The integrated density of states of one-dimensional random Schrödinger operator with white noise potential and background (Q875683)

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 each \((p,q,r)\in{\mathbb R}^{3}\), the function \(u(x)=u(x;p,q,r)\), \(0<x<\infty\), is the bounded solution of the equation \[ \tfrac{1}{2}\sigma^{2}(x)u^{\prime}(x)+b(x;p,q,r)u(x)=1,\qquad 0<x<\pi, \] \(\sigma(x):=c\sin^{2}x\) and \(b(x;p,q,r):=p\cos^{2}x+(-q+\lambda r)\sin^{2}x+c^{2}\sin^{3}x\cos x\).