On spectra of random Schrödinger operators with magnetic fields
Let \(b_ \omega= \sum_{j=1}^ d b^ j_ \omega (x) dx^ j\) be a 1- form valued random field and \(V_ \omega= V_ \omega(x)\) be a real valued random field defined on a probability space \(\Omega \) \((\omega\in\Omega\), \(x\in\mathbb{R}^ d)\). For each \(\omega\in\Omega\), let \(L(b_ \omega, V_ \omega)\) be the Schrödinger operator with the magnetic field, \[ L(b_ \omega, V_ \omega)= -1/2 \sum_{j=1}^ d (\partial_ j- \sqrt{-1} b^ j_ \omega (x))^ 2+ V_ \omega(x). \] Under the condition that the pair \((db_ \omega(x), V_ \omega(x))_{x\in \mathbb{R}^ d}\) is stationary and ergodic on \(\mathbb{R}^ d\), several known results on the spectra of the family of operators \(\{L (0, V_ \omega)\}_{\omega\in \Omega}\) [cf. \textit{R. Carmona} and \textit{J. Lacroix}, Spectral theory of random Schrödinger operators (1990; Zbl 0717.60074)] are extended for the family of operators \(\{L (b_ \omega, V_ \omega)\}_{\omega\in \Omega}\). In particular, it is shown that the asymptotics of the integrated density of states \(N(\lambda)\), \(\lambda\in \mathbb{R}\), as \(\lambda\downarrow -\infty\), is independent of \(b_ \omega\) if \(V_ \omega\) is a Gaussian random field.
spectral theory of random Schrödinger operators
scientific article; zbMATH DE number 639563