[Paper Review] Asymptotics of the eigenvalues of the Anderson Hamiltonian with white noise potential in two dimensions

Khalil Chouk, Willem van Zuijlen|arXiv (Cornell University)|Jul 2, 2019

Spectral Theory in Mathematical PhysicsMathematics30 references20 citations

TL;DR

This paper establishes the almost sure asymptotic behavior of eigenvalues for the Anderson Hamiltonian with space-time white noise potential on a two-dimensional box [0, L]² under Dirichlet boundary conditions. Using paracontrolled distributions and large deviation principles, it proves that the n-th eigenvalue λₙ(L) divided by log L converges almost surely to a deterministic constant given by a variational formula involving the L² and H¹ norms of test functions. The key result is the universal convergence of eigenvalue scaling to a constant determined by Ladyzhenskaya's inequality.

ABSTRACT

In this paper we consider the Anderson Hamiltonian with white noise potential on the box $[0,L]^2$ with Dirichlet boundary conditions. We show that all the eigenvalues divided by $\log L$ converge as $L ightarrow \infty$ almost surely to the same deterministic constant, which is given by a variational formula.

Motivation & Objective

• To establish the almost sure asymptotic behavior of eigenvalues for the Anderson Hamiltonian with space-time white noise on a 2D box with Dirichlet boundary conditions.
• To extend the construction of the Anderson Hamiltonian from periodic to Dirichlet boundary conditions using paracontrolled distributions.
• To prove that the n-th eigenvalue λₙ(L) divided by log L converges almost surely to a deterministic constant determined by a variational formula.
• To connect the eigenvalue asymptotics to the parabolic Anderson model and the total mass of the solution.

Proposed method

• The authors use paracontrolled distributions to define the Anderson Hamiltonian H = Δ + ξ on H₀¹([0, L]²) with Dirichlet boundary conditions.
• They construct Dirichlet and Neumann Besov spaces Bᵈ,αₚ,ₚ and Bⁿ,αₚ,ₚ to handle the singular white noise potential and define para- and resonance products between them.
• A large deviation principle is derived for the enhanced white noise, which is used to control the tail behavior of eigenvalues.
• The eigenvalue convergence is proven via scaling and translation invariance, comparison of eigenvalues on different-sized boxes, and infima over the large deviation rate function.
• The proof relies on Bony-type estimates for para- and resonance products in the Besov spaces, extended to mixed Dirichlet-Neumann settings.
• The key technical step involves bounding the covariance of Gaussian processes related to the white noise approximation, using trigonometric identities and integral estimates.

Experimental results

Research questions

• RQ1 What is the almost sure asymptotic behavior of the eigenvalues of the Anderson Hamiltonian with space-time white noise on a 2D box with Dirichlet boundary conditions as L → ∞?
• RQ2 How can the Anderson Hamiltonian be rigorously defined on a bounded domain with Dirichlet conditions when the potential is white noise, which is too irregular for standard operator theory?
• RQ3 What is the limiting constant to which λₙ(L)/log L converges almost surely, and how is it characterized?
• RQ4 How does the eigenvalue asymptotics relate to the total mass of the solution to the parabolic Anderson equation?

Key findings

• The n-th eigenvalue λₙ(L) divided by log L converges almost surely to a deterministic constant χ as L → ∞, where χ is given by a variational formula involving the L² and H¹ norms of test functions.
• The limiting constant χ is the smallest C > 0 such that ‖f‖₄⁴ ≤ C‖∇f‖₂²‖f‖₂² for all f ∈ H¹(R²), which is Ladyzhenskaya’s inequality.
• The convergence holds for all n ∈ ℕ and is uniform over compact sets of eigenvalues.
• The eigenvalue asymptotics are derived via a large deviation principle for the enhanced white noise, which controls the tail behavior of the eigenvalues.
• The proof relies on the construction of Dirichlet and Neumann Besov spaces and the extension of Bony estimates to mixed Dirichlet-Neumann products.
• The key technical estimate bounds the covariance of Gaussian processes related to the white noise approximation, showing that E[|Xεₖ,ᵣYεₗ,ᵣ|] ≲ (1 + |k − r/ε|)¹⁻ᵟ + (1 + |l − r/ε|)¹⁻ᵟ for δ > 0.