arXiv subjects
Publications and source records attributed to N. D. Filonov.
The equation $- Δu + V u = 0$ in the cylinder $\mathbb{R} \times (0,2π)^d$ with periodic boundary conditions is considered. The potential $V$ is assumed to be bounded, and both functions $u$ and $V$ are assumed to be real-valued. It is shown that the fastest rate of decay at infinity of non-trivial solution $u$ is $O\left(e^{-c|w|}\right)$ for $d=1$ or $2$, and $O\left(e^{-c|w|^{4/3}}\right)$ for $d\ge 3$. Here $w$ is the axial variable.
We study the Schrödinger operator on $L_2(\mathbb R^3)$ with periodic variable metric, and periodic electric and magnetic fields. It is assumed that the operator is reflection symmetric and the (appropriately defined) flux of the magnetic field is rational. Under these assumptions it is shown that the spectrum of the operator is absolutely continuous. Previously known results on absolute continuity for periodic operators were obtained for the zero magnetic flux.