Machine Learning-Assisted Analytical–Numerical Framework for Bound-State Spectra of Screened and Perturbed Potentials

Abstract

Analytical constraints, numerical convergence, and uncertainty calibration establish a framework for reliable machine learning predictions of quantum bound-state energies. A screened cosine Coulomb interaction and a Pöschl–Teller well with a regularized odd perturbation are solved using finite-difference diagonalization. The Nikiforov–Uvarov reduction provides exact eigenvalues and eigenfunctions for the unperturbed well, while the radial Coulomb limit supplies a benchmark. The dataset contains 4306 accepted levels from 4000 sampled Hamiltonians, separated into training, validation, calibration, and test subsets at the Hamiltonian level. Five-network ensembles achieve test mean absolute errors of 0.00820 and 0.00707, with prediction-interval coverage of 96.54% and 95.88% for the screened and perturbed-well families, respectively. Direct numerical tests of 600 Hamiltonians reveal parameter-dependent extrapolation accuracy and coverage. Reflection symmetry, ground-state concavity, second-order perturbation theory, and Hellmann–Feynman derivatives explain spectral trends. A grid-resolved regularization scan confirms that smoothing changes the spectrum and constitutes a model specification. These results support accurate interpolation of resolved bound states while identifying limitations associated with extrapolation, regularization, and the sparse representation of higher excited states. The framework distinguishes numerical convergence diagnostics from statistical prediction uncertainty and separates Hamiltonian properties from constraints explicitly imposed on the learned energy models through their symmetry-based input descriptors.