Summary: A fundamental solution supported in a strip \([-T,T]_t\times\mathbb{R}_x\) is constructed for a general dispersive operator in one space dimension. As a direct application we obtain the behavior as \(x\to\infty\) for the trajectories arising in the (exact) boundary controllability problem in the half-line both for the Korteweg-de Vries equation and for the Schrödinger equation.