There exist two limiting cases for the finite chain, either v = 0 or w = 0 . In both of these cases, the chain is clearly an insulator as the chain is broken into dimers (dimerized). However one of the two cases would consist of N / 2 dimers, while the other case would consist of ( N − − 2 ) / 2 dimers and two unpaired sites at the edges of the chain. In the latter case, as there is no on-site energy, if an electron finds itself on any of the two edge sites, its energy would be zero. So either the case v = 0 or the case w = 0 would necessarily have two eigenstates with zero energy, while the other case would not have zero-energy eigenstates. Contrary to the bulk case, the two limiting cases are not symmetrical in their spectrum.