Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Projection Operator Formalisms and the Nuclear Shell Model

Jennings, B K

Published Apr 13, 2005• 6 citations
OpenAlex

Abstract

The shell model solve the nuclear many-body problem in a restricted model space and takes into account the restricted nature of the space by using effective interactions and operators. In this paper two different methods for generating the effective interactions are considered. One is based on a partial solution of the Schrodinger equation (Bloch-Horowitz or the Feshbach projection formalism) and other on linear algebra (Lee-Suzuki). The two methods are derived in a parallel manner so that the difference and similarities become apparent. The connections with the renormalization group are also pointed out.

Related papers

Projection operator formalisms and the nuclear shell model

B. K Jennings

The shell model solves the nuclear many-body problem in a restricted model space and takes into account the restricted nature of the space by using effective interactions and operators. In this paper two different methods for generating the effective interactions are considered. One is based on a partial solution of the Schrödinger equation (Bloch-Horowitz or the Feshbach projection formalism) and other on linear algebra (Lee-Suzuki). The two methods are derived in a parallel manner so that the difference and similarities become apparent. The connections with the renormalization group are also pointed out.

arXiv:nucl-th/0504042•Europhysics Letters (EPL)•Oct 1, 2005•6 citations

The Canonical Nuclear Many-Body Problem as a Rigorous Effective Theory

W. C. Haxton, C.-L. Song

The shell model is the standard tool for addressing the canonical nuclear many-body problem of nonrelativistic nucleons interacting through a static potential. We discuss several of the uncontrolled approximations that are made in this model to motivate a different approach, one based on an exact solution of the Bloch-Horowitz equation. We argue that the necessary self-consistent solutions of this equation can be obtained efficiently by a Green's function expansion based on the Lanczos algorithm. The resulting effective theory is carried out for the simplest nuclei, d and 3He, using realistic NN interactions such as the Argonne v18 and Reid93 potentials, in order to contrast the results with the shell model. We discuss the wave function normalization, the evolution of the wave function as the "shell model" space is varied, and the magnetic elastic effective operator. The numerical results show a simple renormalization group behavior that differs from effective field theory treatments of the two- and three-body problems. The likely origin of this scaling is discussed.

arXiv:nucl-th/9906082•Jun 27, 1999•0 citations

Effective Field Theory and the No-Core Shell Model

Ionel Stetcu

In finite model space suitable for many-body calculations via the no-core shell model (NCSM), I illustrate the direct application of the effective field theory (EFT) principles to solving the many-body Schrödinger equation. Two different avenues for fixing the low-energy constants naturally arising in an EFT approach are discussed. I review results for both nuclear and trapped atomic systems, using effective theories formally similar, albeit describing different underlying physics.

EPJ Web of Conferences•Jan 1, 2010•0 citations

Formulation of an effective interaction in terms of renormalized vertices and propagators

Kenji Suzuki, Hiroo Kumagai, Masayuki Matsuzaki +1 more

One of the useful and practical methods for solving quantum-mechanical many-body systems is to recast the full problem into a form of the effective interaction acting within a model space of tractable size. Many of the effective-interaction theories in nuclear physics have been formulated by use of the so-called $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Q}$ box introduced by Kuo et al. It has been one of the central problems how to calculate the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Q}$ box accurately and efficiently. We first show that, introducing new basis states, the Hamiltonian is transformed to a block-tridiagonal form in terms of submatrices with small dimension. With this transformed Hamiltonian, we next prove that the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Q}$ box can be expressed in two ways: One is in the form of a continued fraction and the other is a simple series expansion up to second order with respect to renormalized vertices and propagators. This procedure ensures derivation of an exact $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Q}$ box, if the calculation converges as the dimension of the Hilbert space tends to infinity. The $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Q}$ box given in this study corresponds to a nonperturbative solution for the energy-dependent effective interaction which is often referred to as the Bloch-Horowitz or the Feshbach form. By applying the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Z}$-box approach based on the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Q}$ box proposed previously, we introduce a graphical method for solving the eigenvalue problem of the Hamiltonian. The present approach has a possibility of resolving many of the difficulties encountered in the effective-interaction theory.

arXiv:1210.5028•Physical Review C•Feb 13, 2013•4 citations

Formulation of effective interaction in terms of the non-perturbative and analytic box

R. Okamoto, K Suzuki, H Kumagai +1 more

One of the useful and practical methods for solving quantum-mechanical many- body systems is to recast the full problem into a form of the effective interaction acting within a model space of tractable size. Many of the effective-interaction theories in nuclear physics have been formulated by use of the so called box introduced by Kuo et. al. It has been one of the central problems how to calculate the box accurately' and efficiently. Introducing new basis states, the Hamiltonian is transformed to a block-tridiagonal form in terms of submatrices with small dimension. With the transformed Hamiltonian we derive a recursion method of calculating the box non-perturbatively and analytically. The box given in this study corresponds to a non-perturbative solution for the energy-dependent. Hamiltonian interaction which is often referred to as the Bloch-Horowitz or the Feshbach form. The present approach has a possibility of resolving many of the theoretical and practical difficulties encountered in the calculation of effective-interaction and/or Hamiltonian.

Journal of Physics Conference Series•Aug 14, 2014•0 citations

A common many-body approach in shell model and Hubbard model

T. Mizusaki

Recently proposed two methods such as Quantum Monte Carlo Diagonalization method in nuclear structure physics and path-integral renormalization group method in condensed matter physics have a similarity as quantum many-body approach, especially in description beyond mean-field approximation. As these two methods have been developed in different fields, there are significant differences as quantum-number projection technique in former method and extrapolation technique in latter method. As these techniques are independent of details of considered system, these can be applied to other fields. In this paper, we discuss how to apply such extrapolation technique to shell model calculations and discuss how to apply quantum-number projection to path-integral renormalization group method.

Journal of Physics Conference Series•Jan 1, 2005•0 citations
The shell model solve the nuclear many-body problem in a restricted model space and takes into account the restricted nature of the space by using effective interactions and operators. In this paper two different methods for generating the effective interactions are considered. One is based on a partial solution of the Schrodinger equation (Bloch-Horowitz or the Feshbach projection formalism) and other on linear algebra (Lee-Suzuki). The two methods are derived in a parallel manner so that the difference and similarities become apparent. The connections with the renormalization group are also pointed out.
The shell model solves the nuclear many-body problem in a restricted model space and takes into account the restricted nature of the space by using effective interactions and operators. In this paper two different methods for generating the effective interactions are considered. One is based on a partial solution of the Schrödinger equation (Bloch-Horowitz or the Feshbach projection formalism) and other on linear algebra (Lee-Suzuki). The two methods are derived in a parallel manner so that the difference and similarities become apparent. The connections with the renormalization group are also pointed out.
The shell model is the standard tool for addressing the canonical nuclear many-body problem of nonrelativistic nucleons interacting through a static potential. We discuss several of the uncontrolled approximations that are made in this model to motivate a different approach, one based on an exact solution of the Bloch-Horowitz equation. We argue that the necessary self-consistent solutions of this equation can be obtained efficiently by a Green's function expansion based on the Lanczos algorithm. The resulting effective theory is carried out for the simplest nuclei, d and 3He, using realistic NN interactions such as the Argonne v18 and Reid93 potentials, in order to contrast the results with the shell model. We discuss the wave function normalization, the evolution of the wave function as the "shell model" space is varied, and the magnetic elastic effective operator. The numerical results show a simple renormalization group behavior that differs from effective field theory treatments of the two- and three-body problems. The likely origin of this scaling is discussed.
One of the useful and practical methods for solving quantum-mechanical many-body systems is to recast the full problem into a form of the effective interaction acting within a model space of tractable size. Many of the effective-interaction theories in nuclear physics have been formulated by use of the so-called $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Q}$ box introduced by Kuo et al. It has been one of the central problems how to calculate the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Q}$ box accurately and efficiently. We first show that, introducing new basis states, the Hamiltonian is transformed to a block-tridiagonal form in terms of submatrices with small dimension. With this transformed Hamiltonian, we next prove that the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Q}$ box can be expressed in two ways: One is in the form of a continued fraction and the other is a simple series expansion up to second order with respect to renormalized vertices and propagators. This procedure ensures derivation of an exact $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Q}$ box, if the calculation converges as the dimension of the Hilbert space tends to infinity. The $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Q}$ box given in this study corresponds to a nonperturbative solution for the energy-dependent effective interaction which is often referred to as the Bloch-Horowitz or the Feshbach form. By applying the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Z}$-box approach based on the $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{Q}$ box proposed previously, we introduce a graphical method for solving the eigenvalue problem of the Hamiltonian. The present approach has a possibility of resolving many of the difficulties encountered in the effective-interaction theory.
One of the useful and practical methods for solving quantum-mechanical many- body systems is to recast the full problem into a form of the effective interaction acting within a model space of tractable size. Many of the effective-interaction theories in nuclear physics have been formulated by use of the so called box introduced by Kuo et. al. It has been one of the central problems how to calculate the box accurately' and efficiently. Introducing new basis states, the Hamiltonian is transformed to a block-tridiagonal form in terms of submatrices with small dimension. With the transformed Hamiltonian we derive a recursion method of calculating the box non-perturbatively and analytically. The box given in this study corresponds to a non-perturbative solution for the energy-dependent. Hamiltonian interaction which is often referred to as the Bloch-Horowitz or the Feshbach form. The present approach has a possibility of resolving many of the theoretical and practical difficulties encountered in the calculation of effective-interaction and/or Hamiltonian.
Recently proposed two methods such as Quantum Monte Carlo Diagonalization method in nuclear structure physics and path-integral renormalization group method in condensed matter physics have a similarity as quantum many-body approach, especially in description beyond mean-field approximation. As these two methods have been developed in different fields, there are significant differences as quantum-number projection technique in former method and extrapolation technique in latter method. As these techniques are independent of details of considered system, these can be applied to other fields. In this paper, we discuss how to apply such extrapolation technique to shell model calculations and discuss how to apply quantum-number projection to path-integral renormalization group method.