Energy bounds for a class of singular potentials and some related series

Abstract

Perturbation expansions up to third order for the generalized spiked harmonic oscillator Hamiltonians H = -d^2/dx^2+ x^2 + A/x^2 + lambda/x^alpha, A >= 0, 2gamma > alpha, gamma=1+(1/2)sqrt(1+4A), and small values of the coupling lambda > 0, are developed. Upper and lower bounds for the eigenvalues are computed by means of the procedure of Burrows et al [J. Phys. A: Math. Gen. 20, 889-897 (1987)] for assessing the accuracy of a truncated perturbation expansion. Closed-form sums for some related perturbation double infinite series then immediately follow as a result of this investigation.

15 pages

References in corpus (10)

• Spiked harmonic oscillators
• Variational analysis for a generalized spiked harmonic oscillator
• Matrix elements for a generalized spiked harmonic oscillator
• Generalized spiked harmonic oscillator
• Asymptotic solvability of an imaginary cubic oscillator with spikes
• Part of the D - dimensional Spiked harmonic oscillator spectra
• Perurbation expansions for the spiked harmonic oscillator and related series involving the gamma function
• Bound states for spiked harmonic oscillators and truncated Coulomb potentials
• Closed-form sums for some perturbation series involving hypergeometric functions
• Closed-form sums for some perturbation series involving associated Laguerre polynomials

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• Study of anharmonic singular potentials
• An effective singular oscillator for Duffin-Kemmer-Petiau particles with a nonminimal vector coupling: a two-fold degeneracy
• Spiked oscillators: exact solution
• A basis for variational calculations in d dimensions
• Perturbation method for non-square Hamiltonians and its application to polynomial oscillators
• Quasi-exact minus-quartic oscillators in strong-core regime
• Green's function for a Schroedinger operator and some related summation formulas