A note on discrete Borg-type theorems

Abstract

The pseudospectral versions of the discrete Borg-type theorems are obtained in this article using some simple linear algebraic techniques. In particular, we prove that the periodic potential of a discrete Schrödinger operator is almost a constant if and only if the pseudospectrum of the operator is connected. This result is further extended to more general settings and the connection to the well-known Ten Martini problem is also established. The numerical illustrations done at the end will justify the results developed. 2010 Mathematics Subject Classification. Primary 47B35; Secondary 47B36.