Limits of quantum graph operators with shrinking edges
Limits of quantum graph operators with shrinking edges
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DOI：
10.1016/j.aim.2019.06.017
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发表时间：
2018-06
影响因子：
1.7
通讯作者：
G. Berkolaiko;Y. Latushkin;Selim Sukhtaiev
中科院分区：
文献类型：
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作者：
G. Berkolaiko;Y. Latushkin;Selim Sukhtaiev
当一些图的边的长度缩小到零时，我们解决了具有一般自伴顶点条件的度量图上薛定谔算子的收敛问题。我们确定限制算子并以合适的范数解析方式研究收敛性。值得注意的是，随着边长趋于零，标准 Sobolev 型估计会崩溃，导致某些图收敛失败。我们结合使用图的边缘上的函数分析边界和顶点条件的拉格朗日几何考虑因素来建立收敛的充分条件。此条件编码了图的拓扑与其顶点数据之间的复杂平衡。特别是，它不取决于势、收缩边收敛速率的差异或未受影响边的长度。
We address the question of convergence of Schrödinger operators on metric graphs with general self-adjoint vertex conditions as lengths of some of graph's edges shrink to zero. We determine the limiting operator and study convergence in a suitable norm resolvent sense. It is noteworthy that, as edge lengths tend to zero, standard Sobolev-type estimates break down, making convergence fail for some graphs. We use a combination of functional-analytic bounds on the edges of the graph and Lagrangian geometry considerations for the vertex conditions to establish a sufficient condition for convergence. This condition encodes an intricate balance between the topology of the graph and its vertex data. In particular, it does not depend on the potential, on the differences in the rates of convergence of the shrinking edges, or on the lengths of the unaffected edges.