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Spectral and Dynamical Analysis of Fractional Discrete Laplacians on the Half-Lattice

Nassim Athmouni

TL;DR

The paper develops a comprehensive spectral and dynamical analysis for anisotropic discrete fractional Laplacians on the half-lattice ${\mathbb{N}}^d$. It proves that the half-space operator is a boundary-restricted version of the full-lattice operator plus a compact boundary correction, so the essential spectrum and interior threshold structure are preserved; it then establishes localized Mourre estimates and a Limiting Absorption Principle (LAP) on interior windows, yielding absence of singular continuous spectrum, finiteness of eigenvalues, and weighted propagation bounds. The framework handles potentially unbounded fractional powers (negative exponents) via form-commutator methods, and extends to scattering theory, including stationary representations, T-operators, wave matrices, Birman–Krein type identities, and ballistic transport in time averages. Fiber decompositions along tangential directions provide uniform LAP and well-behaved scattering matrices on full-measure sets of parameters, with continuity in the fractional exponents. These results demonstrate that interior-energy spectral and dynamical properties established for the full lattice survive on the half-lattice under precise boundary corrections, enabling robust analysis of nonlocal, anisotropic discrete dynamics and lattice scattering.

Abstract

We investigate discrete fractional Laplacians defined on the half-lattice in several dimensions, allowing possibly different fractional orders along each coordinate direction. By expressing the half-lattice operator as a boundary restriction of the full-lattice one plus a bounded correction that is relatively compact with respect to it, we show that both operators share the same essential spectrum and the same interior threshold structure. For perturbations by a decaying potential, the conjugate-operator method provides a strict Mourre estimate on any compact energy window inside the continuous spectrum, excluding threshold points. As a consequence, a localized Limiting Absorption Principle holds, ensuring the absence of singular continuous spectrum, the finiteness of eigenvalues, and weighted propagation (transport) bounds. The form-theoretic construction also extends naturally to negative fractional orders. Overall, the relative compactness of the boundary correction guarantees that the interior-energy spectral and dynamical results obtained on the full lattice remain valid on the half-lattice without modification.

Spectral and Dynamical Analysis of Fractional Discrete Laplacians on the Half-Lattice

TL;DR

The paper develops a comprehensive spectral and dynamical analysis for anisotropic discrete fractional Laplacians on the half-lattice . It proves that the half-space operator is a boundary-restricted version of the full-lattice operator plus a compact boundary correction, so the essential spectrum and interior threshold structure are preserved; it then establishes localized Mourre estimates and a Limiting Absorption Principle (LAP) on interior windows, yielding absence of singular continuous spectrum, finiteness of eigenvalues, and weighted propagation bounds. The framework handles potentially unbounded fractional powers (negative exponents) via form-commutator methods, and extends to scattering theory, including stationary representations, T-operators, wave matrices, Birman–Krein type identities, and ballistic transport in time averages. Fiber decompositions along tangential directions provide uniform LAP and well-behaved scattering matrices on full-measure sets of parameters, with continuity in the fractional exponents. These results demonstrate that interior-energy spectral and dynamical properties established for the full lattice survive on the half-lattice under precise boundary corrections, enabling robust analysis of nonlocal, anisotropic discrete dynamics and lattice scattering.

Abstract

We investigate discrete fractional Laplacians defined on the half-lattice in several dimensions, allowing possibly different fractional orders along each coordinate direction. By expressing the half-lattice operator as a boundary restriction of the full-lattice one plus a bounded correction that is relatively compact with respect to it, we show that both operators share the same essential spectrum and the same interior threshold structure. For perturbations by a decaying potential, the conjugate-operator method provides a strict Mourre estimate on any compact energy window inside the continuous spectrum, excluding threshold points. As a consequence, a localized Limiting Absorption Principle holds, ensuring the absence of singular continuous spectrum, the finiteness of eigenvalues, and weighted propagation (transport) bounds. The form-theoretic construction also extends naturally to negative fractional orders. Overall, the relative compactness of the boundary correction guarantees that the interior-energy spectral and dynamical results obtained on the full lattice remain valid on the half-lattice without modification.

Paper Structure

This paper contains 34 sections, 32 theorems, 150 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries and framework on the half-lattice $\mathbb{N}^d$
  3. Graph structure
  4. Topology and the boundary of ${\mathbb{N}}^d$
  5. Restriction, extension, and projection
  6. Shifts, position, and discrete differences
  7. Discrete fractional Laplacians from $\mathbb{Z}^d$ to $\mathbb{N}^d$
  8. Extension of the Lattice Laplacian to the hypercubic lattice quadrant
  9. Fractional Laplacians on the Discrete Half-line
  10. Heat kernels and method of images on $\mathbb{N}^d$
  11. Kernel on $\mathbb{Z}^d$.
  12. Method of images on $\mathbb{N}$.
  13. Operator--kernel dictionary.
  14. Inclusion--exclusion on $\mathbb{N}^d$.
  15. Spectral analysis and Mourre estimate
  16. ...and 19 more sections

Key Result

Theorem 1.1

Let $H=\Delta_{\mathbb{N}^d}^{\vec{\mathrm r}}+W(Q)$ with real $W$ satisfies the anisotropic decay assumptions $H0$--$H1$. Then $H\in\mathcal{C}^{1,1}_{\mathrm{loc}}(A_{\mathbb{N}^d})$ on each $\mathcal{I}\Subset (0,\lambda_{\vec{\mathrm{r}}})\setminus \operatorname{Thr}(\Delta_{\mathbb{Z}^d}^{\vec{

Theorems & Definitions (76)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Proposition 2.5: Boundary correction: support and compactness
  • proof
  • Lemma 3.1
  • ...and 66 more