It is increasingly being realised that localised deviations from the average structure can play an important role in a material's properties, and hence an understanding of these deviations is essential when constructing a coherent picture of a system. In this thesis, both neutron and X-ray total scattering data have been collected and used to reveal information on three canonical systems: the high-temperature superconductor YBa₂Cu₃O_7-x; the parent compound of the colossal magnetoresistive manganites, LaMnO₃; and the oldest known magnetic material, Fe₃O₄. Reverse Monte Carlo refinements-using the RMCProfile implementation of the algorithm-have been used as the principal analysis technique, and the functionality of the RMCProfile program has been extended to allow the refinement of magnetic systems involving substitutional disorder. For YBa₂Cu₃O_7-x, the focus of this thesis is on the apical Cu{O bond length. This is shown to have a bimodal distribution, but correlations in the displacements of both atoms disguise this fact in the average structure, thus resolving the apparent controversy that had existed between local- and average-structure probes. LaMnO₃ displays (what was thought to be) a simple order{disorder transition that results in the Jahn-Teller distortion becoming invisible in the average structure above ~ 750 K. Here it is shown that in fact the transition is more complicated, and involves a change in the symmetry of the Jahn-Teller distortion, whereby the long Mn-O bonds move from being opposite one another in the octahedra to being adjacent to one another. This new distortion still breaks the degeneracy of the system, and is consistent with a wide range of existing observations. Finally the low temperature structure of Fe₃O₄ is examined using the updated RMCProfile code. The refinements show sensitivity to local structure variations, producing a bimodal Fe atom bond valence distribution. In addition, the refined magnetic spin configuration is presented, providing the first detailed description of the low-temperature magnetic structure. It is found to be a canted ferrimagnet, and appears to be consistent with Cc symmetry.

In the first part of this thesis the spectrum of a matrix operator is determined. For this the coefficients of the matrix operator are assumed to satisfy rather general properties which combine smoothness and decay. With this the asymptotics of the eigenfunctions can be determined. This in turn leads to properties of the spectra with the aid of the M-matrix. In the second part it will be shown that if a discrete canonical system has absolutely continuous spectrum of a certain multiplicity, then there is a corresponding number of linearly independent solutions y which are bounded in a weak sense.