Interior Spectral Windows and Transport for Discrete Fractional Laplacians on d-Dimensional Hypercubic Lattices

2025/09/07 by Nassim Athmouni, Athmouni, Nassim
Mathematics · #35P25 #39A70 #47A10 #47A40 #47B25 #81Q10 #F.1.1 #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #G.2.2 #Graph theory and applications #Holomorphic and Operator Theory #Mathematical Physics (math-ph) #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2509.06117

openalex publication_date 2025/09/07 · openalex created_date 2025/10/12 · openalex updated_date 2026/07/28

Abstract

We study anisotropic fractional discrete Laplacians Δ_ℤ^d^r with exponents r∈ℝ^d∖\0\ on ℓ²(ℤ^d). We establish a Mourre estimate on compact energy intervals away from thresholds. As consequences we derive a Limiting Absorption Principle in weighted spaces, propagation estimates (minimal velocity and local decay), and the existence and completeness of local wave operators for perturbations H=Δ_ℤ^d^r+W(Q), where W is an anisotropically decaying potential of long--range type. In the stationary scattering framework we construct the on--shell scattering matrix S(λ), prove the optical theorem, and, under a standard trace--class assumption on W, establish the Birman--Krein formula det S(λ)=exp(-2πi ξ(λ)).

Citations

• Spectral analysis of The magnetic Laplacian acting on discrete funnels
• Limiting absorption principle for long-range perturbation in a graphene setting
• Limiting absorption principle for long-range perturbation in the discrete triangular lattice setting
• Spectral analysis of the Laplacian acting on discrete cusps and funnels
• A Schatten -von Neumann class criterion for the magnetic Weyl calculus
• The magnetic Laplacian acting on discrete cusps
• On the a.c. spectrum of 1D discrete Dirac operator
• A new look at Mourre's commutator theory

Related

• Spectral and Dynamical Analysis of Fractional Discrete Laplacians on the Half-Lattice
• Discrete Laplacian in a half-space with a periodic surface potential I: Resolvent expansions, scattering matrix, and wave operators
• Limiting absorption principle for long-range perturbation in the discrete triangular lattice setting
• Spectral and scattering properties at thresholds for the Laplacian in a half-space with a periodic boundary condition
• The fractional Laplacian with reflections