Spiked harmonic oscillators
Richard L. Hall, Nasser Saad, Attila B. von Keviczky
Abstract
A complete variational treatment is provided for a family of spiked-harmonic oscillator Hamiltonians H = -d2/dx2 + B x2 + lambda/xalpha, B > 0, lambda > 0, for arbitrary alpha > 0. A compact topological proof is presented that the set S = psin of known exact solutions for alpha = 2 constitutes an orthonormal basis for the Hilbert space L2(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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