CS45: Lévy processes and random walks in random and deterministic environments and their spectral theory

Organizer: Kamil Kaleta & Katarzyna Pietruska-Pałuba (Wrocław University of Science and Technology & University of Warsaw)

Lifshitz singularity for random Levy-Schroedinger operators with long range interactionsa

Katarzyna Pietruska-Pałuba (University of Warsaw)

In \([2]\) we have addressed the asymptotic behaviour of the integrated density of states (IDS) for random Lévy-Schrödinger operators \[H^\omega=\Phi(-\Delta)+V^\omega,\] where \(V^\omega\) is an alloy-type potential \[V^\omega(x)=\sum_{\mathbf i\in\mathbb R^d} q_{\mathbf i}(\omega) W(x-\mathbf i),\;\; x\in\mathbb R^d.\] Here \(q_{\mathbf i}\) are i.i.d. random variables and \(W\) is the potential profile. Our previous results \([1]\) were concerned with compactly supported profiles, and in present work we consider profiles of unbounded support. When \(W(|x|)\leq |x|^{-(d+\alpha)}\) for large \(|x|\) (\(\alpha\) depends on \(\Phi),\) then this behaviour is similar to that when \(W\) was of compact support. However, if this is not the case, the behaviour can depend on properties of random variables \(q_{\mathbf i}.\) In any case, Lifschitz singularity is present.

Both our previous and current results extend - to the case of nonlocal kinetic terms \(\Phi(-\Delta)\) - the results for the Laplacian of Kirsch and Simon from \([3]\).

Bibliography

\([1]\) Kaleta, K., Pietruska-Pałuba, K. (2025). "Lifshitz tail for long-range alloy-type models with Lévy operators", preprint.

\([2]\) Kaleta, K., Pietruska-Pałuba, K. "Lifschitz tail for continuous Anderson models driven by Lévy operators." Comm. Contemp. Math. 2020, 2050065 (46 pages).

\([3]\) Kirsch, W., Simon, B., "Lifshitz tails for periodic plus random potentials." J. Stat. Phys. vol. 42, no. 5/6, 1986, pp. 799-808.