I was reading in Wikipedia about how the Dirac Equation and the Klein-Gordon Equation where built to introduce in the Schrödinger equation the relativistic description of the Energy–momentum relation: $$E = m^2c^4+m^2p^2$$
Where:
- Klein-Gordon works directly with the squared version: $\left(\frac{1}{c^2}\frac{\partial^2}{\partial t^2} -\nabla^2 -\frac{m^2c^2}{\hbar^2}\right)\psi(t,\vec{x})=0$
- And Dirac equation made a clever matrix decomposition, as stated in Wikipedia, to avoid using $E=c\sqrt{p^2+m^2c^2}$ and "expand the square root in an infinite series of derivative operators, set up an eigenvalue problem, then solve the equation formally by iterations." Finally this lead him find his equation $(i{\partial\!\!\!\big /} - m) \psi = 0$
But recently, I learned through the answers to this question that a differential equation $\ddot{x}=F(\dot{x},x)$ could have solutions that vanishes at a finite ending time only if the equation $F(\cdot,\cdot)$ have a point where is Non-Lipschitz.
As example, the equation $\dot{x}=-\text{sgn}(x)\sqrt{|x|},\,x(0)=1$ could admit the solution $x(t)=\frac{1}{4}\left(1-\frac{t}{2}+\left|1-\frac{t}{2}\right|\right)^2$.
And since I believe that both Klein-Gordon and Dirac Equations are locally Lipschitz so their solutions at best vanishes at infinity, I am wonder if, similar to the example equation where the square root introduce a non-Lipschitz point, If there exist solutions with finite extinction times to the problem of pairing the Schrödinger equation directly with the term $E=c\sqrt{p^2+m^2c^2}$ (without using expansions), that are not considered by Kleing-Gordon or Dirac models.
As example of a modified version of the Schrödinger equation where a Non-Lipschitz component is introduced to achieve solutions with finite extinction times, I found the following paper:
