# Day 15

Thursday, March 30, 2017

# Section 5.5, cont'd

## 3 proofs of Sturm-Liouville facts

orthogonality of eigenfunctions

reality of eigenvalues

uniqueness of eigenfunctions

# Section 5.6: Rayleigh quotient for SL problems

Derivation

Conditions guaranteeing positivity of eigenvalues

Minimization principle: statement, utility, proof

Correction: the class of functions we're minimizing over can be extended to functions that are continuous on [a,b], have a piecewise continuous derivative on (a,b), and satisfy the BCs.

Next: Examples and proof of minimization principle