Accurate analytic presentation of solution for the spiked harmonic oscillator problem

Abstract

High precision approximate analytic expressions of the ground state energies and wave functions for the spiked harmonic oscillator are found by first casting the correspondent Schrödinger equation into the nonlinear Riccati form and then solving that nonlinear equation analytically in the first iteration of the quasilinearization method (QLM). In the QLM the nonlinear differential equation is treated by approximating the nonlinear terms with a sequence of linear expressions. The QLM is iterative but not perturbative and gives stable solutions to nonlinear problems without depending on the existence of a smallness parameter. The choice of zero iteration is based on general features of exact solutions near the boundaries. Comparison of our approximate analytic expressions for binding energies and wave functions with the exact numerical solutions demonstrates their high accuracy in the wide range of parameters. The accuracy ranging between 10 −3 and 10 −7 for the energies and, correspondingly, 10 −2 and 10 −7 for the wave functions in the regions, where they are not extremely small is reached. The derived formulas enable one to make accurate analytical estimates of how variation of different interactions parameters affects the correspondent physical systems.