Convergence properties of {\itshape ab initio} calculations of light nuclei in a harmonic oscillator basis

Abstract

We study recently proposed ultraviolet and infrared momentum regulators of the model spaces formed by construction of a variational trial wavefunction which uses a complete set of many-body basis states based upon three-dimensional harmonic oscillator (HO) functions. These model spaces are defined by a truncation of the expansion characterized by a counting number ($\mathcal{N}$) and by the intrinsic scale ($\hslash \omega$) of the HO basis; in short by the ordered pair ($\mathcal{N},\hslash \omega$). In this study we choose for $\mathcal{N}$ the truncation parameter $N_{max}$ related to the maximum number of oscillator quanta, above the minimum configuration, kept in the model space. The ultraviolet (uv) momentum cutoff of the continuum is readily mapped onto a defined uv cutoff in this finite model space, but there are two proposed definitions of the infrared (ir) momentum cutoff inherent in a finite-dimensional HO basis. One definition is based upon the lowest momentum difference given by $\hslash \omega$ itself and the other upon the infrared momentum which corresponds to the maximal radial extent used to encompass the many-body system in coordinate space. Extending both the uv cutoff to infinity and the ir cutoff to zero is prescribed for a converged calculation. We calculate the ground state energy of light nuclei with "bare" and "soft" $NN$ interactions. By doing so, we investigate the behaviors of the uv and ir regulators of model spaces used to describe ${}^2$H, ${}^3$H, ${}^4$He and ${}^6$He with $NN$ potentials Idaho N${}^3$LO and JISP16. We establish practical procedures which utilize these regulators to obtain the extrapolated result from sequences of calculations with model spaces characterized by ($\mathcal{N},\hslash \omega$).

32 pages, 11 figures

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