Self-consistent-field method and τ-functional method on group manifold in soliton theory. II. Laurent coefficients of soliton solutions for sl̂n and for sûn

Nishiyama, Seiya; Providência, João da; Komatsu, Takao

doi:10.1063/1.2734864

Cited by 1 publication

(2 citation statements)

References 42 publications

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“…We reconstruct a υ-dependent SCFM in a F ∞ and study a relation between soliton equation and υ-dependent HF equation [41]. We start from a single-particle Schrödinger equation with a υ-dependent and a Υ-periodic potential V (r , υ) = V (r , υ + Υ)…”

Section: Scf Methods In F ∞mentioning

confidence: 99%

“…A finite-dimensional Grassmannian is identified with a Gr ∞ which is affiliated with the group manifold obtained by reducting gl(∞) to sl(N ) and su(N ). We have given explicit expressions for Laurent coefficients of soliton solutions for sl(N ) and su(N ) on the Gr ∞ using Chevalley bases for sl(N ) and su(N ) [41]. As an illustration we will attempt to make a υ-HFT approach to an infinite-dimensional matrix model extended from the finite-dimensional su(2) Lipkin-Meshkov-Glick (LMG) model [42].…”

Section: Viewpoint Of Symmetry Of Evolution Equationsmentioning

confidence: 99%

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Self-Consistent-Field Method and τ-Functional Method on Group Manifold in Soliton Theory: a Review and New Results

Nishiyama¹

2009

SIGMA

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Abstract. The maximally-decoupled method has been considered as a theory to apply an basic idea of an integrability condition to certain multiple parametrized symmetries. The method is regarded as a mathematical tool to describe a symmetry of a collective submanifold in which a canonicity condition makes the collective variables to be an orthogonal coordinate-system. For this aim we adopt a concept of curvature unfamiliar in the conventional time-dependent (TD) self-consistent field (SCF) theory. Our basic idea lies in the introduction of a sort of Lagrange manner familiar to fluid dynamics to describe a collective coordinate-system. This manner enables us to take a one-form which is linearly composed of a TD SCF Hamiltonian and infinitesimal generators induced by collective variable differentials of a canonical transformation on a group. The integrability condition of the system read the curvature C = 0. Our method is constructed manifesting itself the structure of the group under consideration. To go beyond the maximaly-decoupled method, we have aimed to construct an SCF theory, i.e., υ (external parameter)-dependent Hartree-Fock (HF) theory. Toward such an ultimate goal, the υ-HF theory has been reconstructed on an affine Kac-Moody algebra along the soliton theory, using infinite-dimensional fermion. An infinite-dimensional fermion operator is introduced through a Laurent expansion of finite-dimensional fermion operators with respect to degrees of freedom of the fermions related to a υ-dependent potential with a Υ-periodicity. A bilinear equation for the υ-HF theory has been transcribed onto the corresponding τ -function using the regular representation for the group and the Schur-polynomials. The υ-HF SCF theory on an infinite-dimensional Fock space F ∞ leads to a dynamics on an infinite-dimensional Grassmannian Gr ∞ and may describe more precisely such a dynamics on the group manifold. A finite-dimensional Grassmannian is identified with a Gr ∞ which is affiliated with the group manifold obtained by reducting gl(∞) to sl(N ) and su(N ). As an illustration we will study an infinite-dimensional matrix model extended from the finite-dimensional su(2) Lipkin-Meshkov-Glick model which is a famous exactly-solvable model.

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Section: Scf Methods In F ∞mentioning

confidence: 99%

Section: Viewpoint Of Symmetry Of Evolution Equationsmentioning

confidence: 99%