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

Birman, M. Sh.; Sulina, T. A.

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 \[ \| M(y)^{-1}\|\leq C(\mu+| y|)^{-1}. \]

Reviewer: Abdumalik Rakhimov (Tashkent)

MSC:

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

Keywords:

absolutely continuous spectrum; magnetic potential