When I am studying some paper dealing with dispersive PDE(e.g. Wave, Schrödinger and Klein-Gordon equations), the potential $\frac {1}{|x|^2}$ which is called critical decay (or inverse square potential) is lots of handled.


They also consider Morrey-Campanato or Fefferman-Phong class to deal with that potential since it is contained those class.


The questions are here. Why is it called 'critical' and important or meaningful?



Why is critical decay important?
"Critical" in the context of the Schrödinger equation (which seems to be what the OP is after) refers to the number of bound states confined by the potential. The Schrödinger equation with potential $-C|x|^{-2+\epsilon}$ and $C>0$ has a discrete spectrum for $E<0$, with finitely many eigenvalues
if $\epsilon < 0$ and infinitely many if $\epsilon > 0$. Hence inverse square decay is the critical threshold for
the existence of infinitely many bound states. For $\epsilon=0$ whether there are finitely or infinitely many discrete eigenvalues depends on the prefactor $C$.


This basically follows from [Kneser's theorem](https://en.wikipedia.org/wiki/Kneser%27s_theorem_(differential_equations)), which also gives $C=1/4$ as the critical prefactor.


