PDE Seminar (2016/7/1): On the Spectra of Schr\”odinger operators on a periodically broken carbon nanotubes, Hiroaki Niikuni ( Maebashi Institute of Technology)

2016-7-1 16:30 - 2016-7-1 18:00
Faculty of Science Building #3, Room 309
Hiroaki Niikuni ( Maebashi Institute of Technology)
Nowadays, we have a lot of opportunities to see that carbon nanotubes play roles as matrials in the fields of mechanical engineering due to their outstanding properties. In this talk, we study the spectra of carbon nanotubes from the point of view of quantum graphs. Namely, we examine the spectral properties of periodic Schr\"odinger operators on the metric graphs corresponding to carbon nanotubes. Especially, we deal with one of the simplest models of periodically broken carbon nanotubes. Utilizing the Floquet-Bloch theory, we show that its spectrum has the band-gap structure. Namely, we notice that its spectrum consists of the absolutely continuous spectrum and the set of eigenvalues with infinite multiplicities. Furthermore, we prove that the absolutely continuous spectrum is characterized by the corresponding discriminants and consists of infinitely many closed intervals. We note that our spectral discriminants are not entire functions but meromorphic functions, whereas the spectral discriminants for the standard Hill operators are entire.