Singularity barriers and Borel plane analytic properties of 1+ difference equations

Abstract

The paper addresses generalized Borel summability of ``$1^+$'' difference equations in ``critical time''. We show that the Borel transform $Y$ of a prototypical such equation is analytic and exponentially bounded for $\Re(p)<1$ but there is no analytic continuation from 0 toward $+\infty$: the vertical line $\ell:=\{p:\Re(p)=1\}$ is a singularity barrier of $Y$. There is a unique natural continuation through the barrier, based on the Borel equation dual to the difference equation, and the functions thus obtained are analytic and decaying on the other side of the barrier. In this sense, the Borel transforms are analytic and well behaved in $\CC\setminus\ell$. The continuation provided allows for generalized Borel summation of the formal solutions. It differs from standard ``pseudocontinuation'' \cite{Shapiro}. This stresses the importance of the notion of cohesivity, a comprehensive extension of analyticity introduced and thoroughly analyzed by Écalle. We also discuss how, in some cases, Écalle acceleration can provide a procedure of natural continuation beyond a singularity barrier.