Spiked harmonic oscillators

Details Spiked harmonic oscillators Spiked harmonic oscillators

2001-09-18

arxiv.org/abs/math-ph/0109014v1

J. Math. Phys. 43, 94-112 (2002)

Physics

Mathematical Physics

32 pages

Scientific paper

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.

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