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

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 $\alpha \ge 3/2$, we analyze asymptotic properties of discrete eigenvalues emerging by coupling $(-\Delta {)}^{\alpha }$ with a localized potential.