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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 $x \in X$).

A Sturm-Liouville problem is regular exactly when its domain is the finite closed interval $X=[a,b]$ and the coefficient $p$ does not vanish anywhere on $X$ ($p(x) \neq 0$ for all $x \in X$). A problem is singular when either the domain $X$ is infinite/unbounded or the coefficient $p$ vanishes at some point $x_0$ in $X$.

Conditions: Regular: $X=[a,b]$ and $p(x) \neq 0$ for all $x \in X$.; Singular: $X$ is infinite, or $p(x_0)=0$ for some $x_0 \in X$.; For unbounded $X$, boundary conditions are interpreted using limits.

A Sturm-Liouville problem is regular exactly when its domain is the finite closed interval $X=[a,b]$ and the coefficient $p$ does not vanish anywhere on $X$ ($p(x) \neq 0$ for all $x \in X$). A problem is singular when either the domain $X$ is infinite/unbounded or the coefficient $p$ vanishes at some point $x_0$ in $X$.

Conditions: Regular: $X=[a,b]$ and $p(x) \neq 0$ for all $x \in X$.; Singular: $X$ is infinite, or $p(x_0)=0$ for some $x_0 \in X$.; For unbounded $X$, boundary conditions are interpreted using limits.