In this talk, I will present and analyse a method to solve self-adjoint eigenvalue problems. This method is based on the Feshbach-Schur map (i.e. a Schur complement) to recast the infinite-dimensional problem as a finite-dimensional one, combined with a perturbative formulation to obtain explicit bounds on the eigenvalues and eigenfunctions.
I will then illustrate this method on two different examples: first, for the estimation of the ground state energy of Helium-type atoms, and second, to compute the eigenvectors and eigenvalues of Schrödinger operators in the context of planewave discretisation. Finally, I will present some numerical results that underline the theoretical findings.
