Guaranteed Lower Eigenvalue Bounds for Spectral Galerkin Methods with Application to Schrödinger Operators

What the math gives to ML

The paper turns a Galerkin eigenvalue approximation into an unconditional, computable lower bound through an explicit projection-error constant, rather than requiring knowledge of neighboring exact eigenvalues. This is transferable to neural spectral solvers: a network can propose eigenfunctions or coefficients while a differentiable Galerkin layer converts its Ritz eigenvalues into certified lower bounds. The most practical setting is a known operator with a Fourier, polynomial, or other spectral trial space, where the projection constant can be obtained analytically from the first omitted eigenvalue. The bound can become a training objective or stopping criterion that prevents a learned eigensolver from reporting spuriously optimistic eigenvalues.