Spectral Analysis of Lattice Schrödinger-Type Operators Associated with the Nonstationary Anderson Model and Intermittency

Abstract: The research explores a high irregularity, commonly referred to as intermittency, of the solution to the non-stationary parabolic Anderson problem: \begin{equation*} \frac{\partial u}{\partial t} = \varkappa \mathcal{L}u(t,x) + \xi_{t}(x)u(t,x) \end{equation*} with the initial condition (u(0,x) \equiv 1), where ((t,x) \in [0,\infty)\times \mathbb{Z}^d). Here, (\varkappa \mathcal{L}) denotes a non-local Laplacian, and (\xi_{t}(x)) is a correlated white noise potential. The observed irregularity is intricately linked to the upper part of the spectrum of the multiparticle Schr\"{o}dinger equations for the moment functions (m_p(t,x_1,x_2,\cdots,x_p) = \langle u(t,x_1)u(t,x_2)\cdots u(t,x_p)\rangle). In the first half of the paper, a weak form of intermittency is expressed through moment functions of order $p\geq 3$ and established for a wide class of operators $\varkappa \mathcal{L}$ with a positive-definite correlator $B=B(x))$ of the white noise. In the second half of the paper, the strong intermittency is studied. It relates to the existence of a positive eigenvalue for the lattice Schr\"odinger type operator with the potential $B$. This operator is associated with the second moment $m_2$. Now $B$ is not necessarily positive-definite, but $\sum B(x)\geq 0$.