Learning quantum eigenspaces under geometric symmetry breaking with Rayleigh-Ritz physics-informed neural networks

Abstract

We develop a Rayleigh–Ritz physics-informed neural formulation for quantum eigenproblems on geometry-parametrized domains, in which the neural representation and its validation diagnostic track the spectral object selected by the physics rather than the eigenvalue index alone. Tested on the one-dimensional well, the square-to-rectangle and disk-to-ellipse deformations, and a controlled avoided crossing, the formulation recovers ordered excited states by explicit projection, degenerate eigenspaces by subspace Rayleigh–Ritz training without imposing a particular basis, symmetry-split branches by parity-adapted sectors, and near-degenerate bands by modal-overlap tracking. Four controls sharpen this picture. Replacing the avoided-crossing potential with a purely geometric shear reproduces the same level repulsion, indicating that the effect is not an artifact of the added potential. A pointwise-residual baseline succeeds only when initialized near the target energy, collapsing onto the wrong state otherwise, unlike the projection-based formulation, which requires no such prior. A calibrated residual formulation with a soft orthogonality penalty recovers the correct excited state in every run but leaves its own trainable eigenvalue unconverged in a minority of them, whereas explicit projection needs no penalty weight to succeed every time. Finally, a carrier-free calculation in a defected, asymmetric quantum dot shows that the same subspace-and-overlap tracking extends to a system with no residual symmetry and no analytical mode to build the ansatz around. These results give a practical diagnostic hierarchy for neural eigensolvers of Hermitian operators under geometric deformation.