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Russia Izhevsk
Year
2012
Issue
3
Pages
25-47
 Section Mathematics Title On the spectrum of a periodic Schrödinger operator with potential in the Morrey space Author(-s) Danilov L.I.a Affiliations Physical Technical Institute, Ural Branch of the Russian Academy of Sciencesa Abstract We consider the periodic Schrödinger operator $\widehat H_A+V$ in ${\mathbb R}^n,$ $n\geqslant 3.$ 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 $$\overline {\lim\limits_{r \to +0}}\ \sup\limits_{x \in {\mathbb R}^n} r^2 \biggl( \frac 1{v(B_r)} \int_{B_r(x)}|V(y)|^p dy\biggr) ^{1/p} \leqslant \varepsilon_0 ,$$ 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 $\widehat H_A(k)+V,$ $k\in 2\pi K^*,$ acting on the space $L^2(K).$ The last operators are also considered for complex vectors $k+ik^{ \prime }\in {\mathbb C}^n.$ 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: $$\| |\widehat H_0(k+ik^{ \prime })|^{-1/2} \bigl( \widehat H_A(k+ik^{ \prime })+V-\lambda \bigr) \varphi \|_{L^2(K)} \geqslant \widetilde C_1 \| |\widehat H_0(k+ik^{ \prime })|^{1/2}\varphi \|_{L^2(K)} ,\quad \varphi \in D(\widehat H_A(k+ik^{ \prime })+V),$$ which holds for some appropriate chosen complex vectors $k+ik^{\prime }\in {\mathbb C}^n$ (depending on $A$, $V,$ and the number $\lambda \in {\mathbb R}$) with sufficiently large imaginary part $k^{\prime },$ where $\widetilde C_1=\widetilde C_1 (n;A)>0$ and $\widehat H_0(k+ik^{\prime })$ is the operator $\widehat H_A(k+ik^{ \prime })$ for $A\equiv 0.$ Keywords Schrödinger operator, absolute continuity of the spectrum, periodic potential, Morrey space UDC 517.958, 517.984.5 MSC 35P05 DOI 10.20537/vm120304 Received 23 December 2011 Language Russian Citation Danilov L.I. 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