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. Additional controls show that the avoided-crossing behavior persists under purely geometric coupling, that explicit projection is more robust than residual and soft-orthogonality baselines, and that carrier-free subspace tracking extends to an asymmetric defected quantum dot. These results give a practical diagnostic hierarchy for neural eigensolvers of Hermitian operators under geometric deformation.