We show that the spectral gap of a random walk on the domain of normal attraction of an $α$-stable law is of order $\mathcal O(n^α)$ when restricted to boxes of size $n$. The proof is based on a comparison principle that may be of independent interest. The comparison principle also allows to derive a sharp bound on the spectral gap of exclusion and zero-range processes with long jumps when restricted to finite boxes in terms of the gap on the complete graph.