I take a very brief look at the paper and I did not see $\Psi DO$ on manifold with boundary being used heavily anywhere (no conormal distribution, multiple blow-ups, heavy handed symbol estimates, etc). The gist of the paper is not the non-compact setting but dispersive PDE. 

I think [the following notes][1] I took may be helpful, but there are plenty of mistakes here and there. There are also notes by [others][2]. Personally, I suggest *against* going into the field as an undergraduate. I think the right strategy should be to isolate pieces of mathematics you cannot understand in that paper, and try to prove it on your own by learning the machinery from books/notes, or from references papers cited whenever possible. The "bottom up" approach simply takes too much time, and you may be aware that there are many approaches for dealing with non-compact manifolds. So learning one may not be enough. The goal of your paper reading should be focusing on the content (dispersive PDE), not the form (the myriad array of techniques or technical detail presented in it). This is the advice offered to me by Atiyah.   


  [1]: https://drive.google.com/file/d/1740PxL9_u6BW34gauNkIXEc32VZ8NpI-/view?usp=sharing
  [2]: https://arxiv.org/abs/math/0010314