# Solutions with a finite support

1. Oct 12, 2015

### naima

I am interested in the solutions of the Klein Gordon equation.
Plane waves solutions are well known in physics. they look like $e^ { i (kx - \sqrt{k^2 + m^2} t)}$ or superpositions of them.
They are finite when t or x go to infinity.
I am looking for the general solution of the problem. In particular are there fast diminishing tempered solutions (in x and t) that could be useful with Schwartz distributions?
Are there solutions whith compact support?

Thanks.

2. Oct 14, 2015

### Geofleur

I think there are solutions for the 1D potential well problem that die off exponentially as $x \rightarrow \pm \infty$. You might also try solving the equation for a 1D particle in a box, using separation of variables just like in the nonrelativistic case. A helpful reference is Wachter's Relativistic Quantum Mechanics.

3. Oct 24, 2015

### naima

I know that i can solve differential equations in (x,t) by Fourier transforming them. I get a simpler equation in (k,E) i solve it and i apply the inverse transformation to get the result in (x,t). Is it valid when i have to solve: eqdif(x,t) = 0 AND |t| > 1 => f(x,t) = 0?
I couls so add the condition for supp(f).

Edit:
it seems that |t| > 1 => f(x,t) = 0 is equivalent to $(1 - rect (t)) f = 0$
As Fourier transforms rect in sinc it should give a second equation in E,p : TF(f) - sinc * TF(f) = 0
it uses the convolution of sinc and TF(f).
To be continued...

Last edited: Oct 24, 2015
4. Nov 2, 2015