Criticality transition for positive powers of the discrete Laplacian on the half line

TL;DR

The paper identifies a sharp threshold for the criticality of positive powers $(-\Delta)^\alpha$ of the discrete half-line Laplacian: $(-\Delta)^\alpha$ is critical if and only if $\alpha\ge 3/2$. It develops a Birman–Schwinger framework together with uniform Green-kernel bounds to prove subcriticality for $\alpha<3/2$ and criticality for $\alpha\ge 3/2$, and derives Hardy-type inequalities in the subcritical regime with explicit weights. It further constructs concrete Hardy weights $V_n$ with explicit decay rates and constants, and shows that the upper bound operator $4^\alpha-(-\Delta)^\alpha$ is always subcritical, highlighting a notable discrete-continuous discrepancy. As an illustrative case, the paper analyzes the bilaplacian perturbed by a localized potential, showing that a unique negative eigenvalue emerges and giving explicit formulas and asymptotics, which demonstrate spectral instability in the critical regime.

Abstract

We study the criticality and subcriticality of powers $(-Δ)^α$ with $α>0$ of the discrete Laplacian $-Δ$ acting on $\ell^2(\mathbb{N})$. We prove that these positive powers of the Laplacian are critical if and only if $α\ge 3/2$. We complement our analysis with Hardy type inequalities for $(-Δ)^α$ in the subcritical regimes $α\in (0,3/2)$. As an illustration of the critical case, we describe the negative eigenvalues emerging by coupling the discrete bilaplacian with an arbitrarily small localized potential.

Theorems & Definitions (35)

• Theorem 1.1
• Remark 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.5
• Proposition 2.1
• proof
• Corollary 2.2
• Remark 2.3
• Lemma 2.4