Perturbation Theory on Finite Domains

[bolbteppa]

In this video (from 27.00 - 50.00, which you don't need to watch!) a guy shows how you can solve the general second order ode $y'' + P(x)y = 0$ using perturbation theory. However he points out that the domain must be finite in order for this to work, I'm wondering how you would phrase a question like this or how you would know when you are working with an infinite domain etc...? If I was just given $y'' + x^2y = 0$ & asked to solve it using perturbation theory I wouldn't know if I was getting into trouble about finite or infinite domains, how do I re-phrase the question so that it makes sense? Are there any other things that cross your mind that one should be careful about? Thanks for your time.

 

[hilbert2]

I suppose you have to rewrite the equation as y''+λP(x)y=0 (λ will be set to have value 1 at the end of calculation), do the substitution y(x) = y₀(x)+λy₁(x)+λ²y₂(x)+... , and then find how to calculate y_n+1 when y_n is known (equating like powers of λ)... Is that what you mean by a perturbation calculation?

Often perturbation expansions in quantum mechanics don't converge. I'm not sure how the domain affects this.

 

[bolbteppa]

Yeah that's what he's talking about. He goes on to spend a lot of the rest of the course dealing with divergent series & relating it to quantum mechanics because you have to deal with infinite dimensional domains due to normalization, but I'm just curious as to solving general second order ode's on finite domains & when you can use this, because he proves that convergence is assured on any finite domain using this method (I just want to be able to identify those problems where I'll be able to use this as a tool).

 

[Office_Shredder]

He's saying that if you are trying to solve the problem for
$$x\in (1, \infty)$$
or some other domain of infinite length then you might run into trouble in theory with convergence. But if your domain is finite, like you want to solve it on (-4,12) then you're fine. The problem is a technical one about the rate of convergence of the infinite series - if you don't know what your domain is, the solution never required knowing it, so you can certainly still plug in values of x and be happy. But if you try to do some operation that required it existing on the whole domain (like integrating over all of R) you might run into theoretical trouble.

 

[bolbteppa]

Thanks a lot, I'll do a bit of work over the next while & hopefully come back with more interesting questions.

 

[WannabeNewton]

Wow, that man is a brilliant lecturer.

 

[bolbteppa]

I know, he's left-handed too I got his book I was so enthralled by him, & it's got the same flavour to it. Just as with my lie groups thread I'll throw out the offer to go through these videos & post summaries, thoughts, ideas etc... if you're up for it, though no pressure. I'd love to compare his approach to divergent series with that in Bromwich for example, & have people there to discuss the stuff with, so whatever anyone thinks, no pressure or time constraints as it's summer.