On the Bloch eigenvalues, band functions and bands of the differential operator of odd order with the periodic matrix coefficients

Abstract

In this paper, we consider the Bloch eigenvalues, band functions and bands of the self-adjoint differential operator L generated by the differential expression of odd order n with the \(m\times m\) periodic matrix coefficients, where \(n>1.\) We study the localizations of the Bloch eigenvalues and continuity of the band functions and prove that each point of the set \(\left[ (2\pi N)^{n},\infty \right) \cup (-\infty ,(-2\pi N)^{n}]\) belongs to at least m bands, where N is the smallest integer satisfying \(N\ge \pi ^{-2}M+1\) and M is the sum of the norms of the coefficients. Moreover, we prove that if \(M\le \ \pi ^{2}2^{-n+1/2}\), then each point of the real line belong to at least m bands.