hi...let me briefly state the problem I am particularly interested in: The PDE: Angular laplcian f(theta,phi) +(- a*cos²(theta)-b*cos(theta))*f(theta,phi) = lambda*f(theta,phi)

Hello, unfortunately I don't have too much time today, but some.

Hi, alright. no problem.

How did you get the equation at the top?

It is a Schrödinger equation.

What I also know is that for the same potential there is a famous ODE: -f''(x) +(-a*cos²(x)-b*cos(x))f(x)=lambda f(x)

lboro.ac.uk/microsites/maths/research/preprints/papers10/…

called the Whittaker Hill equation

I was wondering now: if we can solve the ODE, can we also solve the PDE?

I assume you want periodic conditions on $f$ in $x$.

you are talking about the ODE?

Yes.

Are you working on an infinite interval?

no, [0,2*pi]

well, the answer is that you can solve this equation for some eigenvalues if you take a=n*t and b=t²; if n is even you take antiperiodic BCs and otherwise odd ones.

Okay. So this is a very regular, periodic, Sturm-Liouville problem.

That means a complete set of eigenfunctions.

yes, but it is known that only some eigenfunctions have a finite representation and these are the solutions I am interested in. Now, since I get some solutions for this equation I was wondering whether we can also say something about the PDE?

do you see any relationship between the ODE and the PDE?

Did you get this from separation of variables or no?

no

if I do separation of variables with f(theta,phi) = y(theta)/sqrt(sin(theta))*e^(i*n*phi) I get: -y''(theta)+(- a*cos(theta)-b*cos²(theta)+(n²-1/4)/sin^2(x))y(theta)=lambda y(theta)

The PDF would take a while to scan. You can certainly expand in a basis of such eigenfunctions. Can you get an equation for the coefficients by doing this?

which basis do you mean?

A basis of the eigenfunctions for the periodic ODE. You can let the coefficients be a function of the other variable not associated with the ODE.

I don't have a full basis of eigenfunctions, I only have some eigenfunctions.

the thing is: the ODE -y''(theta)+(- a*cos(theta)-b*cos²(theta)+(n²-1/4)/sin^2(x))y(theta)=lambda y(theta) is pretty similar to the ODE -f''(x) +(-a*cos²(x)-b*cos(x))f(x)=lambda f(x) that I can solve for a few eigenfunctions/eigenvalues. My question is: Is this of any help?

You could still expand in a basis of eigenfunctions $\sum_{n}C_{n}(\theta)f_{n}(x)$ but the equations could be fairly ugly for $C_{n}$.

Just assume that you know all of the eigenfunctions and look at the equations.

The equations for the coefficients.

well, I don't know ALL the eigenfunctions to the regular ODE, just some of them.

more precisely I said that I can solve it for a=n*t and b=t²

That may help you with a few of the coefficients. But you could assume you know all of them, assume certain eignevalues $\lambda_{n}$ and plug it in.

if I have n=1 I get 1 eigenfunctions, if I have n=2 I get 2 eigenfunctions, etc.

True eigenfunctions where the functions are periodic/

?

for n=1 I have the eigenfunction e^(t*cos(x)), it is an actual eigenfunction, to the eigenvalue -t²

By the way, have you tried the SLEIGN2 package? I've wanted to get some time to try.

I was not sure what you meant there...

*about

does it give analytic solutions?

SLEIGN2 is a computational code to find eigenvalues and eigenvectors, and it deals with singular and regular Sturm-Liouville systems using fairly advanced features. Teschl has published a lot in Sturm-Liouville theory as well as Evereitt (sp?) and they put a lot of this together.

It gives computational representations.

I have a feeling its very good. The FORTRAN source code is provided by them with instructions on compiling and using it.

They even have methods for approximating situations of continuous spectrum.

well, I currently don't have access to a UNIX computer

I'm sure it would work under Linux.

I have windows^^

You can get Oracle VBOX an run Linux in a VM. That's what I do for other reasons. Just make sure to install all Linux packages--that will include GNU Fortran.

They can even share a file system.

All of that is free.

CentOS is free version of RedHat Linux. SuSE is another option. Ubuntu still another.

alright, so I will try...thank you....see you then...

By the way, Teschl has a good book on ODES and another on Mathematical Methods for Quantum Mechanics.

He's a primary contributor to SLEIGN2

I just wanted to get that in.

yes, I found these two books as he suggested them to me on stackexchange, unfortunately, he is seldom online...

I've been learning many near things from his book. This is just information. Back to the discussion at hand.

Do you have some additional ideas about my problem?

Even if $sin$ and $cos$ are not the right basis functions, you can still expand $f(x,y)$ as $f(x,y)=\sum_{n}C_{n}(x)e^{iny}$, just as an example.

Then you can plug such a representation into an equation and you'll get an ugly mess, but sometimes you get lucky and the equations have a resonable form for at least getting an approximate $C_{n}$.

This is the hope if the $e^{inx}$ are somehow related to the equation in $x$.

That's what I am suggesting here, but with the eigenfunctions of the modified periodic condition.

equation (not condition)

The eigenfunctions you found when plugged into the original equation will give an eigenvalue times the eigenfunction but with a left-over term.

alright, maybe i try the things you suggest first and then come back to you, especially since you do not have much time today, so I don't want to busy you too much ;-)

Maybe I am not understanding your problem fully either. Don't be afraid to correct me. I'm learning, too.

Later.

Thank you...wish you a nice day/evening and would love to see you again in this chat...:-)