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

1. ## 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 ( $$\lambda _{n}$
$
) and eigenfunctions ( $$\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.

2. 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

$\frac{\phi_{n}}{||\phi_{n}||}$

3. 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?