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

Dan Han (), Stanislav Molchanov and Boris Vainberg
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Dan Han: Department of Mathematics, University of Louisville, Louisville, KY 40292, USA
Stanislav Molchanov: Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC 28223, USA
Boris Vainberg: Department of Mathematics and Statistics, University of North Carolina at Charlotte, Charlotte, NC 28223, USA

Mathematics, 2025, vol. 13, issue 5, 1-21

Abstract: We investigate the nonstationary parabolic Anderson problem ∂ u ∂ t = ϰ L u ( t , x ) + ξ t ( x ) u ( t , x ) , u ( 0 , x ) ≡ 1 , ( t , x ) ∈ [ 0 , ∞ ) × Z d where ϰ L denotes a nonlocal Laplacian and ξ t ( 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 m p ( t , x 1 , x 2 , ⋯ , x p ) = ⟨ u ( t , x 1 ) u ( t , x 2 ) ⋯ u ( t , x p ) ⟩ . 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 m 2 , revealing a more pronounced intermittent regime.

Keywords: spectral analysis; nonstationary; Anderson model; intermittency (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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