Variational neural-network solution of the two-body $^{17}\mathrm{F}$ proton-halo problem with a Coulomb--Whittaker tail

Abstract

We present a variational artificial neural-network (VANN) solution of $^{17}$F in a two-body $^{16}$O$+p$ potential model. The calculation uses a standard interaction from the literature as a controlled benchmark for testing whether a neural variational ansatz can reproduce not only bound-state energies and interior wave functions, but also the Coulomb--Whittaker tails that control halo and peripheral-capture observables. The reduced radial wave function is obtained by minimizing the Rayleigh quotient of the radial Schrödinger Hamiltonian with the constraints required by each partial wave. Because the variational energy can converge before the asymptotic normalization is correct, the ansatz combines a neural interior with the charged-particle Coulomb--Whittaker form. In the $s_{1/2}$ channel, the Pauli-forbidden $0s_{1/2}$ component is computed and the physical one-node branch is checked independently for forbidden-state contamination. The Coulomb--Whittaker-constrained VANN reproduces independent Numerov benchmarks for the compact $d_{5/2}$ ground state and the extended $s_{1/2}$ halo state in energy, nodes, rms radius, and overlap. The compact-state ANC agrees to within one percent, while the halo ANC differs by about $4.2\%$, within the larger numerical sensitivity of the asymptotic extraction. The continuum scattering states are obtained by standard Numerov integration with Coulomb matching; only the bound states are represented by the neural ansatz. Combined with these $p$-wave scattering states, the VANN bound states yield astrophysical $S$ factors consistent with published benchmarks and data within the accuracy of the adopted two-body model. The results demonstrate the usefulness of physically constrained neural wave functions for tail-sensitive nuclear calculations and identify the asymptotic region as the most sensitive part of the calculation.