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Zeitschrift für Analysis und ihre Anwendungen


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Volume 3, Issue 4, 1984, pp. 357–366
DOI: 10.4171/ZAA/113

Published online: 1984-08-31

Infinite Representability of Schrödinger Operators with Ergodic Potential

Harald Englisch[1] and Klaus-Detlef Kürsten[2]

(1) Universität Leipzig, Germany
(2) Universität Leipzig, Germany

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.

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Englisch Harald, Kürsten Klaus-Detlef: Infinite Representability of Schrödinger Operators with Ergodic Potential. Z. Anal. Anwend. 3 (1984), 357-366. doi: 10.4171/ZAA/113