Explicit asymptotic properties of the integrated density of states N(λ) with respect to the spectrum for the random Schrödinger operator Hω = (−Δ)α/2 + Vω are established, where α ∈ (0, 2] and <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10473_2025_523_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="219" /> </InlineMediaObject> <EquationSource Format="TEX">$V^\omega(x)=\sum\nolimits_{i \in {\mathbb Z}^{d}}\, \xi_i(\omega) W(x-i)$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msup> <mi>V</mi> <mi>ω</mi> </msup> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mo movablelimits="false">∑</mo> <mrow> <mi>i</mi> <mo>∈</mo> <msup> <mrow> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> <mrow> <mi>d</mi> </mrow> </msup> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mi>ξ</mi> <mi>i</mi> </msub> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mi>W</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>i</mi> <mo stretchy="false">)</mo> </math></EquationSource> </InlineEquation> is a random potential term generated by a sequence of independent and identically distributed random variables <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10473_2025_523_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">$\{\xi_i\}_{i\in {\mathbb Z}^d}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>ξ</mi> <mi>i</mi> </msub> <msub> <mo fence="false" stretchy="false">}</mo> <mrow> <mi>i</mi> <mo>∈</mo> <msup> <mrow> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> </mrow> <mi>d</mi> </msup> </mrow> </msub> </math></EquationSource> </InlineEquation> and a non-negative measurable function W(x). In particular, the exact order of asymptotic properties of N(λ) depends on the decay properties of the reference function W(x) and the spectrum properties of the first Dirichlet eigenvalue of (−Δ)α/2.

Asymptotic properties of the integrated density of states for random Schrödinger operators

期刊论文
发表日期：

2025年8月4日

查看全文

Longteng Zhang,
Jin Chen

摘要

Explicit asymptotic properties of the integrated density of states N(λ) with respect to the spectrum for the random Schrödinger operator H^ω = (−Δ)^α/2 + V^ω are established, where α ∈ (0, 2] and $V^\omega(x)=\sum\nolimits_{i \in {\mathbb Z}^{d}}\, \xi_i(\omega) W(x-i)$ $V^{ω} (x) = \sum_{i \in Z^{d}} ξ_{i} (ω) W (x - i)$ is a random potential term generated by a sequence of independent and identically distributed random variables $\{\xi_i\}_{i\in {\mathbb Z}^d}$ ${ξ_{i}}_{i \in Z^{d}}$ and a non-negative measurable function W(x). In particular, the exact order of asymptotic properties of N(λ) depends on the decay properties of the reference function W(x) and the spectrum properties of the first Dirichlet eigenvalue of (−Δ)^α/2.