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Math Help - Eigenvalue Problem (Sturm-Liouville): Normaliztion of the eigenfunctions

  1. #1
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    Eigenvalue Problem (Sturm-Liouville): Normaliztion of the eigenfunctions

    Let's say I'm am given the Sturm-Lioville problem

    \[{y}''+\lambda y=0\]

    with a particular set of boundary conditions.

    From this I can determine the set of eigenvalues ( \[\lambda _{n}\]<br />
) and eigenfunctions ( \[\phi _{n}\]) which satisfy the equation.

    The part I am stuck at is determing the normalization constant needed to no to normalize the eigenfunctions. Any suggestions. Thanks.
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  2. #2
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    To normalize the eigenfunctions just divide each of them by their length.
    The length (or norm) of a vector can be found using the given inner product.
    The normalization constant depends on the inner product that you are using.
    Then your orthonormal set is

    \frac{\phi_{n}}{||\phi_{n}||}
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  3. #3
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    What if I would determine the normalization ( N_{n}^{2}) constant using the following condition:

    \int_{a}^{b}N_{n}^{2}\left | \phi _{n}\right |^{2}dx=1

    where (a,b)=(0,L). Would this lead to the correct normalization constant?
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