# zbMATH — the first resource for mathematics

Absolute continuity of a two-dimensional magnetic periodic Schrödinger operator with potentials of the type of measure derivative. (English. Russian original) Zbl 1040.35049
J. Math. Sci., New York 115, No. 6, 2862-2882 (2003); translation from Zap. Nauchn. Semin. POMI 271, 276-312 (2000).
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).

##### MSC:
 35P05 General topics in linear spectral theory for PDEs 35J10 Schrödinger operator, Schrödinger equation 35Q40 PDEs in connection with quantum mechanics 35Q60 PDEs in connection with optics and electromagnetic theory 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
Full Text: