量子图：谱的$ \mathcal{PT}$-对称性和反射对称性
Quantum Graphs: $ \mathcal{PT}$-symmetry and reflection symmetry of the spectrum
摘要：
不一定自伴量子图-微分算子度量图-被认为是。此外，假设基础度量图具有自同构（对称性）$ \mathcal P $。如果微分算子是$ \mathcal P \mathcal T$-对称的，那么它的谱关于实数线具有反射对称性。我们的目标是了解是否相反的说法成立，即是否反射对称的量子图的频谱意味着潜在的度量图具有非平凡的自同构和微分算子是$ \mathcal P \mathcal T$-对称的。我们通过考虑等边星图给出了这个问题的部分答案。相应的具有Robin顶点条件的Laplace算子具有反射对称谱当且仅当该算子是$ \mathcal P \mathcal T$-对称的，其中$ \mathcal P $是度量图的自同构.
英文摘要：
Not necessarily self-adjoint quantum graphs -- differential operators on metric graphs -- are considered. Assume in addition that the underlying metric graph possesses an automorphism (symmetry) $ \mathcal P $. If the differential operator is $ \mathcal P \mathcal T$-symmetric, then its spectrum has reflection symmetry with respect to the real line. Our goal is to understand whether the opposite statement holds, namely whether the reflection symmetry of the spectrum of a quantum graph implies that the underlying metric graph possesses a non-trivial automorphism and the differential operator is $ \mathcal P \mathcal T$-symmetric. We give partial answer to this question by considering equilateral star-graphs. The corresponding Laplace operator with Robin vertex conditions possesses reflection-symmetric spectrum if and only if the operator is $ \mathcal P \mathcal T$-symmetric with $ \mathcal P $ being an automorphism of the metric graph.
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