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We study recently proposed ultraviolet and infrared momentum regulators of the model spaces formed by construction of a variational trial wavefunction which uses a complete set of many-body basis states based upon three-dimensional harmonic oscillator (HO) functions.
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Schwartz gives an example of a variational calculation which at the early, or even at intermediate, stages the convergence is governed by a general shape fitting to the many-body wavefunction and appears exponentially convergent. But it occurs that eventually this part of the problem is well satisfied and the convergence settles down to a slower power rate in response to some weak singularity that contributes only a very small part of the entire problem. C. Schwartz, “Estimating convergence rates of variational calculations” in “Methods in Computational Physics”, volume 2, ed. B. Alder, S. Fernbach and M. Rotenberg, (Academic Press, New York, 1963) pp 241-266
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One should be aware that the variational principle guarantees only an extrema which could be a stationary point rather than a minima. Examples of cases in which the ground state energy does not increase with an increase in 𝒩 \mathcal{N} for a limited range of ℏ ω \hbar\omega are discussed by L. Majling, J. R̆izek, Z. Pluhaur and Yu. F. Smirnov, “On some peculiarities of the variational calculations in the harmonic oscillator basis” J. Phys. G: Nucl. Phys 2
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This subject is reviewed by J. D. Morgan III, “The analytic structure of atomic and molecular wavefunctions and its impact on the rate of convergence of variational calculations”, in “Numerical Determination of the Electronic Structure of Atoms, Diatomic and Polyatomic Molecules”, edited by M. Defranceschi and J. Delhalle, NATO Advanced Study Institutes, Series C 271 (Kluwer, Dordrecht, 1989), pp. 49-84
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1991
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1991
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S. Gould, “Variational Methods for Eigenvalue Problems: An Introduction to the Methods of Rayleigh, Ritz, Weinstein, and Aronszajn” (Dover, New York, 1995)
1995
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Point nucleon radii of nuclei are calculated with the operator r 2 r^{2} . However, unlike the Hamiltonian, r 2 r^{2} is not a bounded operator and therefore has no convergence theorems. See, for example, W. N. Polyzou, “Nucleon-Nucleon Interactions and Observables”, Phys. Rev. C58
1998
Cited alongside, same era.
P. Navrátil, J. P. Vary and B. R. Barrett, “Properties of C-12 in the ab initio
2000
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P. Navrátil, G. P. Kamuntavic̆ius and B. R. Barrett, “Few-nucleon systems in a translationally invariant harmonic oscillator basis ”, Phys. Rev. C 61
A. Kievsky, S. Rosati, M. Viviani, L. E. Marcucci and L. Girlanda, “A high-precision variational approach to three- and four-nucleon bound and zero-energy scattering states”, J. Phys. G: Nucl. Part. Phys. 35, 063101 (2008)
2008
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C. Forssen, J. P. Vary, E. Caurier and P. Navrátil, “Converging sequences in the ab initio
2008
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I. Stetcu, C.-P. Liu J. L. Friar, A. C. Hayes, P. Navrátil, “Nuclear electric dipole moment of 3 He”, Phys. Lett. B 665
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P. Maris, J. P. Vary and A. M. Shirokov, “ ab initio
2009
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P. Navrátil, S. Quaglioni, I. Stetcu, and B. R. Barrett, “Recent developments in no-core shell-model calculations”, J. Phy. G: Nucl. Phys. 36
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2000
Cited alongside, same era.
Steven C. Pieper and R. B. Wiringa, “Quantum Monte Carlo Calculations of Light Nuclei”, Annual Review of Nuclear and Particle Science 51
2001
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O. Portilho, “ Λ Λ 10 {}^{10}_{\Lambda\Lambda} Be hypernucleus in the four-body model: effect of correlation functions”, J. Phys. G: Nucl. Part. 28
2002
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N. Barnea, W. Leidemann and G. Orlandini, “Improved effective interaction for the hyperspherical formalism”, Phys. Rev. C 67
2003
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2003
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S. R. Beane and M. J. Savage, “Pions in the pionless effective field theory”, Nucl. Phys. A 717
2003
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P. Navrátil and E. Caurier, “Nuclear Structure with accurate chiral perturbation theory nucleon-nucleon potential: Application to 6 Li and 10
2004
Cited alongside, same era.
A. M. Shirokov, A. I. Mazur, S. A. Zaytsev, J. P. Vary and T. A. Weber, “Nucleon-nucleon interaction in the J-matrix inverse scattering approach and few-nucleon systems”, Phys. Rev. C 70
2004
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S. Bacca, A. Schwenk, G. Hagen and T. Papenbrock, “Helium halo nuclei from low-momentum interactions” Eur. Phys. J. A 42
2009
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Simen Kvaal, “Harmonic oscillator eigenfunction expansions, quantum dots, and effective interactions”, Phys. Rev. B 80
2009
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S. Vaintraub, N. Barnea and D. Gazit, “ 6 He β \beta -decay rate and the suppression of the axial constant in nuclear matter”, Phys. Rev. C 79
2009
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Doron Gazit, Sofia Quaglioni and Petr Navrátil, “Three-Nucleon Low-Energy Constants from the Consistency of Interactions and Currents in Chiral Effective Field Theory”, Phys. Rev. Lett. 103
2009
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I. Stetcu, S. Quaglioni, J. L. Friar, A. C. Hayes and Petr Navrátil, “Electric dipole polarizabilities of hydrogen and helium isotopes”, Phys. Rev. C 79
2009
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N. Barnea, W. Leidemann and G. Orlandini, “Hyperspherical effective interaction for nonlocal potentials”, Phys. Rev. C 81
2010
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J. P. Vary, “The Many-Fermion-Dynamics Shell-Model Code,” Iowa State University, 1992 (unpublished); J. P. Vary and D. C. Zheng, ibid
2010
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G. Hagen, T. Papenbrock, D. J. Dean and M. Hjorth-Jensen, “ ab initio
2010
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E. D. Jurgenson, P. Navrátil and R. J. Furnstahl, “Evolving nuclear many-body forces with the similarity renormalization group”, Phys. Rev C 83
2011
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P. Navrátil and S. Quaglioni, “ ab initio
2011
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S. Kreuzer and H.-W. Hammer, “The triton in a finte volume”, Phys. Lett B 694
2011
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2011
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The six-nucleon problem for the bound state has been formulated in the Yakubovsky scheme but numerical results have not yet appeared. W. Glöckle and H. Witala, “The Six-Nucleon Yakubovsky Equations for 6 He”, Few-Body Syst. 51
2011
Later among the works it cites.
I. Stetcu, B. R. Barrett, U. van Kolck and J. P. Vary, “Effective theory for trapped few-fermion systems”, Phys. Rev. A 76
2012
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