Explicit asymptotic properties of the integrated density of states $N(\lambda)$ with respect to the spectrum for the random Schrödinger operator $H^{\omega}=(-\Delta)^{\alpha/2}+V^{\omega}$ are established, where $\alpha\in (0,2]$ and $V^\omega(x)=\sum_{i \in \mathbb{Z}^{d}} \xi_i(\omega) W(x-i)$ is a random potential term generated by a sequence of independent and identically distributed random variables $\{\xi_i\}_{i\in \mathbb{Z}^d}$ and a non-negative measurable function $W(x)$. In particular, the exact order of asymptotic properties of $N(\lambda)$ depends on the decay properties of the reference function $W(x)$ and the spectrum properties of the first Dirichlet eigenvalue of $(-\Delta)^{\alpha/2}$.
Longteng ZHANG
,
Jin CHEN
. ASYMPTOTIC PROPERTIES OF THE INTEGRATED DENSITY OF STATES FOR RANDOM SCHRÖDINGER OPERATORS[J]. Acta mathematica scientia, Series B, 2025
, 45(5)
: 2251
-2263
.
DOI: 10.1007/s10473-025-0523-x
[1] Carmona R, Lacroix J.Spectral Theory of Random Schrödinger Operators. Basel: Birkhäuser Verlag, 1990
[2] Friedberg R, Luttinger J M. Density of electronic energy levels in disordered systems. Physical Review B, 1975, 12(10-15): 4460-4474
[3] Fukushima M. On the spectral distribution of a disordered system and the range of a random walk. Osaka Journal of Mathematics, 1974, 11(1): 73-85
[4] Fukushima M, Nagai H, Nakao S. On an asymptotic property of spectra of a random difference operator. Proceedings of the Japan Academy, 1975, 51(2): 100-102
[5] Kaleta K, Pietruska-Pałuba K. Lifschitz tail for alloy-type models driven by the fractional Laplacian. Journal of Functional Analysis, 2020, 279(5): Art 108575
[6] Kaleta K, Pietruska-Pałuba K. Lifshitz tail for continuous Anderson models driven by Lévy operators. Communications in Contemporary Mathematics, 2021, 23(6): Art 2050065
[7] Kirsch W, Martinelli F. Large deviations and Lifshitz singularity of the integrated density of states of random Hamiltonians. Communications in Mathematical Physics, 1983, 89: 27-40
[8] Kirsch W, Martinelli F. On the density of states of Schrödinger operators with a random potential. Journal of Physics A: Mathematical and General, 1982, 15(7): 2139-2156
[9] Kirsch W, Simon B. Lifschitz tails for periodic plus random potentials. Journal of Statistical Physics, 1986, 42(5/6): 799-808
[10] Kirsch W, Veselić I. Lifshitz tails for a class of Schrödinger operators with random breather-type potential. Letters in Mathematical Physics, 2010, 94(1): 27-39
[11] Klopp F. Weak disorder localization and Lifshitz tails. Communications in Mathematical Physics, 2002, 232(1): 125-155
[12] Klopp F. Weak disorder localization and Lifshitz tails: continuous Hamiltonians. Annales Henri Poincaré, 2002, 3(4): 711-737
[13] Lifshitz I M. Energy spectrum structure and quantum states of disordered condensed systems. Soviet Physics Uspekhi, 1965, 7(4): 549-573
