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Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2012, Issue 3, Pages 25–47 (Mi vuu334)  

This article is cited in 3 scientific papers (total in 3 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
$$ \overline{\lim_{r\to+0}}\sup_{x\in\mathbb R^n}r^2(\frac1{v(B_r)}\int_{B_r(x)}|V(y)|^p dy)^{1/p}\leqslant\varepsilon_0, $$
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}(\widehat H_A(k+ik')+V-\lambda)\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.

Full text: PDF file (353 kB)
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UDC: 517.958+517.984.5
MSC: 35P05
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

Citation in format AMSBIB
\Bibitem{Dan12}
\by L.~I.~Danilov
\paper On the spectrum of a~periodic Schr\"odinger operator with potential in the Morrey space
\jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki
\yr 2012
\issue 3
\pages 25--47
\mathnet{http://mi.mathnet.ru/vuu334}


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    This publication is cited in the following articles:
    1. L. I. Danilov, “O spektre dvumernogo obobschennogo periodicheskogo operatora Shredingera”, Izv. IMI UdGU, 2013, no. 1(41), 78–95  mathnet
    2. L. I. Danilov, “O spektre dvumernogo obobschennogo periodicheskogo operatora Shredingera. II”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 2014, no. 2, 3–28  mathnet
    3. L. I. Danilov, “O spektre dvumernogo operatora Shredingera s odnorodnym magnitnym polem i periodicheskim elektricheskim potentsialom”, Izv. IMI UdGU, 51 (2018), 3–41  mathnet  crossref  elib
  • Вестник Удмуртского университета. Математика. Механика. Компьютерные науки
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