Spectral analysis of fourth order differential operators I

Abstract

We study the spectral theory of differential operators of the form tau y = w(-1)[(ry `')'' - (py')' qy - i((my `')' (my')'' - ny' - (ny)')] on L-w(2) (0,infinity). By means of asymptotic integration, estimates for the eigenfunctions and M-matrix are derived. Since the M-function is the Stieltjes transform of the spectral measure, spectral properties of tau are directly related to the asymptotics of the eigenfunctions. The method of asymptotic integration, however, excludes coefficients which are too oscillatory or whose derivatives decay too slowly. Consequently there is no singular continuous spectrum in all our cases. This was found earlier for Sturm-Liouville operators, for which the WKB method provides a good approximation. (c) 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.