Variational and perturbative schemes for a spiked harmonic oscillator

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Abstract
A variational analysis of the spiked harmonic oscillator Hamiltonian operator −d2/ dx2+x2+l(l+1)/x2+λ‖x‖−α , where α is a real positive parameter, is reported in this work. The formalism makes use of the functional space spanned by the solutions of the Schrödinger equation for the linear harmonic oscillator Hamiltonian supplemented by a Dirichlet boundary condition, and a standard procedure for diagonalizing symmetric matrices. The eigenvalues obtained by increasing the dimension of the basis set provide accurate approximations for the ground state energy of the model system, valid for positive and relatively large values of the coupling parameter λ. Additionally, a large coupling perturbative expansion is carried out and the contributions up to fourth-order to the ground state energy are explicitly evaluated. Numerical results are compared for the special case α=5/2.

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