Structure of the Spectra of Disordered Systems. I

Two theorems are presented, which play a fundamental role in discussing the spectra of disordered systems. The first theorem is valid for any mixed lattice which is described by a set of a number of 2×2 non-singular transfer-metrices with real traces, arranged regularly or randomly. It states that, if a value of frequency or energy lies in one of the spectral gaps for every constituent regular lattice, and if the fixed points of the transformations induced by these matrices are so arranged that there exists a trapping region, the density of frequency- or energy-spectrum of the mixed lattice vanishes at this frequency or energy-value. For some special forms of the transfer-matrices, which often appear in physical problems, the latter condition may be stated in terms of a simple notion “phase”, so that we get the second theorem. Various theorems of Saxon-Hutner-type are to be deduced from these theorems, according the the nature, or the manner of description, of individual systems.