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

Author

Listed:
  • 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)

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.

Suggested Citation

  • Dan Han & Stanislav Molchanov & Boris Vainberg, 2025. "Spectral Analysis of Lattice Schrödinger-Type Operators Associated with the Nonstationary Anderson Model and Intermittency," Mathematics, MDPI, vol. 13(5), pages 1-21, February.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:5:p:685-:d:1595509
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    References listed on IDEAS

    as
    1. Archer, Eleanor & Pein, Anne, 2023. "Parabolic Anderson model on critical Galton–Watson trees in a Pareto environment," Stochastic Processes and their Applications, Elsevier, vol. 159(C), pages 34-100.
    2. Avena, Luca & Gün, Onur & Hesse, Marion, 2020. "The parabolic Anderson model on the hypercube," Stochastic Processes and their Applications, Elsevier, vol. 130(6), pages 3369-3393.
    3. Raluca M. Balan & Le Chen, 2018. "Parabolic Anderson Model with Space-Time Homogeneous Gaussian Noise and Rough Initial Condition," Journal of Theoretical Probability, Springer, vol. 31(4), pages 2216-2265, December.
    4. Comets, Francis & Cranston, Michael, 2013. "Overlaps and pathwise localization in the Anderson polymer model," Stochastic Processes and their Applications, Elsevier, vol. 123(6), pages 2446-2471.
    Full references (including those not matched with items on IDEAS)

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