Effective and intermediate Hamiltonians obtained by similarity transformations

A simple similarity transformation is used to derive equations for effective and intermediate Hamiltonians in a lucid way. Effective and intermediate Hamiltonians based on the wave operator formalism provide only a subset of all eigenvalues while the similarity transform technique divides the eigenvalue problem into two subproblems that can be solved separately. This means that the complete spectrum of the Hamiltonian remains well defined and this proves to be advantageous in the formal analysis and may be useful in many applications. Moreover both left and right hand eigenvectors of the transformed Hamiltonian can be obtained and this allows a convenient evaluation of properties. Rayleigh–Schrödinger and Brillouin–Wigner perturbation expansions of the intermediate Hamiltonians are discussed and a comparison is made of the various possible schemes.