Energy-embedded neural solvers for one-dimensional quantum systems

Physics-informed neural networks (PINNs) have emerged as powerful tools for solving partial differential equations (PDEs) in computational physics. However, most existing approaches rely on problem-specific architectures or require retraining for each new equation or boundary condition. In this work, we present an energy-embedded PINN framework for solving the one-dimensional time-independent Schr"odinger equation, capable of obtaining both ground- and excited-state wave functions and energy eigenvalues within a unified, parameter-free architecture. By treating energy as a continuous input and incorporating an embedding layer, the network efficiently learns the entire quantum spectrum without training separate models for each eigenstate. Numerical results show that this method achieves a high precision across various potentials---including infinite wells, harmonic oscillators, Woods-Saxon, and double-well potentials---with minimal errors in both energies and wave functions, where orthogonality emerges naturally during training. As a general-purpose, structure-free approach, the method reported in this study provides an alternative strategy for analyzing quantum observables and for scanning energy spectra under arbitrary one-dimensional interaction potentials. Its successful extension to the radial Coulomb problem demonstrates its adaptability to broader classes of Schr"odinger-type systems and may open new paths toward solving quantum many-body and nuclear problems in the future.