Recursion method for deriving effective interaction and its application to eigenvalue problem

概要

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It is often useful to recast the full many-body problem of a quantum system described by a Hamiltonian H in the form of the effective interaction acting within a chosen model space. The central problem of the effective-interaction theory is how to calculate the so-called Q^^^ box introduced by Kuo et al. We first show that the Hamiltonian H is transformed to a block tri-diagonal form in terms of submatrices of small dimensions. With this transformed Hamiltonian, we next show that, making use of recursion methods, the Q^^^ box can be expressed as a continued fraction form and/or a simple perturbative form with the renormalized vertices and propagators. This procedure for the Q^^^ box ensures the exact calculation of the Q^^^ box if the dimension of the relevant Hilbert space is finite. We apply this approach to solving the eigenvalue problem for a given Hamiltonian H. We introduce a function g(E) of an energy variable E. This function is determined by the Q^^^ box and has a characteristic that the eigenvalues are represented as "resonance" positions of g(E). We discuss a possibility of applying this method to solving an eigenvalue problem with a huge dimension.
2012-02-20