Local cubic convergence of Rayleigh quotient iteration for unbounded self-adjoint operators

Prove local cubic convergence of Rayleigh quotient iteration when applied to unbounded self-adjoint Hamiltonians on infinite-dimensional Hilbert spaces. Specifically, let H be a densely defined self-adjoint operator on a Hilbert space and define the Rayleigh quotient E(ψ)=⟨ψ,Hψ⟩/⟨ψ,ψ⟩ and the iteration ψ_{k+1} = ((H − E(ψ_k) I)^{-1} ψ_k) / ||(H − E(ψ_k) I)^{-1} ψ_k||. Establish that, under appropriate local conditions near an eigenpair, the iterates converge with cubic rate to the eigenfunction corresponding to the eigenvalue nearest to E(ψ_0).

Background

Within the paper’s framework, the Riemannian Newton method on the unit sphere for minimizing the Rayleigh quotient corresponds in function space to Rayleigh quotient iteration. In finite-dimensional settings, Rayleigh quotient iteration is known to exhibit locally cubic convergence to the eigenpair closest in energy to the initial iterate.

However, many quantum Hamiltonians of interest in variational quantum Monte Carlo are unbounded self-adjoint operators on infinite-dimensional Hilbert spaces. The authors explicitly note that, while cubic local convergence is established in finite dimensions, they are not aware of a proof for the unbounded operator case, indicating a gap in the theoretical foundation for extending these convergence guarantees to the infinite-dimensional setting relevant to ab initio problems.