Spectral analysis of higher-order differential operators II: Fourth-order equations

We investigate the location and nature of the spectrum of the fourth-order self-adjoint equation (p(0)y ")" + (p(1)y')' + qy = zwy subject to certain asymptotic assumptions on the coefficients. The main tools are the theory of asymptotic integration and the Titchmarsh-Weyl M-matrix. Asymptotic integration yields asymptotic formulae for the solutions of the differential equation which are then used to derive properties of the M-matrix. The characterisation of spectral properties in terms of the boundary behaviour of M leads to the desired results.