La. Dmitrieva et Ma. Khlabystova, SPECTRAL PROPERTIES AND IDENTITIES FOR THE POLAR OPERATOR WITH NONSMOOTH PERIODIC DENSITY, letters in mathematical physics, 39(4), 1997, pp. 355-366
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 coefficie
nts, to be asymptotically constant if the coefficient derivative is pi
ecewise 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 c
orresponding classical string equation are derived.