The Seke self-consistent projection-operator approach for the calculation of quantum-mechanical eigenvalues and eigenstates

Originally, the Seke self-consistent projection-operator method has been developed for treating non-Markovian time evolution of probability amplitudes of a relevant set of state vectors. In the so-called Born approximation the method leads automatically to an Hamiltonian restricted to a subspace and thus enables the construction of effective Hamiltonians. In the present paper, in order to explain the efficiency of Sekeʹs method its algebraic operator structure is analyzed and a new successive approximation technique for the calculation of eigenstates and eigenvalues of an arbitrary quantum-mechanical system is developed. Since, unlike most perturbative techniques, in the present approach a well-defined effective (restricted) Hamiltonian in each order of the applied approximation exists, the self-consistency of all obtained results is self-evident. Finally, to illustrate the efficiency of the developed formalism, the optical polaron model is investigated.