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

TL;DR

This paper investigates the criticality of positive powers of the discrete Laplacian on the half line, establishing a threshold at alpha=3/2, and explores Hardy inequalities and eigenvalue emergence in subcritical regimes.

Contribution

It precisely characterizes when powers of the discrete Laplacian become critical and provides Hardy inequalities and eigenvalue analysis in subcritical regimes.

Findings

Powers of the discrete Laplacian are critical if and only if alpha ≥ 3/2.

Hardy type inequalities are established for alpha in (0, 3/2).

Negative eigenvalues emerge with small localized potentials in the critical case.

Abstract

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