Why do some bound states disappear in a discontinuous way? 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:
http://journals.aps.org/prl/cited-by/10.1103/PhysRevLett.52.755
Personally, we had a similar finding:
http://journals.aps.org/pra/abstract/10.1103/PhysRevA.87.023613
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.  
 A: It's important to be more precise when you mean 'bound state of finite size'.
Recall that bound state solutions to the Schrödinger equation behave at large distances as $\sim\exp(-\kappa x)$, where $\kappa$ is dependent on potential strength.  Typically as potential strength weakens, $\kappa$ becomes smaller.
Conventionally it is said that $\kappa$ is the scale that sets the size of the bound state.  With this precise meaning of bound state size, no unexpected behavior occurs at threshold potential strength: at this point $\kappa$ is exactly zero, and an infinitesimally weaker potential makes $\kappa$ become purely imaginary (turning it into an oscillatory scattering state).
The confusion of bound state size often arises because when $\kappa$ vanishes, a power-law behavior at large distances is exposed e.g. $\sim x^{- \alpha}$ with alpha being a dimensionless positive number depending on number of dimensions of the system.  While such a state appears to have finite size, it must be kept in mind that the wavefunction exhibits a very different functional form on account of vanishing $\kappa$.
A: I can not quickly follow the detailed reasoning in your paper.  But, usually, it is possible to use perturbation theory to calculate the derivatives of many quantities in quantum mechanics.  The first derivative of an the energy of a bound state with respect to a change in the parameters of quantum mechanics is, in fact, very, very simple, just the expectation value of the change in the hamiltonian in that state.  So, can you tell me?  Does this expectation value somehow diverge for your states?  It would seem to have to, if what you say is correct.  This, in turn might help to understand your result.
