Spectral Analysis of Lattice Schr\"{o}dinger-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: {equation*} ∂ u/∂ t = Lu(t,x) + ξₜ(x)u(t,x) {equation*} with the initial condition \(u(0,x) ≡ 1\), where \((t,x) ∈ [0,∞)× Z^d\). Here, \( L\) denotes a non-local Laplacian, and \(ξₜ(x)\) 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 \(m_p(t,x_1,x_2,⋯,x_p) = u(t,x_1)u(t,x_2)⋯ u(t,x_p)\). In the first half of the paper, a weak form of intermittency is expressed through moment functions of order p≥ 3 and established for a wide class of operators 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 m₂. Now B is not necessarily positive-definite, but ∑ B(x)≥ 0.