Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2022 / Paper 1 / 10

Choose homogeneous Dirichlet boundary conditions $\psi (a)=\psi (b)=0$, and take a nonzero real admissible trial $\psi$. Write the numerator as $F[\psi ]$ and denominator as $G[\psi ]>0$. For an endpoint-zero variation $\eta$, integration by parts givesThus the first variation of the Rayleigh quotient isBy the fundamental lemma of the calculus of variations, stationarity is equivalent toConversely, multiplying an eigenfunction equation by $\psi$ and integrating gives $F=\lambda G$, so its first variation vanishes. This establishes the equivalence for the chosen Sturm-Liouville problem. Homogeneous Neumann boundary conditions, or a mixed Dirichlet-Neumann pair, also cancel the boundary term; general Robin boundary conditions require adding the corresponding boundary contribution to the numerator.

Use the normalized eigenfunctions and their weighted inner product ${\int }_a^{b}w\overline{{\psi }_m}{\psi }_{n}dx={\delta }_{mn}$. Orthogonality givesFor complex trials, $F$ means the corresponding Hermitian quadratic form with $|{\psi }^{\prime }{|}^2,|\psi {|}^2$; it agrees with the displayed functional on real trials. If $a_0\ne 0$, division by $|a_0{|}^2$ yieldsSince ${\lambda }_n\ge {\lambda }_0$, this is at least ${\lambda }_0$. The Rayleigh-Ritz variational principle therefore gives an upper bound from any admissible trial, improved by minimizing over trial parameters or enlarging the trial space. Equality holds when the trial is entirely in the lowest eigenspace.

The displayed ratio using $b_n$ presupposes $a_0\ne 0$. If $a_0=0$, use the undivided quotient $\sum {\lambda }_n|a_n{|}^2/\sum |a_n{|}^2$ instead; the same upper-bound argument still applies. No ground-state coefficient can be divided out in that case.

For the two sine modes, orthogonality givesThe diagonal potential integrals vanish, while the mixed one isThus ${\int }_0^1\cos (\pi x){\tilde{\psi }}^{2}dx=\alpha /2$, giving the two-sine variational bound for a Dirichlet Mathieu equation:Its derivative vanishes when $1+6{\pi }^2\alpha -{\alpha }^2=0$. Hence the two stationary parameters are $3{\pi }^2\pm \sqrt{9{\pi }^4+1}$. To identify the minimum, use the normalized sine basis: the restricted self-adjoint operator has real symmetric matrixIts smaller eigenvalue is the minimum of the Rayleigh quotient, attained at the negative stationary parameter. ThereforeThe estimate improves on the single-sine value ${\pi }^2$. The Rayleigh-Ritz variational principle remains valid here even though $q(x)=\cos (\pi x)$ changes sign: on this finite interval the Sturm-Liouville operator is self-adjoint and bounded below, which is sufficient for the upper bound.