The eigenfunctions of a regular Sturm-Liouville problem are mutually orthogonal and total (meaning they form a complete basis). Because they are already orthogonal and complete, normalizing each eigenfunction to unit length upgrades the basis to an orthonormal basis (ONB) of eigenfunctions for the differential operator.
Conditions: The Sturm-Liouville problem is regular.; The eigenfunctions are mutually orthogonal and total.
The eigenfunctions of a regular Sturm-Liouville problem are mutually orthogonal and total (meaning they form a complete basis). Because they are already orthogonal and complete, normalizing each eigenfunction to unit length upgrades the basis to an orthonormal basis (ONB) of eigenfunctions for the differential operator.
Conditions: The Sturm-Liouville problem is regular.; The eigenfunctions are mutually orthogonal and total.