It is known in Korotyaev and Lobanov (Ann Henri Poincare 8:1151–1176, 2007) and Parchment (Commun Math Phys 275:805–826, 2007) that spectra of periodic Schrodinger operators on the metric graph corresponding to a carbon nanotube have the band-gap structure. The band-gap spectrum consists of not only infinitely many closed intervals but also the so-called flat bands (the set of eigenvalue with infinite multiplicities). In this paper, we study the spectrum of the free Schrodinger operators on a zigzag carbon nanotube with finite number of impurities expressed as $$\delta $$-interactions with strength $$\alpha \in \mathbf{R}$$. Assume that our impurities are symmetrically located in two ways. For a suitable $$\alpha $$, we construct eigenvalues embedded in the interior of spectral bands (not in spectral gaps).
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