Absence of diffusion near the bottom of the spectrum for a random Schrödinger operator on \(L^2(\mathbb{R}^3)\) (Q1290988)

Let \((\Omega,F,\mathbb{P})\) be a probability space. \(\mathbb{E}\) denotes the expectation with respect to \(\mathbb{P}\). \(L^2_2(R^3)\) is the totality of \(f\in L^2(R^3)\) such that \((1+| x|^2) f\in L^2(R^3)\). Consider Anderson type random Schrödinger operator on \(L^2(R^3)\): \(H_\omega= -\Delta+ V_\omega(x)\), \(\omega\in \Omega\) with \(V_\omega(x) =\sum_{i\in \mathbb{Z}^3} q_i(\omega) f(x-i)\). \(\{q_i\}\)'s are i.i.d. random variables with uniform distribution on \([0,1]\). \(f\) satisfies \(C_0\leq f(x)\leq C_1\) \((\exists C_0, C_1>0)\) for \(x\in [0,1)^3\) and \(f(x)=0\) for \(x\notin [0,1)^3\). When \(\sigma(H_\omega)\) is the spectrum of \(H_\omega\), it is well known that \(\sigma(H_\omega)\) a.s., see \textit{W. Kirsch} and \textit{F. Martinelli} [Commun. Math. Phys. 85, 329-350 (1982; Zbl 0506.60058)]. Suppose that \(g_E\in C_0^\infty(R)\) and \(\text{supp} g_E \subset (0,E)\) for \(E>0\), and set \[ r^2_E(t)= \mathbb{E}\int_{R^3} | x|^2 \bigl| e^{-itH_\omega} g_E(H_\omega) \psi(x)\bigr |^2dx \] for \(\psi\in L^2_2 (R^3)\). Physically, \(H_\omega\) is considered to be the operator corresponding to the Hamiltonian of the electron in random metallic media. \(g_E (H_\omega) \psi\) is a wave function of an electron which is well localized in \(L^2_2(R^3)\)-sense and has energy near the bottom of the spectrum. \(r^2_E(t)\) represents the mean square distance from the origin of the time evolution of the electron with the initial wave function \(g_E(H_\omega)\psi\). When \(V\equiv 0\) or \(V\) is periodic, \(r^2_E(t)\) behaves asymptotically as \(r^2_E(t)\sim Ct^2\) \((t\to \infty)\). While, when \(V\) is random, physical consideration allows us to expect that \(r^2_E(t) \sim Dt\) \((t\to \infty)\), with diffusion constant \(D\). \textit{J.-M. Combes} and \textit{P. D. Hislop} proved Anderson localization (AL) [J. Funct. Anal. 124, No. 1, 149-180 (1994; Zbl 0801.60054)], i.e., there exists \(E^* >0\) such that in \([0,E^*]\) the spectrum is pure point and the corresponding eigenfunctions decay asymptotically. Hence, it is quite natural to expect that \(D=0\) when \(E\) is sufficiently small. However, this does not follow directly from (AL). Here is the main result in this paper: There exists \(E^*>0\) such that if \(0<E<E^*\), then \(\lim_{T\to\infty}T^{-1}\int_{[1,T]} r^2_E(t)/t dt=0\). The leading ideas of the proof are greatly due to \textit{J. Fröhlich} and \textit{T. Spencer} [Commun. Math. Phys. 88, 151-184 (1983; Zbl 0519.60066)] and \textit{F. Martinelli} and \textit{H. Holden} [ibid. 93, 197-217 (1984; Zbl 0546.60063)], where similar results on absence of diffusion can be found in different settings.