- #1
moso
- 14
- 0
Hey Guys, Trying to figure out how to replicate the following from an article, but can not understand their notations;
The main points are:
The bounce action can be written as the equation
$$\left( n^2 \Theta^2 + 2\alpha n\Theta -1\right)R_n = 2 \sum_{m=1}^\infty R_{n+m}R_m + \sum_{m=1}^n R_{n-m}R_m,$$
where we chose a value of $\Theta$ between 0 and 1 and $\alpha$ between 0 and 10. The paper states that it does successive iterations starting with a zero-order approximation of $R_n \propto \exp(-n)$. It is section VII in this paper (https://journals.aps.org/prb/abstract/10.1103/PhysRevB.36.1931). page 35-36.
The problem is then to calculate the coefficients $R_n$, which is stated to decay exponentially fast with increasing n. My problem with this is I do not see how the iteration occur, as you normally have something on the form $x_i=f(x_{i-1})$ and you can then iteration the equation, but here you have a sum to infinity.
But in this example, I can not seem to identify the correct procedure to find the coefficient R_n in the equation using successive iteration. An example from the article is that for $\alpha=0.1$ and $\Theta=0.3$, the sum of the coefficients $R_n$ should be 8.44.
It anyone can see the trick or procedure to implement this numerically I would very much appreciate it.
The main points are:
The bounce action can be written as the equation
$$\left( n^2 \Theta^2 + 2\alpha n\Theta -1\right)R_n = 2 \sum_{m=1}^\infty R_{n+m}R_m + \sum_{m=1}^n R_{n-m}R_m,$$
where we chose a value of $\Theta$ between 0 and 1 and $\alpha$ between 0 and 10. The paper states that it does successive iterations starting with a zero-order approximation of $R_n \propto \exp(-n)$. It is section VII in this paper (https://journals.aps.org/prb/abstract/10.1103/PhysRevB.36.1931). page 35-36.
The problem is then to calculate the coefficients $R_n$, which is stated to decay exponentially fast with increasing n. My problem with this is I do not see how the iteration occur, as you normally have something on the form $x_i=f(x_{i-1})$ and you can then iteration the equation, but here you have a sum to infinity.
But in this example, I can not seem to identify the correct procedure to find the coefficient R_n in the equation using successive iteration. An example from the article is that for $\alpha=0.1$ and $\Theta=0.3$, the sum of the coefficients $R_n$ should be 8.44.
It anyone can see the trick or procedure to implement this numerically I would very much appreciate it.