Condition for a forbidden energy in an alloy chain

A study is made of the conditions in which an energy forbidden to periodic arrays of the constituents is also forbidden in the random alloy chain. Linear symmetric sources of potential are considered, each type of source having its characteristic 2 × 2 transfer matrix T. It is shown that, for `well-type' chains (all Im T12 >or= 0) or for `barrier-type' chains (all Im T12 0 common to the nth gap of each type is also forbidden in the random alloy chain, provided that, for each pair of constituent types specified by transfer matrices T(E) and T'(E) the following relation holds: (Im T'12 - Im T12)(arg T'11/1 + T'12 - arg T11/1 + T12) >or= 0. Simpler versions, sacrificing some information, are also given. The condition means roughly that the types of well must obey the rule: the deeper the well, the longer its range. The alloy chain as defined here is more general than, but includes, the more customary periodically-spaced model. The property introduced, of being a `well-like' or a `barrier-like' chain, is shown invariant throughout the energy range treated.