Periodic Schrödinger operators with (non-periodic) magnetic and electric potentials
We consider perturbations of a periodic Schrödinger operator \(-\Delta +V_ p\) in \(L^ 2({\mathbb{R}}^ N)\), with \(V_ p\) periodic, by singular magnetic and electric potentials (\(a_ j)_{1\leq j\leq N}\) and V, respectively, and compare the essential spectrum of \(-\Delta +V_ p\) with the essential spectrum of \(H=-\sum_{j}(\partial_ j-ia_ j)^ 2+V_ p+V,\) where H is defined via quadratic forms. H describes a (spin- less) electron on a crystal in a magnetic or electric field. In section 1, Corollary 1.11, we show that \(\sigma_{ess}(H)=\sigma_{ess}(-\Delta +V_ p),\) provided \(a^ 2_ j+| V|\) tends to 0 at infinity (in a weak sense). To obtain the basic relative compactness properties, a new and very elementary argument (see Lemma 1.2) is employed. In section 2, we only require that \(\chi_{\Omega}(a^ 2_ j+| V|)\) tends to 0 at infinity, for some (quasi-)conical region \(\Omega \subset {\mathbb{R}}^ N\); then it is still possible to show \(\sigma_{ess}(H)\supset \sigma_{ess}(-\Delta +V_ p)\) (Theorem 2.3).
scientific article; zbMATH DE number 3903142
scientific article