# Thread: Multidimensional eigenfunctions and eigenvalues

1. ## Multidimensional eigenfunctions and eigenvalues

Hi there. I was wondering about the problem of finding the eigenfunctions and eigenvalues for multidimensional functions. What I mean is, let's suppose I have some differential operator $\displaystyle \mathcal{\hat L}$, and I want to find it's eigenfunctions and eigenvalues $\displaystyle \lambda$ and $\displaystyle u(\mathbf{r})$:

$\displaystyle \mathcal{\hat L}u(\mathbf{r})=\lambda u(\mathbf{r})$

Where for example $\displaystyle \mathcal{\hat L}$ could be some differential operator of the Sturm-Liouville form:

$\displaystyle \mathcal{\hat L}=-\nabla \cdot \left(p(\mathbf{r})\nabla \right)+q(\mathbf{r})$, so in particular I could be concerned with the multidimensional Sturm-Liouville problem:

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

I wanted to know if it is possibly to solve this multidimensional eigenvalue problems, either numerically or analytically analytically, and if there is some bibliography in this type of problems reporting how to treat them. I am aiming to tackle the general case, avoiding to look to particular cases like when the problem can be factorized in one dimensional problems (which I think is only restricted for some specific cases).