A general framework for localization of classical waves. II: Random media (Q1876639)

[Part I cf. ibid. 4, No.~2, 97--130 (2001; Zbl 0987.35154)]. The authors prove the following main results: Let \(W_{g,\omega}\) be a random second-order classical wave operator satisfying the assumption, there exist \(a,b\in \sigma(W_0)\), \(0< a<b\), such that \((a,b)\cap \sigma(W_0)= \emptyset\). There exists \(g_0\), with \[ \begin{multlined} \max\left\{\frac{1}{U_+},\,\frac{1}{V_+}\right\} \left(1-\left(\frac ab\right)^{1/4}\right)\leq g_0\leq\\ \min\left\{\frac{1}{Z_+},\,\frac{1}{U_+}\left(\left(\frac ba\right)^{U_+/4U_-}-1\right),\;\frac{1}{V_+}\left(\left(\frac ba\right)^{V_+/4V_-}-1\right)\right\}, \end{multlined} \] and increasing, Lipschitz continuous real valued functions \(a(g)\) and \(-b(g)\) on the interval \([0,1/Z_+\)), with \(a(0) = a\), \(b(0) = b\), and \(a(g)\leq b(g)\) such that: (i) \(\Sigma_g\cap [a,b] = [a,a(g)]\cup [b(g),b].\) (ii) If \(g <g_0\) we have \(a(g)<b(g)\) and \((a(g),b(g))\) is a gap in the spectrum \(\Sigma_g\) of the random operator \(W_{g,\omega}\). Moreover, we have \[ a(1+gU_+)^{U_-/U_+}(1+gV_+)^{V_-/V_+}\leq a(g)\leq \frac{a}{(1-gU_+)(1-gV_+)} \] and \[ b(1- gU_+)(1- gV_+)\leq b(g)\leq \frac {b}{(1+gU_+)^{U_-/U_+}(1-gV_+)^{V_-/V_+}}. \] In addition, if \(0\leq g_1 < g_2< g_0\) we have \[ \frac12(\sigma_-(g_2)+\eta_-(g_2))(a(g_1) +a(g_2))\leq \frac{a(g_2)-a(g_1)}{g_2-g_1}\leq \tfrac12(\delta_+(g_2)+\eta_+(g_2))(a(g_1)+a(g_2)), \] \[ \tfrac12(\delta_-(g_2)+\eta_-(g_2))(b(g_1)+b(g_2))\leq \frac{b(g_1)-b(g_2)}{g_2-g_1}\leq \tfrac12(\delta_+(g_2)+\eta_+(g_2))(b(g_1)+b(g_2)). \] (iii) If \(g_0 < 1/Z_+\) we have \(a(g)=b(g)\) for all \(g\in[g_0,1/Z_+)\), and the random classical wave operator \(W_{g,\omega}\) has one spectral gap inside the gap \((a,b)\) of the periodic classical wave operator \(W_0\), i.e., we have \([a,b]\subset\Sigma_g\).