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From Orthogonalizing Pseudopotential to the Feshbach-Schur Projection

M. M. Nishonov

Abstract

The Pauli exclusion principle imposes an important structural constraint in cluster descriptions of light nuclei and is commonly taken into account using methods such as the Resonating Group Method (RGM), the Orthogonality Condition Model (OCM), and the Orthogonalizing Pseudopotential (OPP) approach. The latter provides a practical implementation for suppressing Pauli-forbidden states in few-body calculations through the introduction of a large auxiliary coupling constant $λ_0$ and an associated limiting procedure. Exact $λ_0$-eliminated formulations have appeared in the literature. It is shown that the OPP method may be interpreted as the singular $λ_0\to\infty$ limit of the Feshbach-Schur projection, and that the Schur complement provides a natural operator-level framework for understanding this connection. In contrast to earlier approaches, the elimination is derived explicitly as a closed Schur-complement operator identity. When formulated in terms of operators and Green's functions, the Feshbach-Schur projection eliminates Pauli-forbidden components algebraically and avoids the explicit introduction of large pseudopotential parameters. This reformulation clarifies the implementation of Pauli projection in cluster models and provides a convenient framework for few-body calculations.

From Orthogonalizing Pseudopotential to the Feshbach-Schur Projection

Abstract

The Pauli exclusion principle imposes an important structural constraint in cluster descriptions of light nuclei and is commonly taken into account using methods such as the Resonating Group Method (RGM), the Orthogonality Condition Model (OCM), and the Orthogonalizing Pseudopotential (OPP) approach. The latter provides a practical implementation for suppressing Pauli-forbidden states in few-body calculations through the introduction of a large auxiliary coupling constant and an associated limiting procedure. Exact -eliminated formulations have appeared in the literature. It is shown that the OPP method may be interpreted as the singular limit of the Feshbach-Schur projection, and that the Schur complement provides a natural operator-level framework for understanding this connection. In contrast to earlier approaches, the elimination is derived explicitly as a closed Schur-complement operator identity. When formulated in terms of operators and Green's functions, the Feshbach-Schur projection eliminates Pauli-forbidden components algebraically and avoids the explicit introduction of large pseudopotential parameters. This reformulation clarifies the implementation of Pauli projection in cluster models and provides a convenient framework for few-body calculations.
Paper Structure (9 sections, 109 equations, 1 figure, 5 tables)

This paper contains 9 sections, 109 equations, 1 figure, 5 tables.

Table of Contents

  1. Introduction
  2. Feshbach–Schur complement for the separable $T$-- mat-rix
  3. Extension to configuration-space formulation
  4. Effect on the resolvent
  5. Numerical illustration: ${}^6$He and ${}^6$Li binding energies
  6. Discussion and conclusions
  7. Schur Complement
  8. Faddeev equations for the ground-state wave function
  9. $NN$ and $\alpha N$ potentials used

Figures (1)

  • Figure 1: The geometric part of the kernel of the Faddeev integral equation.