# 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.

• How about analogus to the band theory of solids hyperphysics.phy-astr.gsu.edu/hbase/solids/band.html – anna v Jun 25 '14 at 14:41
• @Jiang-min Zhang perhaps is analogous to first order phase transitions where you have a discontinuous spectrum at the critical point because two degenerate minima aren't continuously connected one to the other as varying the external parameters that break the degeneracy. – TwoBs Jun 26 '14 at 5:26

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 $\kappa$.