量，归一化基态不存在，这表明一维分量可以主导变分问题并导致紧致性的丧失。最后，我们证明该机制对图的局部几何形状敏感：适当的额外缺陷保留了小质量不存在性，但恢复了大质量基态的存在性。我们的分析结合了小质量能量渐近与大质量爆破论证。
英文摘要
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.
发表机构
- Sapienza Università di Roma（罗马大学）
- SIS Paolino d’Aquileia（保利诺·达阿奎莱亚 SIS）
- Politecnico di Torino（都灵理工大学）
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