We consider a family of Schrödinger operators supported by a periodic chain of loops connected either tightly or loosely through connecting links of the length [Formula: see text] with the vertex coupling which is non-invariant with respect to the time reversal. The spectral behavior of the model illustrates that the high-energy behavior of such vertices is determined by the vertex parity. The positive spectrum of the tightly connected chain covers the entire halfline while the one of the loose chain is dominated by gaps. In addition, there is a negative spectrum consisting of an infinitely degenerate eigenvalue in the former case, and of one or two absolutely continuous bands in the latter. Furthermore, we discuss the limit [Formula: see text] and show that while the spectrum converges as a set to that of the tight chain, as it should in view of a result by Berkolaiko, Latushkin, and Sukhtaiev, this limit is rather non-uniform.

中文翻译：

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具有优选方向的顶点耦合的环链

我们考虑一系列由周期性循环链支持的薛定谔算子，这些循环通过长度为 [公式：见文本] 的连接链接紧密或松散地连接起来，顶点耦合在时间反转方面是非不变的。该模型的谱行为说明了这些顶点的高能行为是由顶点奇偶性决定的。紧密连接链的正谱覆盖了整个半线，而松散链的正谱则以间隙为主。此外，在前一种情况下，有一个由无限退化特征值组成的负谱，在后一种情况下，有一个或两个绝对连续带组成。此外，我们讨论了极限 [公式：见正文]，并表明虽然频谱收敛为紧链的集合，