the Rayleigh quotient
/ RAY-lee /
Suppose you want to know how big the smallest eigenvalue of a Sturm-Liouville problem is, but you cannot solve for it exactly. The Rayleigh quotient is a single recipe that turns any trial function into a number — and the smallest eigenvalue is the smallest number you can ever produce this way. It converts an eigenvalue problem into a minimization problem.
For the operator L = -(p u')' + q u with weight w, the Rayleigh quotient of a function u is R[u] = (integral of [p (u')^2 + q u^2] dx) / (integral of w u^2 dx), where the boundary terms from integration by parts are arranged to fit the boundary conditions. The numerator is, physically, the energy stored (bending plus potential) and the denominator is the weighted size. If u happens to be the n-th eigenfunction, R[u] equals exactly lambda_n. For any other trial function, R[u] sits above the lowest eigenvalue: lambda_1 = the minimum of R[u] over all admissible nonzero u.
This is enormously practical. Plug in any reasonable guess that respects the boundary conditions and you instantly get a rigorous upper bound on lambda_1 — and a clever guess gives a sharp one. This is the heart of the Rayleigh-Ritz and Galerkin methods used everywhere from quantum chemistry (estimating ground-state energies) to structural engineering (estimating buckling and vibration frequencies). The deeper structure — minimizing over subspaces to reach higher eigenvalues — is the min-max characterization.
For -X'' = lambda X on [0, 1] with X(0) = X(1) = 0, the true lambda_1 = pi^2 = 9.87. Try the simple polynomial u = x(1 - x): R[u] = (integral of (u')^2) / (integral of u^2) = (1/3) / (1/30) = 10, an upper bound just 1.3% above the exact value — from a crude guess.
A crude trial function gives a rigorous, surprisingly tight upper bound on the lowest eigenvalue.
The Rayleigh quotient always gives an upper bound for lambda_1, never a lower one — so it tells you the true eigenvalue is at most R[u], but not how much less. Getting lower bounds is genuinely harder and needs different tools.