  A variational and perturbative treatment is provided for a family of
generalized spiked harmonic oscillator Hamiltonians H = -(d/dx)^2 + B x^2 +
A/x^2 + lambda/x^alpha, where B > 0, A >= 0, and alpha and lambda denote two
real positive parameters. The method makes use of the function space spanned by
the solutions |n> of Schroedinger's equation for the potential V(x)= B x^2 +
A/x^2. Compact closed-form expressions are obtained for the matrix elements
<m|H|n>, and a first-order perturbation series is derived for the wave
function. The results are given in terms of generalized hypergeometric
functions. It is proved that the series for the wave function is absolutely
convergent for alpha <= 2.
