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

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) + ξ_{t}(x)u(t,x) \end{equation*} with the initial condition $u(0,x)\equiv 1$, where $(t,x)\in [0,\mathrm{\infty })\times {\mathbb{Z}}^d$. Here, $\mathrm{\varkappa }\mathcal{L}$ 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 Schrödinger equations for the moment functions $m_p(t,x_1,x_2,\cdots ,x_p)=⟨u(t,x_1)u(t,x_2)\cdots u(t,x_p)⟩$. In the first half of the paper, a weak form of intermittency is expressed through moment functions of order $p\ge 3$ and established for a wide class of operators $\mathrm{\varkappa }\mathcal{L}$ 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 Schrödinger type operator with the potential $B$. This operator is associated with the second moment $m_2$. Now $B$ is not necessarily positive-definite, but $\sum B(x)\ge 0$.