Spectral Properties and Identities for the Polar Operator with Nonsmooth Periodic Density

Abstract

Spectral properties of the polar operator depending on the smoothness of the periodic coefficient are studied. The width of the far gaps in the Bloch spectrum is shown to grow for piecewise continuous coefficients, to be asymptotically constant if the coefficient derivative is piecewise continuous, and to decrease in the more smooth cases. The high energy asymptotics of the Lyapunov function, of the quasimomentum and of ‘effective masses’ are obtained. The spectral identities for the corresponding classical string equation are derived.