Spectral theory of sparse potentials (Q2702442)

Sparse potentials are functions \(V(x)\) which do not decay to zero as \(|x|\to \infty\) but which become small near infinity in an averaged sense. The authors consider Schrödinger operators \(H=-\Delta+V\) with the following (both random and deterministic) sparse potentials NEWLINE\[NEWLINE\begin{aligned} V_\omega (x) &= \sum_{i\in{\mathbb Z}^d} \xi_i(\omega) f(x-i),\\ V_\omega (x) &= \sum_{i\in{\mathbb Z}^d} q_i(\omega) \xi_i(\omega) f(x-i),\\ V_\omega (x) &= \sum_{i\in{\mathbb Z}^d} a_i q_i(\omega) f(x-i),\\ V_\omega(x) &= \sum_{i\in{\mathbb Z}^{d-1}\times\{0\}} q_i(\omega) f(x-i),\end{aligned}NEWLINE\]NEWLINE where \(f\leq 0\) is a bounded function of compact support; \(\{\xi_i\}_{i\in {\mathbb Z}^d}\) are independent random variables with values in \(\{0,1\}\); \(q_i\) are independent identically distributed random variables; and \(\{a_i\}_{i\in{\mathbb Z}^d}\) is a deterministic sequence decaying (fast enough) at infinity. NEWLINENEWLINENEWLINEFor a huge class of potentials with decaying randomness it is proved that the positive half axis belongs to the absolutely continuous spectrum and that the corresponding wave operators exist. The main result is probabilistic in nature, that is, it claims the existence of the wave operators for a set of potentials of full measure. NEWLINENEWLINENEWLINEMoreover, it is shown that many of the above potentials admit an essential spectrum below zero. Although the spectrum itself is a random set the essential spectrum is not. In various situations this part of the spectrum is pure point. Under certain conditions it is shown that the essential spectrum below zero is a countable or even finite set. In other situations there is a dense point spectrum.NEWLINENEWLINEFor the entire collection see [Zbl 0953.00048].