Journal of Physics A Mathematical and General

Comparison between the projection operator and continued fraction approaches to perturbation theory

R I Jackson, S. Swain

Published Dec 1, 1981• 7 citations

Abstract

The connection between the projection operator approach and the continued fraction approach to perturbation theory is investigated. A concise solution to the linear operator equation A mod x>= mod b> is found in terms of the level shift operator using projection operator techniques. The analogous process to the use of projection operators in the continued fraction method of solving the same problem is identified, and a parallel development performed. The connection between the two approaches is thereby established, and continued fraction expressions for the level shift operator obtained. The abstract equation is then specialised to deal with (i) the eigenvalue problem and (ii) the calculation of transition probabilities for quantum mechanical systems described by a time-independent Hamiltonian. Particular attention is paid to the problem of degeneracy and it is shown that the most convenient expressions are found by a hybrid of the two approaches.

DOI: 10.1088/0305-4470/14/12/014

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