Influence of singularities on perturbation theory for the effective Hamiltonian

Journal Article · Sun Aug 01 00:00:00 EDT 1976 · Phys. Rev., C; (United States)

DOI:https://doi.org/10.1103/PhysRevC.14.660· OSTI ID:7354427

Vincent, C M

The energy-independent effective Hamiltonian H (z) is considered as an analytic function of the coupling parameter z. It is shown that a point z/subb/ is not a singularity of H (z) unless the energy of some state that is selected for representation in the model space (L/subP/) coincides, at z=z/subb/, with the energy of a state that is excluded from representation in L/subP/. This result establishes the correctness of previous conjectures by Schucan and Weidenmuller, and implies that perturbation theory for H (z) will generally have a larger radius of convergence than perturbation theory for the individual energies. For the case of a cut (''intruder-state cut'') joining an isolated pair of branch points close to the real axis, the singular behavior of H (z) is examined in detail. The residue of the cut is expressed in terms of quantities that can be calculated by diagonalization of a real, symmetric modification (''minimal smoothing'') of the Hamiltonian matrix. A formula is given for the contribution of an intruder-state cut to the error incurred by nth order perturbation theory for the physical quantity H (1). Consideration of a numerical example shows that if the values of sufficiently high orders of perturbation theory are known, the residues of intruder-state cuts may be evaluated. This allows estimation of the errors that intruder-state cuts produce in perturbation theory summed to finite order, and yields a criterion for the optimal truncation of divergent perturbation series. (AIP)

• Research Organization: Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
• OSTI ID: 7354427
• Journal Information: Phys. Rev., C; (United States), Vol. 14:2
• Country of Publication: United States
• Language: English

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