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Lp boundedness, r-nuclearity and approximation of pseudo-differential operators on $\hbar\mathbb{Z}^n$

Juan Pablo Lopez

TL;DR

The paper develops symbol-order conditions that guarantee boundedness, compactness, and $r$-nuclearity for pseudo-differential operators on the lattice $\hbar\mathbb{Z}^n$, using an associated matrix and decay bounds on the symbol’s Fourier coefficients. It builds a matrix-analytic framework to translate symbol classes $S^{\mu}_{\rho,\delta}$ into concrete operator ideals on $\ell^p$ spaces, and derives $r$-nuclearity criteria with explicit eigenvalue-decay rates. The results are applied to discrete Schrödinger-type operators, showing the resolvent is $r$-nuclear and establishing growth rates for eigenvalues, with a first-order diagonal-approximation giving sharp perturbative insights. The framework extends to general elliptic operators in $\Psi^{\mu}_{\rho,\delta}$, providing broad spectral consequences for discrete quantum systems and lattice differential operators.

Abstract

In this work sufficient conditions on the order of the symbol are developed to ensure boundedness, compactness and r-nuclearity of pseudo-differential operators in $\hbar\mathbb{Z}^n$. In addition, these conditions allow us to obtain growth estimates for the eigenvalues of some elliptic operators, in particular perturbed discrete Schrödinger operator.

Lp boundedness, r-nuclearity and approximation of pseudo-differential operators on $\hbar\mathbb{Z}^n$

TL;DR

The paper develops symbol-order conditions that guarantee boundedness, compactness, and -nuclearity for pseudo-differential operators on the lattice , using an associated matrix and decay bounds on the symbol’s Fourier coefficients. It builds a matrix-analytic framework to translate symbol classes into concrete operator ideals on spaces, and derives -nuclearity criteria with explicit eigenvalue-decay rates. The results are applied to discrete Schrödinger-type operators, showing the resolvent is -nuclear and establishing growth rates for eigenvalues, with a first-order diagonal-approximation giving sharp perturbative insights. The framework extends to general elliptic operators in , providing broad spectral consequences for discrete quantum systems and lattice differential operators.

Abstract

In this work sufficient conditions on the order of the symbol are developed to ensure boundedness, compactness and r-nuclearity of pseudo-differential operators in . In addition, these conditions allow us to obtain growth estimates for the eigenvalues of some elliptic operators, in particular perturbed discrete Schrödinger operator.

Paper Structure

This paper contains 5 sections, 16 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Matrix analysis
  3. r-Nuclearity conditions
  4. First order approximation
  5. Discrete Schrödinger operators

Key Result

Proposition 1

Let $\sigma$ be a measurable function such that $|(\mathcal{F}_{\mathbb{T}} \sigma)(k,m)|\leq C(k)\omega(m)$. The following conditions are sufficient to guarantee that operator $T_\sigma:L^p(\mathbb{Z})\to L^p(\mathbb{Z})$ with $1\leq p<\infty$ belong to different ideals of operators as correspondin

Theorems & Definitions (27)

  • Proposition 1
  • Remark 1
  • Proposition 2
  • proof
  • Proposition 3
  • Theorem 4
  • proof
  • Remark 2
  • Remark 3
  • Lemma 1
  • ...and 17 more