= Solution
For a regular <self-adjoint> problem with a lowest <eigenvalue> and a complete <orthonormal eigenbasis> in the <weighted inner product>, expand an admissible trial function as $\psi=\sum_jc_j\psi_j$. Then
$$
\frac{Q[\psi]}{N[\psi]}=\frac{\sum_j\lambda_j|c_j|^2}{\sum_j|c_j|^2}\geq\lambda_0.
$$
The <Rayleigh-Ritz variational principle> therefore provides the upper bound
$$
\boxed{\lambda_0\leq\min_{\text{chosen trial family}}\frac{Q[\psi]}{N[\psi]}.}
$$
Choose a family satisfying the essential boundary conditions and finite energy, evaluate the quotient, and minimize over its parameters. Enlarging the trial family can only improve or preserve the bound; equality is attained if it includes an exact ground eigenfunction. For a finite linear trial space, this becomes a generalized matrix eigenvalue problem.
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