Absolute Continuity of the Spectrum of a Periodic 3D Magnetic Schr\"{o}dinger Operator with Singular Electric Potential

L. I. Danilov

Matematičeskie zametki, Tome 110 (2021) no. 4, pp. 507-523

Résumé

We prove that the spectrum of a periodic 3D magnetic Schrödinger operator whose electric potential $V=d\mu/dx$ is the derivative of a measure is absolutely continuous provided that the distribution $d|\mu|/dx$ is $(-\Delta)$-bounded in the sense of quadratic forms with bound not exceeding some constant $C(A)\in(0,1)$, and the periodic magnetic potential $A$ satisfies certain conditions, which, in particular, hold if $A\in H^q_{\mathrm{loc}}(\mathbb R^3;\mathbb R^3)$ for some $q>1$ or $A\in C(\mathbb R^3;\mathbb R^3)\cap H^q_{\mathrm{loc}}(\mathbb R^3;\mathbb R^3)$ for some $q>1/2$.

Détail

Keywords: absolutely continuous spectrum, periodic Schrödinger operator.