Infinite representability of Schrödinger operators with ergodic potential
This paper contains a few results about the spectrum of Schrödinger operators corresponding to ergodic potentials. The method by which the results are obtained is the infinite representability of one operator, say A in another, say B. This means, for self-adjoint operators, that there is an increasing sequence \(\{D_ k\}_ 1^{\infty}\), of subspaces of dom A and a corresponding sequence \(\{U_ k\}_ 1^{\infty}\) of unitary operators such that i) \(\cup_{k}D_ k\) is dense in dom A with respect to the topology defined by \(<z,z>+<Az,Az>\) ii) \(U_ k(D_ k)\subset dom B\) and for \(\epsilon >0\), there is a natural number \(k_ 0\) such that for all \(k>k_ 0\) and all \(z\in D_ k,<(A-U^*_ kBU_ k)z,z>\leq \epsilon (\| Az\| +\| z\|)^ 2\), and iii) for \(k\neq \ell\), \(U_ k(D_ k)\) is orthogonal to \(U_{\ell}(D_{\ell}).\) As a main tool throughout the paper, the authors prove: Theorem. If A is infinitely representable in B, then \(\sigma (A)\subset \sigma_{ess}(B).\) Now let (\(\Omega\),B,\(\nu)\) be a measure space where \(\Omega \subset L^ 2_{loc,unif}(R^ n)\), \(n=1,2,3\), B is the \(\sigma\)-algebra generated by cylindrical sets, and \(\nu\) is a probability measure. Also let \(\Delta\) denote the Laplacian on \(L^ 2(R^ n)\). Then Theorem. i) If \(\nu\) is stationary with respect to translations in \(R^ n\), then the operator \(H^{\omega}=-\Delta +V^{\omega}\) for \(V^{\omega}\in \Omega\) has no discrete spectrum a.e. \(\omega\), and (ii) if \(\nu\) is ergodic, then almost all \(H^{\omega}\) have the same spectrum as a set. Next let d\({\mathcal N}\) denote the density of states measure. Then Theorem. i) If \(V^{\omega}\) is ergodic and \(E\in \sigma (H^{\omega})\) for a.e. \(V^{\omega}\), then \({\mathcal N}(E+\epsilon)-{\mathcal N}(E- \epsilon)>0\) for all \(\epsilon >0\), and (ii) if the space dimension is 1 and \({\mathcal N}(E+\epsilon)-{\mathcal N}(E-\epsilon)>0\) for all \(\epsilon >0\), and (ii) if the space dimension is 1 and \({\mathcal N}(E+\epsilon)-{\mathcal N}(E-\epsilon)>0\) for each \(\epsilon >0\), then \(E\in \sigma (H^{\omega})\) for a.e. \(V^{\omega}\).
spectrum of Schrödinger operators corresponding to ergodic potentials
\(\sigma \) -algebra generated by cylindrical sets
scientific article; zbMATH DE number 3917050