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Thread: Multidimensional eigenfunctions and eigenvalues

  1. #1
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    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 \mathcal{\hat L}, and I want to find it's eigenfunctions and eigenvalues \lambda and u(\mathbf{r}):

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

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

    \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:

    -\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).

    Thanks in advance.
    Last edited by Ulysses; Jul 5th 2017 at 11:50 AM.
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  2. #2
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    Re: Multidimensional eigenfunctions and eigenvalues

    Essentially, you are asking for solutions of a linear partial differential equation with variable coefficients. The solutions depend heavily on what those coefficients, p, q, and w, are.
    Thanks from Ulysses
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  3. #3
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    Re: Multidimensional eigenfunctions and eigenvalues

    Yes, I know. But what I would like to know is if there are techniques for (in my preference) numerically diagonalizing a discrete version of this problem, for particular p, q and w functions, but treating the problem as general as possible. In particular, any previous attempt in the bibliography, books, or papers for this type of problem could be of great help (I haven't found much).
    Last edited by Ulysses; Jul 5th 2017 at 11:58 AM.
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