2025/07/07 by David Damanik, Damanik, David, Zhiyan Zhao +1
Mathematics · #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Numerical methods in inverse problems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2507.04988
openalex publication_date 2025/07/07 · openalex created_date 2025/10/20 · openalex updated_date 2026/08/01
We consider the discrete Schrödinger operator H = -Δ+ V on ℓ2(ℤd) with a decaying potential, in arbitrary lattice dimension d∈ℕ^*, where Δ is the standard discrete Laplacian and Vn = o(|n|-1) as |n| → ∞. %We prove the absence of singular continuous spectrum for H. For the unitary evolution e-i tH, we prove that it exhibits ballistic transport in the sense that, for any r > 0, the weighted ℓ2-norm ‖e-i tHu‖r:=(∑n∈ℤd (1+|n|2)r |(e-i tHu)n|2)^\frac12 grows at rate ≃ tr as t→ ∞, provided that the initial state u is in the absolutely continuous subspace and satisfies ‖u‖r<∞. The proof relies on commutator methods and Mourre estimate, which yields quantitative lower bounds on transport for operators with purely absolutely continuous spectrum over appropriate spectral intervals.