Absolute continuity of a two-dimensional magnetic periodic Schrödinger operator with potentials of the type of measure derivative
The author discusses the absolute continuity of the spectrum for the periodic (with respect to the lattice \(\Gamma\)) Schrödinger operator \(H\) in \(L_2({\mathbb R}^2)\) with magnetic potential, electric potential and variable metric \[ H = (i\overrightarrow{\nabla} - \overrightarrow{A}(x))^* g(x) (i\overrightarrow{\nabla} - \overrightarrow{A}(x)) + V(x). \] An electric potential \(V\) is assumed to be a distribution formally given by an expression \({d\nu} \over {dx}\), where \(\nu\) is a periodic signed measure with a locally finite variation form bounded with respect to \(\Delta\). The magnetic potential \(\overrightarrow{A}\) is assumed to be a real \(\Gamma\)-periodic function from \(L_r, r > 2\). The metric \(g\) is defined by a \((2\times 2)\)-matrix valued function with \(\Gamma\)-periodic real-valued entries from Sobolev space \(W^2_r\), \(r > 2\), such that \(c_1 I \leq g \leq c_2I\). The proof of absolute continuity is based on the method due to L. Thomas (1973).
scientific article; zbMATH DE number 1971380