One-dimensional Schrödinger Operators with Ergodic Potential

Abstract

In the first part of the paper the results of Luttinger and Dworin concerning the following problem of Saxon/Hutner are generalized: Which conditions guarantee that an energy value $E$ lies in the resolvent set of the Hamiltonian for an alloy, presupposing that $E$ lies in the resolvent set of the Hamiltonians of all pure components. Symmetric potentials play an particular role in this. In the third part the ground state energy is investigated for different types of the ergodie potential $V$. For comparison, examples with almost periodic potentials are given. E.g. Cordon’s result concerning eigenvalues for almost periodic potentials is generalized in the second part.