Floer trajectories with immersed nodes and scale-dependent gluing

We define an enhanced compactification of Floer trajectories under Morse background using the adiabatic degeneration and the scale-dependent gluing techniques. The compactification reflects the 1-jet datum of the smooth Floer trajectories nearby the limiting nodal Floer trajectories arising from adiabatic degeneration of the background Morse function. This paper studies the gluing problem when the limiting gradient trajectories has length zero through a renomalization process. The case with limiting gradient trajectories of non-zero length will be treated elsewhere. An immediate application of our result is a proof of the isomorphism property of the PSS map : A proof of this isomorphism property was first outlined by P\"unihikin-Salamon-Schwarz \cite{PSS} in a way somewhat different from the current proof in its details. This kind of scale-dependent gluing techniques was initiated in [FOOO07] in relation to the metamorphosis of holomorphic polygons under Lagrangian surgery and is expected to appear in other gluing and compactification problem of pseudo-holomorphic curves that involves `adiabatic' parameters or rescales the targets.