Sturm-Liouville problems are the general class of boundary value problems that arise when the separation of variables technique is applied to a PDE or an initial-boundary value problem. They are also situated as part of the extension of the spectral theorem to infinite dimensions.
Conditions: The context is solving PDEs or initial-boundary value problems using separation of variables.
Sturm-Liouville problems are the general class of boundary value problems that arise when the separation of variables technique is applied to a PDE or an initial-boundary value problem. They are also situated as part of the extension of the spectral theorem to infinite dimensions.
Conditions: The context is solving PDEs or initial-boundary value problems using separation of variables.
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.