Lossless propagation bounds for long-range fermionic interactions

We prove propagation estimates for extensive fermionic many-body interactions with polynomial time growth, no loss of spatial decay, and an optimal threshold for the polynomial spatial exponent. More precisely, we construct local representatives of evolved fermionic many-body interactions on metric spaces with polynomial volume growth of order $d_{\mathrm{vol}}$ and suitable cell partitions. For the even Hamiltonian interaction $Φ_u$ and the even interaction $Ψ$ being evolved, we assume bounds, uniform in the site, on sums of term norms over finite supports $X$ containing a given site, with weights $e^{κ|X|}(1+\operatorname{diam}X)^a$ and $e^{κ|X|}(1+\operatorname{diam}X)^p$, respectively. Here $a>d_{\mathrm{vol}}$, $0\le p\le a$, $κ$ is sufficiently large, and the Hamiltonian bound is uniform in time. We construct local representatives whose interaction norms with weight $e^{c(1+|t-s|)^{-b}|X|}(1+\operatorname{diam}X)^p$ grow at most polynomially in $|t-s|$, for suitable $c,b>0$. In particular, $p=a$ is allowed: the Hamiltonian need not decay faster than the interaction being evolved. The threshold $a>d_{\mathrm{vol}}$ is optimal in general: uniform polynomial time bounds at $p=a$ fail for $0