Questions: Eigenfunction Expansions and Sturm-Liouville Theory

Question 1

In a regular Sturm-Liouville problem, what is guaranteed about the eigenvalues?

• A. They form an increasing sequence tending to infinity, and are all real
• B. They are all positive
• C. They are complex conjugate pairs
• D. There are finitely many

Answer: (A)

Regular Sturm-Liouville problems have countably many real eigenvalues λ₁ < λ₂ < λ₃ < ... → ∞. This follows from the self-adjointness of the Sturm-Liouville operator in a weighted L² space.

Question 2

Eigenfunctions corresponding to distinct eigenvalues of a Sturm-Liouville problem are orthogonal with respect to the weight function w(x).

• T. True
• F. False

Answer: True

The orthogonality ∫ φ_m(x)φ_n(x)w(x)dx = 0 for m ≠ n follows from the self-adjointness of the operator. This generalizes the orthogonality of sin(nπx/L) that appears in Fourier series, which is the Sturm-Liouville problem y'' = -λy with w(x) = 1.

Question 3

What classical special functions arise as eigenfunctions of Sturm-Liouville problems?

Model answer: Legendre polynomials, Bessel functions, Hermite polynomials, Laguerre polynomials, and Chebyshev polynomials

Each of these arises from a Sturm-Liouville problem on an appropriate interval with specific coefficient functions p(x), q(x), and w(x). They form orthogonal bases for function spaces relevant to problems with spherical, cylindrical, or other specific geometries.

Question 4

The nth eigenfunction of a regular Sturm-Liouville problem has exactly n-1 zeros in the interior of the interval.

• T. True
• F. False

Answer: True

This is the Sturm oscillation theorem: the eigenfunction φ_n corresponding to the nth eigenvalue λ_n has exactly n-1 interior zeros. Higher eigenvalues correspond to more oscillatory eigenfunctions, generalizing the fact that sin(nπx/L) has n-1 zeros in (0,L).