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Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2012, Issue 3, Pages 25–47
(Mi vuu334)
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This article is cited in 5 scientific papers (total in 5 papers)
MATHEMATICS
On the spectrum of a periodic Schrödinger operator with potential in the Morrey space
L. I. Danilov Physical–Technical Institute, Ural Branch of the Russian Academy of Sciences, Izhevsk, Russia
Abstract:
We consider the periodic Schrödinger operator $\widehat H_A+V$ in $\mathbb R^n$, $n\geqslant3$. The vector potential $A$ is supposed to satisfy some conditions which are fulfilled whenever the potential $A$ belongs to the Sobolev class $H^q_\mathrm{loc}(\mathbb R^n;\mathbb R^n)$, $q>\frac{n-1}2$, and also in the case where $\sum\|A_N\|_{\mathbb C^n}<+\infty$. Here $A_N$ are the Fourier coefficients of the potential $A$. We prove absolute continuity of the spectrum of the periodic Schrödinger operator $\widehat H_A+V$ provided that the scalar potential $V$ belongs to the Morrey space $\mathfrak L^{2,p}(\mathbb R^n)$, $p\in(\frac{n-1}2,\frac n2]$, and
$$
\overline{\lim_{r\to+0}}\sup_{x\in\mathbb R^n}r^2\biggl(\frac1{v(B_r)}\int_{B_r(x)}|V(y)|^p\,dy\biggr)^{1/p}\leqslant\varepsilon_0,
$$
where the number $\varepsilon_0=\varepsilon_0(n,p;A)>0$ depends on the vector potential $A$, $B_r(x)$ is a closed ball of radius $r>0$ centered at the point $x\in\mathbb R^n$, $v(B_r)$ is the $n$-dimensional volume of the ball $B_r=B_r(0)$. Let $K$ be the fundamental domain of the period lattice (which is common for the potentials $A$ and $V$), $K^*$ the fundamental domain of the reciprocal lattice. The operator $\widehat H_A+V$ is unitarily equivalent to the direct integral of operators $\widehat H_A(k)+V$, $k\in2\pi K^*$, acting on the space $L^2(K)$. The last operators are also considered for complex vectors $k+ik'\in\mathbb C^n$. To prove absolute continuity of the spectrum of the operator $\widehat H_A+V$, we use the Thomas method. The main ingredient in the proof is the following inequality:
\begin{gather*}
\|\,|\widehat H_0(k+ik')|^{-1/2}\bigl(\widehat H_A(k+ik')+V-\lambda\bigr)\varphi\|_{L^2(K)}\geqslant\widetilde C_1\|\,|\widehat H_0(k+ik')|^{1/2}\varphi\|_{L^2(K)},\\
\varphi\in D(\widehat H_A(k+ik')+V),
\end{gather*}
which holds for some appropriate chosen complex vectors $k+ik'\in\mathbb C^n$ (depending on $A,V$, and the number $\lambda\in\mathbb R$) with sufficiently large imaginary part $k'$, where $\widetilde C_1=\widetilde C_1 (n;A)>0$ and $\widehat H_0(k+ik')$ is the operator $\widehat H_A(k+ik')$ for $A\equiv0$.
Keywords:
Schrödinger operator, absolute continuity of the spectrum, periodic potential, Morrey space.
Received: 23.12.2011
Citation:
L. I. Danilov, “On the spectrum of a periodic Schrödinger operator with potential in the Morrey space”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2012, no. 3, 25–47
Linking options:
https://www.mathnet.ru/eng/vuu334 https://www.mathnet.ru/eng/vuu/y2012/i3/p25
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