On the spectrum of a~periodic Schr\"odinger operator with potential in the Morrey space

L. I. Danilov

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, no. 3 (2012), pp. 25-47

Résumé

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 $A\equiv0$.

Détail

Keywords: Schrödinger operator, absolute continuity of the spectrum, periodic potential, Morrey space.