The periodic Schrödinger operators with potentials in the Morrey class. (English) Zbl 1119.35316

Summary: We consider the periodic Schrödinger operator \(H = - \Delta + V(x)\) in \(\mathbb R^d\), \(d\geq 3\) with potential \(V\) in the Morrey class. Let \(\Omega\) be a periodic cell for \(V\). We show that, for \(p\in ((d-1)/2,d/2]\), there exists a positive constant \(\varepsilon\) depending only on the shape of \(\Omega\), \(p\) and \(d\) such that, if \[ \limsup_{r\to 0} \sup_{x \in \Omega} r^2 \left\{{1 \over r^d} \int_{Q(x,r)} | V(y)|^p dy \right\}^{1/p} < \varepsilon_0,\tag{\({*}\)} \] then the spectrum of \(- \Delta + V(x)\) is purely absolutely continuous. We obtain this result as a consequence of certain weighted \(L^2\) Sobolev inequalities on the \(d\)-torus. It improves an early result by the author for potentials in \(L^{d/2}\) or weak-\(L^{d/2}\) space.


35J10 Schrödinger operator, Schrödinger equation
47F05 General theory of partial differential operators
35B10 Periodic solutions to PDEs
35P05 General topics in linear spectral theory for PDEs
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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