The integrated density of states of one-dimensional random Schrödinger operator with white noise potential and background
type: Item
instance of: scholarly article
title: The integrated density of states of one-dimensional random Schrödinger operator with white noise potential and background
zbMATH Open document ID: 1126.34053
author: Katsumi Nagai
published in: Tsukuba Journal of Mathematics
publication date: 2007-04-13T00:00:00Z
review text: The author discusses the integrated density of states \(N(\lambda)\), \(\lambda\in{\mathbb R}\), of the random Schrödinger operator \[ H=-\frac{1}{r(t)}\frac{d}{dt}\left(\frac{1}{p(t)}\frac{d}{dt}\right) +\frac{q(t)}{r(t)}+\frac{cB^{\prime}(t)}{r(t)},\qquad 0\leq t<\infty. \] Here, we suppose that the coefficients of \(H\) satisfy the following conditions: \((p_{\omega}(t))_{t\geq 0}\), \((q_{\omega}(t))_{t\geq 0}\), \((r_{\omega}(t))_{t\geq 0}\) are continuous semi-martingales on a probability space \((\Omega, {\mathcal F}, P)\) with a filtration \(({\mathcal F}_{t})_{t\geq 0}\), namely, \(p(t)\) is expressed as \(p_{\omega}(t)=p(t)=p(0)+M^{p}(t)+A^{p}(t)\), where \(M^{p}\) (\(M(0)=0\) a.s.) is a continuous local (\({\mathcal F}_{t}\))-martingale and \(A^{p}(t)\) (\(A(0)=0\) a.s.) is a continuous (\({\mathcal F}_{t}\))-adapted process whose sample functions (\(t\mapsto A^{p}\)) are of bounded variation on any finite interval a.s., and \(p(0)\) is an (\({\mathcal F}_{0}\))-measurable random variable. \((B_{\omega}(t))_{t\geq 0}\) is an (\({\mathcal F}_{t}\))-Brownian motion, and \(B^{\prime}(t)\) is the derivative of its sample function. Moreover, \(p(t)\), \(M^{p}(t)\) and \(A^{p}(t)\) satisfy the following conditions: (A.1) The limits \(M^{p}(\infty):=\lim_{t\to\infty}M^{p}(t)\), \(A^{p}(\infty):=\lim_{t\to\infty}A^{p}(t)\) exist a.s. (A.2) For some \(0<c_{1}<c_{2}\), \(c_{3}\in{\mathbb R}\), which are independent of \(\omega\), the inequalities \(c_{1}\leq p(t)\), \(r(t)\leq c_{2}\), \(|q(t)|\leq c_{3}\) hold true. (A.3) \(\int^{l}_{0}t|dA^{p}|=o(l)\) as \(l\to\infty\), \(\int^{\infty}_{0}t^{2}d\langle M^{p}\rangle=O(l^{\delta})\) for some \(0<\delta<2\) as \(l\to\infty\). We also suppose that \(q(t)\) and \(r(t)\) satisfy the above conditions. The main result of this paper is then stated as follows. Theorem. Under the assumptions (A.1), (A.2) and (A.3), we have \[ N(\lambda)=\left(\int^{\pi}_{0}u(x;p(\infty),q(\infty),r(\infty))\,dx\right)^{-1}, \] where, for ea…
reviewed by: Kazushi Yoshitomi
Mathematics Subject Classification ID: 34L40
Mathematics Subject Classification ID: 47B80
Mathematics Subject Classification ID: 60H25
zbMATH DE Number: 5142561
zbMATH Keywords: random Schrödinger operator
zbMATH Keywords: integrated density of states
zbMATH Keywords: white noise potential
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496165069
OpenAlex ID: W248220956
DOI: 10.21099/TKBJM/1496165069
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description: scientific article; zbMATH DE number 5142561