RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki: Year: Volume: Issue: Page: Find

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

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)
References: PDF file   HTML file
UDC: 517.958+517.984.5
MSC: 35P05

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} 

• http://mi.mathnet.ru/eng/vuu334
• http://mi.mathnet.ru/eng/vuu/y2012/i3/p25

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
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
3. L. I. Danilov, “O spektre dvumernogo operatora Shredingera s odnorodnym magnitnym polem i periodicheskim elektricheskim potentsialom”, Izv. IMI UdGU, 51 (2018), 3–41
•  Number of views: This page: 228 Full text: 61 References: 42 First page: 1