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Abstract: In a paper from 2025 C. Bourne introduced a general framework to study mixtures of physical systems brought together on a discrete interface. The theory sits within an operator algebraic framework, in which an operator T describing the dynamics of a one dimensional interface of physical systems acts on the discrete Hilbert space l²(Z^m)=l²(Z, l²(Z^m-1)) and recovers the operators at the boundary via spatial asymptotics in the first variable. Refining to a C*-algebra of observables, instead of considering bounded operators on the Hilbert space l²(Z, l²(Z^m-1)) one studies the more general space of adjointable operators on the standard Hilbert B-module l²(Z,B) where B is a C*-algebra of bounded operators on l²(Z^m-1). For those operators in this space describing the dynamics of a mixture of physical systems it can be shown that certain spectral and topological properties can be fully obtained from knowing the operators which are being mixed, i.e. the spatial asymptotics at infinity. We investigate whether there are similar equality and existence results relating the density of states of the mixing operator to the density of states of the spatial asymptotics. To do so, we first remain within the classical framework of bounded operators on the Hilbert space l²(Z, l²(Z^m-1)), where such formulas may be proven quite directly. To state the results within the Hilbert module framework an adequate notion of density of states must first be introduced.