A remark on the Lifshitz tail for Schrödinger operator with random magnetic field
In his previous paper [Commun. Math. Phys. 214, 565-572 (2000; Zbl 1018.81011)], the author studied the integrated density \(k(E)\) of states for the magnetic Schrödinger operator \((-i\partial_x- A(x))^2\) in \(L^2(\mathbb{R}^d)\) with ergodic random magnetic field \(B^\omega(x)\) to prove that, assuming that \(B^\omega(x)\) and \(B^\omega(y)\) become almost independent if two points \(x\) and \(y\) are sufficiently far from each other, one has \[ \limsup_{E\downarrow 0} \log(-\log k(E))/\log E\leq -d/2, \] i.e. for any \(\varepsilon> 0\), \(k(E)\leq \exp(- E^{-d/2+\varepsilon})\) for sufficiently small \(E\geq 0\). But to prove this theorem, he had to assume in addition uniform boundedness of random magnetic field \(B^\omega(x)\) for all \(\omega\) and \(x\in\mathbb{R}^d\). In the note under review he removes this additional condition with a brief proof.
scientific article; zbMATH DE number 1751267