论文标题
无界度量图的光谱最小分区
Spectral minimal partitions of unbounded metric graphs
论文作者
论文摘要
我们研究了无限制的度量图的光谱最小分区的存在或不存在，其中操作员应用于每个分区元素是$-Δ+ v $的schrödingeroberator，其固定的（电动）潜在$ v $，该元素是固定的，在整个图形上被固定为固定。 我们表明，一方面，光谱最小分区与虚拟分区能量之间存在着很强的联系，另一方面，相应的Schrödinger操作员的基本光谱的最大$σ$另一方面是整个图。也就是说，我们表明，对于任何$ k \ in \ mathbb {n} $中的任何$ k \，所有可允许的$ k $ - 分区之间的虚拟能量从上面限制在上面的$σ$，如果严格低于$σ$，则存在频谱最小$ k $ - 分区存在。我们用有或没有电势的无限制和无限图的最小分区存在的几个示例和不存在的示例来说明我们的结果。 证明的性质是量子图的Persson定理的一种关键要素，强烈表明，基于Schrödingeroperator的相应结果应保留欧几里得空间中无界域的基于无界域的分区。
We investigate the existence or non-existence of spectral minimal partitions of unbounded metric graphs, where the operator applied to each of the partition elements is a Schrödinger operator of the form $-Δ+ V$ with suitable (electric) potential $V$, which is taken as a fixed, underlying ``landscape'' on the whole graph. We show that there is a strong link between spectral minimal partitions and infimal partition energies on the one hand, and the infimum $Σ$ of the essential spectrum of the corresponding Schrödinger operator on the whole graph on the other. Namely, we show that for any $k\in\mathbb{N}$, the infimal energy among all admissible $k$-partitions is bounded from above by $Σ$, and if it is strictly below $Σ$, then a spectral minimal $k$-partition exists. We illustrate our results with several examples of existence and non-existence of minimal partitions of unbounded and infinite graphs, with and without potentials. The nature of the proofs, a key ingredient of which is a version of Persson's theorem for quantum graphs, strongly suggests that corresponding results should hold for Schrödinger operator-based partitions of unbounded domains in Euclidean space.
