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

We investigate the nonstationary parabolic Anderson problem ∂u∂t=ϰLu(t,x)+ξt(x)u(t,x),u(0,x)≡1,(t,x)∈[0,∞)×Zd where ϰL denotes a nonlocal Laplacian and ξt(x) is a correlated white-noise potential. The irregularity of the solution is linked to the upper spectrum of certain multiparticle Schrödinger operators that govern the moment functions mp(t,x1,x2,⋯,xp)=⟨u(t,x1)u(t,x2)⋯u(t,xp)⟩. First, we establish a weak form of intermittency under broad assumptions on L and on a positive-definite noise correlator B=B(x). We then examine strong intermittency, which emerges from the existence of a positive eigenvalue in a related lattice Schrödinger-type operator with potential B. Here, B does not have to be positive definite but must satisfy ∑B(x)≥0. The presence of such an eigenvalue intensifies the growth properties of the second moment m2, revealing a more pronounced intermittent regime.