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MATHEMATICS
On the spectrum of a multidimensional periodic magnetic Shrödinger operator with a singular electric potential
L. I. Danilov Udmurt Federal
Research Center, Ural Branch of the Russian Academy of Sciences, ul. T. Baramzinoi, 34, Izhevsk,
426067, Russia
Abstract:
We prove absolute continuity of the spectrum of a periodic
$n$-dimensional Schrödinger operator for
$n\geqslant 4$. Certain conditions on the magnetic potential
$A$ and the electric potential
$V+\sum f_j\delta _{S_j}$ are
supposed to be fulfilled. In particular, we can assume that the following conditions are satisfied.
(1) The magnetic potential
$A\colon{\mathbb{R}}^n\to {\mathbb{R}}^n$ either has an absolutely convergent Fourier series or belongs to
the space $H^q_{\mathrm {loc}}({\mathbb{R}}^n;{\mathbb{R}}^n)$,
$2q>n-1$, or to the space $C({\mathbb{R}}^n;{\mathbb{R}}^n)\cap
H^q_{\mathrm {loc}}({\mathbb{R}}^n;{\mathbb{R}}^n)$,
$2q>n-2$.
(2) The function
$V\colon{\mathbb{R}}^n\to \mathbb{R} $ belongs to Morrey space
${\mathfrak L}^{2,p}$,
$p\in \big( \frac {n-1}2,
\frac n2\big] $, of periodic functions (with a given period lattice), and
$$
\lim\limits_{\tau\to+0}
\sup\limits_{0<r\leqslant\tau}\sup\limits_{x\in{\mathbb{R}}^n}r^2\bigg(\big(v(B^n_r)\big)^{-1}\int_{B^n_r(x)}|{\mathcal{V}}(y)|^p dy\bigg)^{1/p}\leqslant C,
$$
where
$B^n_r(x)$ is a closed ball of radius
$r>0$ centered at a point
$x\in{\mathbb{R}}^n$,
$B^n_r=B^n_r(0)$,
$v(B^n_r)$ is
volume of the ball
$B^n_r$,
$C=C(n,p;A)>0$.
(3)
$\delta_{S_j}$ are
$\delta$-functions concentrated on (piecewise)
$C^1$-smooth periodic hypersurfaces
$S_j$,
$f_j\in L^p_{\mathrm {loc}}(S_j)$,
$j=1,\dots ,m$. Some additional geometric conditions are imposed on the hypersurfaces
$S_j$, and these conditions determine the choice of numbers
$p\geqslant n-1$. In particular, let hypersurfaces
$S_j$ be
$C^2$-smooth, the unit vector
$e$ be arbitrarily taken from some dense set of the unit sphere
$S^{n-1}$ dependent on the magnetic potential
$A$, and the normal curvature of the hypersurfaces
$S_j$ in the direction of the unit vector
$e$ be nonzero at all points of tangency of the hypersurfaces
$S_j$ and the lines
$\{x_0+te\colon t\in\mathbb{R}\}$,
$x_0\in{\mathbb{R}}^n$. Then we can choose the number
$p>\frac {3n}2-3$,
$n\geqslant 4$.
Keywords:
absolute continuity of the spectrum, periodic Schrödinger operator.
UDC:
517.958,
517.984.56
MSC: 35P05 Received: 19.05.2021
DOI:
10.35634/2226-3594-2021-58-02
Fulltext:
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