Generalized spiked harmonic oscillator

A variational and perturbative treatment is provided for a family of generalized spiked harmonic oscillator Hamiltonians H = -d²/dx² + Bx² + A/x² + λ/x^α, where B>0, A⩾0, and α and λ denote two real positive parameters. The method makes use of the function space spanned by the solutions |n⟩ of Schrödinger's equation for the potential V(x) = Bx² + A/x². Compact closed-form expressions are obtained for the matrix elements ⟨m|H|n⟩, and a first-order perturbation series is derived for the wavefunction. The results are given in terms of generalized hypergeometric functions. It is proved that the series for the wavefunction is absolutely convergent for α⩽2.