  A complete variational treatment is provided for a family of spiked-harmonic
oscillator Hamiltonians H = -d^2/dx^2 + B x^2 + lambda/x^alpha, B > 0, lambda >
0, for arbitrary alpha > 0. A compact topological proof is presented that the
set S = {psi_n} of known exact solutions for alpha = 2 constitutes an
orthonormal basis for the Hilbert space L_2(0, infinity). Closed-form
expressions are derived for the matrix elements of H with respect to S. These
analytical results, and the inclusion of a further free parameter, facilitate
optimized variational estimation of the eigenvalues of H to high accuracy.
