Generally, we have the picture that as the parameter (say, the depth of a trap) of a system varies, the bound state gets more and more extended and disappears eventually at some critical parameter value. The point is that, the whole process is a continuous one.
This is the case for the defect mode induced by a site-defect in the 1d tight binding model.
However, Mattis discovered an exotic bound state which disappears discontinuously:
Personally, we had a similar finding:
The point is that, even at the critical parameter value, the bound state is still of a FINITE size, and then, as the parameter changes by an infinitesimal value, it is gone.
What is the mathematical reason behind some a discontinuous behavior? It has baffled me for a long time.
I would like to draw an analogy of this behavior with the first-order phase transition, while the more common continuous behavior with the second-order phase transition.