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

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