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: ∂ u/∂ t = ϰℒu(t,x) + ξ_t(x)u(t,x) with the initial condition u(0,x) ≡ 1, where (t,x) ∈ [0,∞)×ℤ^d. Here, ϰℒ denotes a non-local Laplacian, and ξ_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,⋯,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 ϰℒ 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 ∑ B(x)≥ 0.