On the density of states for the periodic Schrödinger operator
The operators considered in the paper under review are of the form \[ H_V=(-\Delta)^l+V \qquad \text{in} L^2(\mathbb{R}^n), n\geq 2, \] with \(l\geq 1/2\). Here \(V\) is the operators of multiplication by a real potential that is periodic with respect to the lattice \(\Gamma=\{(a_1k_1,\dots, a_nk_n)\mid (k_1,\dots k_n)\in \mathbb{Z}^n\}\), with \(a_1, \dots, a_n>0\); \(V\) is supposed to be of the form \[ V(x)=\sum\limits_{{m\in \mathbb{Z}^n}\atop{m\neq 0}}v_m\exp{(ip_m(0)\cdot x)}, \] with \(p_m(0)=2\pi (m_1a_1^{-1}, \dots, m_na_n^{-1})\). Moreover, \(V\) satisfies the following condition of smoothness: \[ \sum\limits_{{m\in \mathbb{Z}^n}\atop{m\neq 0}}|v_m||m|^{\nu}<\infty, \quad \text{for some }\nu>0. \tag{1} \] Let \(Q\), \(K\) be the elementary cells of the lattice \(\Gamma\) and of the dual lattice, respectively. For \(t\in K\), \(H(t)\) is, as usual, the reduced operator acting in \(L^2(Q)\) with quasiperiodic boundary conditions depending on \(t\). Let \(N_V(\lambda, t)\) be the counting function of the eigenvalues associated to \(H(t)\). The paper deals with the density of states of \(H_V\), that is, \[ D_V(k^{2l})=\int\limits N_V(k^{2l}, t) \text{d}t. \] The main results of the paper are the following. (i) The estimate \[ D_V(k^{2l})=\omega_nk^n+{\mathcal O}(k^{-\xi}\log k), \quad \xi=4l-n-1 \] holds provided \(l>1/2\) and \(V\) satisfies (1) with \(\nu>\frac{1}{2}(n-1)\), if \(n\geq 3\), or \(\nu>\frac{2}{3}(l+1)\), if \(n=2\). (Here \(\omega_n\) is a constant that depends on \(n\) and the volume of \(K\).) This estimate is proved using a perturbation technique and is a generalization of a previous result of \textit{B. Helffer} and \textit{A. Mohamed} [Duke Math. J. 92, 1-60 (1998; Zbl 0951.35104)]. The estimate also holds for \(V\) belonging to a certain class of periodic differential operator; yet, in the author remarks that, in the particular case of magnetic perturbations, the estimate is not optimal, and that, for this particular case, a better estimate was given by \textit{A. Mohamed} [J. Math. Phys. 38, No. 8, 4023-4051 (1997; Zbl 0883.47045)]. (ii) The second term of the asymptotic of \(D_V(k^{2l})\) is computed for the special case \(n=3\), \(l=1\), and \(V\) satisfies (1) with \(\nu=9000\). It turns out that \[ D_V(k^2)=\omega_3k^3+d_V^0+{\mathcal O}(k^{-\xi}),\quad \xi<1/130, \] \(d_V^0\) being a constant, which is given in the paper. Again, the method is perturbative, and the idea is to compare \(H_V(t)\) with an appropriate well-chosen perturbation of \(H_0(t)\).
scientific article; zbMATH DE number 1928891
scientific article