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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.