What spectral properties do regular Sturm-Liouville problems have?

Conditions

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

Reasoning, step by step

1. Identify that the spectrum consists of an infinite sequence of eigenvalues $\lambda_n$.
2. Note that the eigenvalues tend to infinity ($\lambda_n \to \infty$).
3. Recognize that the corresponding eigenfunctions are mutually orthogonal.
4. Understand that the eigenfunctions are total, meaning they form a complete basis.
5. Conclude that normalizing the eigenfunctions yields an orthonormal basis.

Example

The board states: Thm Regular SLPs have an infinite spectrum $\lambda_n \to \infty$ whose eigenfunctions are orthogonal and total (so can form an ONB).

Common misconceptions

• Thinking that because the theorem says the eigenfunctions form a basis, they are automatically an orthonormal basis as written; normalization is the extra step required.
• Assuming the spectrum is finite.

Watch the explanation

Connected concepts

Explore next

• Why can the eigenfunctions of a regular SLP be used to form an orthonormal basis?
• Why do Sturm-Liouville problems appear when solving PDEs by separation of variables?

Related questions