Sturm-Liouville theory in multiple dimensions

[Telemachus]

Hi there. It is well known from the theory of differential equation the Sturm-Liouville theory, concerning with the eigenvalue problem:

$\displaystyle \frac{d}{dx} \left ( p(x) \frac{dy_n}{dx} \right ) - q(x) y_n(x)=\lambda_n w(x) y_n(x)$

with prescribed boundary condtitions for $a\leq x \leq b$, let's say $y_n(a)$ and $y_n(b)$.

$w(x)$ is the weight function. The eigenfunctions $y_n$ obeys the weighted orthonormality relation:

$\int_a^b w(x) y_n(x) y_m(x)dx=\delta_{n,m}$,

Also the solutions of the Sturm-Liouville problem provide a complete set of functions.

What I want to know is if all the theory regarding the Sturm-Liouville problem in one dimension holds in multidimensional problems, id est, for the eigenvalue problem:

$\displaystyle \nabla \cdot \left ( p(\mathbf{r}) \nabla \phi(\mathbf{r}) \right ) - q(\mathbf{r}) \phi(\mathbf{r})=\lambda w(\mathbf{r}) \phi(\mathbf{r})$,

Are the solutions to this problem orthogonal for prescribed boundary conditions? do they form a complete set?

In the affirmative case, I would also like to know if there are any references in the bibliography to the Sturm-Liouville problem in multiple dimensions (specially in 2D and 3D, which I think are the most important).

Thanks in advance.

 

[NFuller]

I don't believe there is a general solution for the two or three dimensional case. The general method for solving partial differential equations is to first try and separate the partial differential equation into ordinary differential equations. It is often the case that at least one of these ODE's will be of the Sturm-Liouville type. If this is the case then the solution can be written as a linear combination of orthogonal solutions satisfying both the PDE and boundary conditions.

 

[Telemachus]

Hi. Thank you very much for posting first of all.

What you mean with 'no general solution'? I am certain that the equation can be solved, perhaps not analytically, but for sure it can be done numerically. My question is if the problem posed for the multidimensional case holds the properties due to the theory of one dimensional Sturm-Liouville problems. In the one dimensional case one can get Legendre polynomials, Chebyshev Polynomials, or even I think the Spherical Harmonics, all are solutions to Sturm-Liouville problems. The first two are purely one dimensional, the spherical harmonics factors by separation of variables into one dimensional cases.

I would like to know if there is a theory of multidimensional Sturm-Liouville eigenvalue problems. I think that there can be some difficulties, for example, regarding the degeneracy of the eigenvalues, but I would like to know if there is some theory developed over this.

Thank you again.

 

[NFuller]

By general solution I mean one that can be written purely in terms of variables and known functions. You could solve this numerically but you would actually have to define the functions $p(\mathbf{r})$, $\phi(\mathbf{r})$, $q(\mathbf{r})$, etc, and your solution would be a set of numbers for those particular functions. I did just come across something where it looks like the solution can be cast as a Schwarzian http://ac.els-cdn.com/S002203961100...t=1499133819_cafe1e74ffaa2864c5cabc4f6ff59ecc. Maybe this is what your looking for?

 

[Telemachus]

Yes, that's it. The thing is I don't know what the Schwarzian is. It seems it's not pretty much a standard problem. I was hoping to find more bibliography on this (in an ideal case, a book treating something like the problem I posed). Thank you.

 

[Orodruin]

Are the solutions to this problem orthogonal for prescribed boundary conditions?

In general the solutions are not orthogonal. There may be several linearly independent eigenfunctions that share the same eigenvalue. However, eigenfunctions with different eigenvalues are still orthogonal. The proof is equivalent to that in one dimension.

That eigenfunctions have distinct eigenvalues for a 1D SL problem is only true for regular SL problems and the usual proof relies on studying the Wronskian.

The first two are purely one dimensional, the spherical harmonics factors by separation of variables into one dimensional cases.

Note that the spherical harmonics have degenerate eigenvalues. This is a result of the rotational symmetry of the sphere. Any rotation of an eigenfunction gives another eigenfunction with the same eigenvalue. In fact, the spherical harmonics of a fixed $\ell$ constitute an irrep of the rotation group $SO(3)$.

do they form a complete set?

In the case that the differential equation is separable, this is rather easy to show as long as you have the 1D SL theorem in place.

 

[Telemachus]

Thank you very much Orodruin. What if the equation isn't separable? for example, if I have some $p(\mathbf{r})$ and $q(\mathbf{r})$ that mixes the x and y coordinates, I think I wouldn't be able to separate the equation in 1D cases. In this situation the solution will provide a complete set?

 

[rajesh_d]

@Telemachus : Please see this : https://mathoverflow.net/q/277687/14414

 

[Telemachus]

Hello rajesh, thank you very much for posting. I can't really help you with that, I'm not a mathematician, and I've been only speculating about Sturm-Liouville theory in higher dimensions, I don't really know how much is actually known on this theory. Do you have any sources, bibliography or links on this topic?

Thanks!