## Absolute continuity of the two-dimensional periodic magnetic Hamiltonian with discontinuous vector-valued potential.(English. Russian original)Zbl 0922.35101

St. Petersbg. Math. J. 10, No. 4, 579-601 (1999); translation from Algebra Anal. 10, No. 4, 1-36 (1998).
The authors consider the two-dimensional magnetic Hamiltonian $M= (i\nabla+ A(x))^2+ V(x),$ where the vector-valued magnetic potential $$A$$ and the electric potential $$V$$ are assumed periodic with the same period. In a previous paper, the authors investigated the Hamiltonian with continuous potentials. In the present article, the potentials are assumed integrable with some degrees in an elementary cell $$\Omega$$ and the vector valued magnetic potential $$A(x)$$ is also assumed to satisfy the gauge conditions $\partial_1 A_1+\partial_2 A_2= 0,\quad \int_\Omega A dx= 0.$ The main result of the paper is the proof that the Hamiltonian $$M$$ is absolutely continuous. There is also a technical result which is independent and important: there exist constants $$b= b(A,V)$$ and $$C= C(A)$$, such that $$M(y)$$ is invertible for $$| y|>b$$ and $\| M(y)^{-1}\|\leq C(\mu+| y|)^{-1}.$

### MSC:

 35P05 General topics in linear spectral theory for PDEs 35J10 Schrödinger operator, Schrödinger equation

### Keywords:

absolutely continuous spectrum; magnetic potential