# How to get the eigenvalues of Hamiltonian in an over complete basis

Let $|\psi_i\rangle$, $i=1...N+m$, be a set of overcomplete basis vector in a $N$-dim Hilbert space. The following are known: (Einstein's summation convention assumed) $$\hat{H}|\psi_i\rangle=H_{ji}|\psi_j\rangle$$, $$\lambda^\alpha_j|\psi_j\rangle=0,\alpha=1..m$$ $$\langle \psi_i|\psi_j\rangle=S_{ij}$$ We already know the numerical values of $H_{ij}, S_{ij}, \lambda^\alpha_j$. Then how to find the eigenvalues of $\hat{H}$ using the common Matlab or Mathematica functions?

• @Lagrenge After giving this few seconds of thought, I'd say 1. diagonalize S, 2. project S and H to eigenvectors of S with non-zero eigenvalues. Sometimes such basis set 'purifications' (I can't remember the official word right now) are done in basis set codes to avoid numerical instabilities with nearly linearly dependent basis. – Mikael Kuisma Aug 15 '16 at 21:38
• Your notation seems a bit odd to me. But, in any case, it seems that you need to solve a generalized eigenvalue problem. – nicoguaro Aug 16 '16 at 21:50