Absolute continuity of the spectrum of a periodic Schrödinger operator
The author considers the Schrödinger operator \[ \widehat{H}= \widehat{H}(A)+V= \sum_{j=1}^n \bigg(-i \frac{\partial}{\partial x_j}-A_j\bigg)^2+V,\tag{1} \] acting in \(L^2(\mathbb R^n)\), \(n\geq 3\). He proves the absolute continuity of the spectrum of (1), under the following assumptions on magnetic and electric potentials, namely, \(A\in C^1(\mathbb R^n,\mathbb R^n)\) for which either \(A\in H_{\text{loc}}^q(\mathbb R^n,\mathbb R^n)\), \(2q>n-2\) or the Fourier serie sof \(A\) converges absolutely, \(V\in L_w^{p(n)}(K)\), where \(K\) is an elementary cell of the lattice \(\Lambda\), \(p(n)= n/2\) for \(n=3,4,5,6\), and \(p(n)=n-3\) for \(n\geq 7\), and \(L_w^{p(n)}\) is the suitably chosen set of measurable functions \(W:K\to\mathbb C\) such that \[ \|W\|_{p,w}:=\sup_{t>0} t\big(v(\{x\in K\mid |W(x)|>t\})\big)^{1/p}<+\infty, \] where \(v(\cdot)\) is the Lebesgue measure.
scientific article; zbMATH DE number 1928360