Schroedinger operators on a periodically broken zigzag carbon nanotube

In this talk, we discuss the spectra of the Schroedinger operators on the zigzag carbon nanotube. Since carbon nanotubes can be broken by acids in the process to refine them, we especially deal with a periodically broken zigzag carbon nanotubes. In order to consider a model of broken zigzag carbon nanotubes, we get rid of edges from the metric graph corresponding to the pure zigzag carbon nanotubes. For a simple model of broken zigzag carbon nanotubes, we study the structure of the spectrum. By using the Floquet-Bloch theory, we see that the spectrum can be characterized by the Lyapunov function. In this talk, it turns out that the spectrum of our quantum graph has the band structure, namely, the spectrum consists of the infinitely many closed intervals and eigenvalues with infinite multiplicities. However, there are a difference as follows: Although the Lyapunov function for the standard Hill operator is entire, the one in our broken case is not entire but melomorphic. By studying the properties of this melomorphic Lyapunov function, we examine the spectral structure of a broken zigzag carbon nanotube.

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