[![enter image description here](https://i.stack.imgur.com/bRexg.png)](https://i.stack.imgur.com/bRexg.png)[![enter image description here](https://i.stack.imgur.com/uGj4w.png)](https://i.stack.imgur.com/uGj4w.png)So I am trying to recreate the numerical treatment in this papir (<https://journals.aps.org/prb/abstract/10.1103/PhysRevB.29.130>), equation 3.7. The main point is to try to solve the following equation

$$z(\omega)=\frac{1}{1+2\*\alpha|\omega|+\omega^2}\*\lambda\*\int\_{-\infty}^{\infty} d\omega' z(\omega-\omega')z(\omega'),$$

where $\lambda$ is a variable we define through and $\alpha$ is a constant we choose. There is a trivial solution to this which is $z(\omega)=0$, which I want to avoid. We know that $z(\pm\infty)=0$ and that it is symmetric around the y-axis. The procedure in the article is an iteration procedure 

1. Start with an initial $z\_0(\omega)$ and $\lambda\_0=\frac{3}{2\sqrt{2}\Pi}$
2. Calculate $z\_1(\omega)=O(\lambda\_0,z\_0(\omega))$, where O is the operator defined by the right hand side of the equation
3. calculate $\lambda\_1 = \lambda\_0\zeta^{-2}$, where $\zeta = \frac{z\_1(\omega=0)}{z\_0(\omega=0)}$
4. repeat, thereafter find $z\_2(\omega)=O(\lambda\_1,z\_1(\omega))$ and so on. untill we have a converged function.

I am trying to perform this for $\alpha=1$, which should give us an shape $\propto exp(-\omega)$. My initial code is below for one iteration:

```
        approxsoln [x_] = 1;
        \[Lambda] = 3/(2*Sqrt[2]*Pi);
         values = Table[{x, 1/(1 + 2*1*x + x^2)*\[Lambda]
          NIntegrate[approxsoln[y] approxsoln[x - y],
             {y, -Infinity, Infinity}]}, {x,0, 2, 1/100}];
        approxsoln1[x_] = InterpolatingPolynomial[values, x];
        \[Lambda]1 = \[Lambda]*(approxsoln1[0]/approxsoln[0])^(-2)

```

And then I will just repeat my code. My problem is to deal with the $|\omega|$ and automating it. 

Hope someone can help 

Edit:

I have uploaded a picture of the solution they got, which is normalised