= Solution
Use the normalized <eigenfunctions> and their <weighted inner product> $\int_a^bw\overline{\psi_m}\psi_n dx=\delta_{mn}$. <Orthogonality> gives
$$
G[\widetilde\psi]=\sum_{n\geq0}|a_n|^2,\qquad F[\widetilde\psi]=\int_a^b\overline{\widetilde\psi}\,\mathcal L\widetilde\psi\,dx=\sum_{n\geq0}\lambda_n|a_n|^2.
$$
For complex trials, $F$ means the corresponding Hermitian <quadratic form> with $|\psi'|^2,|\psi|^2$; it agrees with the displayed functional on real trials. If $a_0\ne0$, division by $|a_0|^2$ yields
$$
\boxed{\Lambda[\widetilde\psi]=\frac{\lambda_0+\sum_{n\geq1}|b_n|^2\lambda_n}{1+\sum_{n\geq1}|b_n|^2},\qquad b_n=\frac{a_n}{a_0}.}
$$
Since $\lambda_n\geq\lambda_0$, 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 $a_0\ne0$. 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.
Back to article page
