Third order operator with small periodic coefficients

Abstract: We consider the third order operator with small 1-periodic coefficients on the real line. The spectrum of the operator is absolutely continuous and covers all real line. Under the minimal conditions on the coefficients we show that there are two possibilities: 1) The spectrum has multiplicity one except for a small spectral nonempty interval with multiplicity three. Moreover, the asymptotics of the small interval is determined. 2) All spectrum has multiplicity one only.