Infinite Representability of Schrödinger Operators with Ergodic Potential

Abstract

Analogous to the notion of finite representability in the theory of Banach spaces, the notions of the representability and the infinite representability of self-adjoint operators are introduced. It is proved that the infinite representability of the operator $A$ in $B$ yields that the essential spectrum of $B$ contains the spectrum of $A$. This result applied to ergodic Schrödinger operators yields a new proof for the nonrandomness of the spectrum and for the connection between the spectrum and the density of states. A formula for the spectrum of the Hamiltonian of a substitutional alloy is presented, which clarifies the bowing effect. Similar results were found independcntly by Kirsch and Martinelli.