Ring chains with vertex coupling of a preferred orientation

TL;DR

This paper analyzes the spectral properties of Schrödinger operators on periodic chains of loops with non-time-reversal-invariant vertex couplings, revealing how vertex parity influences high-energy behavior and spectral gaps.

Contribution

It introduces a detailed spectral analysis of vertex-coupled loop chains with non-invariant couplings, highlighting the impact of vertex parity and the non-uniform limit as link length approaches zero.

Findings

High-energy spectrum depends on vertex parity.

Tightly connected chain spectrum covers the entire positive halfline.

Loose chain spectrum exhibits spectral gaps and negative eigenvalues.

Abstract

We consider a family of Schr\"odinger operators supported by a periodic chain of loops connected either tightly or loosely through connecting links of the length $\ell>0$ 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 $\ell\to 0$ 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.