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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.

Regular Sturm-Liouville problems have an infinite spectrum $\lambda_n \to \infty$, and their eigenfunctions are orthogonal and total. Because they are orthogonal and total, they can form an orthonormal basis (ONB) after normalization.

Conditions: The Sturm-Liouville problem is regular ($X=[a,b]$ and $p(x) \neq 0$ for all $x \in X$).

Regular Sturm-Liouville problems have an infinite spectrum $\lambda_n \to \infty$, and their eigenfunctions are orthogonal and total. Because they are orthogonal and total, they can form an orthonormal basis (ONB) after normalization.

Conditions: The Sturm-Liouville problem is regular ($X=[a,b]$ and $p(x) \neq 0$ for all $x \in X$).

The Sturm-Liouville operator is the second-order differential expression $L y = (p y')' + q y$. The coefficient function $p$ must be continuously differentiable, meaning $p \in C^1(X)$, and the coefficient function $q$ must be continuous, meaning $q \in C(X)$.

Conditions: The operator acts on a function $y$.; $p$ must be continuously differentiable.; $q$ must be continuous.

The Sturm-Liouville operator is the second-order differential expression $L y = (p y')' + q y$. The coefficient function $p$ must be continuously differentiable, meaning $p \in C^1(X)$, and the coefficient function $q$ must be continuous, meaning $q \in C(X)$.

Conditions: The operator acts on a function $y$.; $p$ must be continuously differentiable.; $q$ must be continuous.