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

arXiv:cond-mat/9309038 · Journal of the Physical Society of Japan

Renormalized Harmonic-Oscillator Description of Confined Electron Systems with Inverse-Square Interaction

Norio Kawakami

Published Dec 15, 1993• 15 citations
arXivHTMLPDF

Abstract

An integrable model for SU($ν$) electrons with inverse-square interaction is studied for the system with confining harmonic potential. We develop a new description of the spectrum based on the {\it renormalized harmonic-oscillators} which incorporate interaction effects via the repulsion of energy levels. This approach enables a systematic treatment of the excitation spectrum as well as the ground-state quantities.

DOI
10.1143/jpsj.62.4163

Related papers

Renormalized Harmonic-Oscillator Description of Confined Electron Systems with Inverse-Square Interaction

Norio Kawakami

An integrable model for SU(ν) electrons with inverse-square interaction is studied for the system with confining harmonic potential. We develop a new description of the spectrum based on the renormalized harmonic-oscillators which incorporate interaction effects via the repulsion of energy levels. This approach enables a systematic treatment of the excitation spectrum as well as the ground-state quantities. 1 Quantum integrable systems with inverse-square (1/x2) interaction have been studied extensively with renewed interest [1–11]. Though various models including multicomponent cases have been studied in detail for periodic boundary conditions, the energy spectrum for confined systems has not been investigated systematically so far. In fact various investigations concerning confined systems have been mainly devoted to the single-component case [1,2]. Besides interest in the integrability there have been some attempts to apply the confined models to conductance-oscillation phenomena in mesoscopic systems [12–14], in which the exact ground state has been found for the electron model [14]. It is desirable to develop systematic treatments of the energy spectrum for the class of the integrable multicomponent models with confining potential. In this letter we wish to propose a systematic description of the energy spectrum for the class of one-dimensional multicomponent inverse-square systems with harmonic confinement. The essence of the idea is that the energy spectrum is expressed in terms of the renormalized harmonic oscillators (RHO) for which the interaction effects are taken into account by the renormalized quantum numbers for oscillators. To illustrate the idea, let us start with the integrable spinless-fermion model with confining harmonic potential [1,2],

•Jan 1, 1993•3 citations

Hierarchical models with inverse-square interaction in harmonic confinement

Norio Kawakami, Yoshio Kuramoto

A hierarchy of quantum models with inverse-square pairwise interaction is studied for the system with a confining harmonic potential. An approach based on renormalized harmonic oscillators is used to construct the energy spectrum, which naturally results in the renormalized Hartree-Fock description of the interaction. In the limit of weak confinement, the model may serve as a canonical model for multiple chiral Tomonaga-Luttinger liquids for the edge states of the fractional quantum Hall effect. For the SU(\ensuremath{\nu}) symmetric case, we find that the excitation spectrum is classified in terms of the simple roots and the weights of SU(\ensuremath{\nu}) Lie algebra.

Physical review. B, Condensed matter•Aug 15, 1994•6 citations

Exactly solvable model of interacting electrons confined by the Morse potential

V. I. Inozemtsev

The integrals of motion are constructed for the Sutherland hyperbolic quantum systems of particles with internal degrees of freedom (SU( n ) spins) interacting with an external field with the Morse potential of an arbitrary strength τ 2 . These systems are confined if some constraint is imposed on τ, the strength λ of the pairwise interaction and the number of particles. The ground state is described by the wave function of the Jastrow form. The known results for the systems with inverse-square interaction within harmonic well are reproduced in the limit τ → ∞ .

Physica Scripta•May 1, 1996•16 citations

Integral representation for the eigenstates of the spin system with inverse square interactions

Kazuhiro Hikami

We consider the Calogero-Sutherland-Moser system with spin - 1/2 . We give the integral representation for the ground state wavefunction. Excited states of the Calogero spin model confined in the harmonic potential are also given.

Journal of Physics A Mathematical and General•Aug 7, 1994•9 citations

Eigenfunctions for SU( nu ) particles with 1/r2interaction in harmonic confinement

K Vacek, A Okiji, N Kawakami

We find a set of exact eigenfunctions which provide the energy spectrum for the quantum N-body Calogero-Sutherland model for fermions or bosons with SU( nu ) spin degrees of freedom moving in a harmonic confinement potential. The eigenfunctions are explicitly constructed as a simple product of the Jastrow wavefunction for the ground state and the Hermite polynomials introduced to generate the excited states. The corresponding energy spectrum is given by a sum of the correlated ground-state energy and of the excited-state energy for SU( nu ) free particles in a harmonic well.

arXiv:cond-mat/9402012•Journal of Physics A Mathematical and General•Apr 7, 1994•27 citations

Quantum harmonic oscillator: an elementary derivation of the energy spectrum

Riccardo Borghi

Abstract An elementary treatment of the quantum harmonic oscillator is proposed. No previous knowledge of linear differential equation theory or Fourier analysis are required, but rather only a few basics of elementary calculus. The pivotal role in our analysis is played by the sole particle localization constraint, which implies square integrability of stationary-state wavefunctions. The oscillator ground-state characterization is then achieved in a way that could be grasped, in principle, even by first-year undergraduates. A very elementary approach to build up and to characterize all higher-level energy eigenstates completes our analysis.

European Journal of Physics•Feb 3, 2017•7 citations
An integrable model for SU(ννν) electrons with inverse-square interaction is studied for the system with confining harmonic potential. We develop a new description of the spectrum based on the {\it renormalized harmonic-oscillators} which incorporate interaction effects via the repulsion of energy levels. This approach enables a systematic treatment of the excitation spectrum as well as the ground-state quantities.
An integrable model for SU(ν) electrons with inverse-square interaction is studied for the system with confining harmonic potential. We develop a new description of the spectrum based on the renormalized harmonic-oscillators which incorporate interaction effects via the repulsion of energy levels. This approach enables a systematic treatment of the excitation spectrum as well as the ground-state quantities. 1 Quantum integrable systems with inverse-square (1/x2) interaction have been studied extensively with renewed interest [1–11]. Though various models including multicomponent cases have been studied in detail for periodic boundary conditions, the energy spectrum for confined systems has not been investigated systematically so far. In fact various investigations concerning confined systems have been mainly devoted to the single-component case [1,2]. Besides interest in the integrability there have been some attempts to apply the confined models to conductance-oscillation phenomena in mesoscopic systems [12–14], in which the exact ground state has been found for the electron model [14]. It is desirable to develop systematic treatments of the energy spectrum for the class of the integrable multicomponent models with confining potential. In this letter we wish to propose a systematic description of the energy spectrum for the class of one-dimensional multicomponent inverse-square systems with harmonic confinement. The essence of the idea is that the energy spectrum is expressed in terms of the renormalized harmonic oscillators (RHO) for which the interaction effects are taken into account by the renormalized quantum numbers for oscillators. To illustrate the idea, let us start with the integrable spinless-fermion model with confining harmonic potential [1,2],
A hierarchy of quantum models with inverse-square pairwise interaction is studied for the system with a confining harmonic potential. An approach based on renormalized harmonic oscillators is used to construct the energy spectrum, which naturally results in the renormalized Hartree-Fock description of the interaction. In the limit of weak confinement, the model may serve as a canonical model for multiple chiral Tomonaga-Luttinger liquids for the edge states of the fractional quantum Hall effect. For the SU(\ensuremath{\nu}) symmetric case, we find that the excitation spectrum is classified in terms of the simple roots and the weights of SU(\ensuremath{\nu}) Lie algebra.
The integrals of motion are constructed for the Sutherland hyperbolic quantum systems of particles with internal degrees of freedom (SU( n ) spins) interacting with an external field with the Morse potential of an arbitrary strength τ 2 . These systems are confined if some constraint is imposed on τ, the strength λ of the pairwise interaction and the number of particles. The ground state is described by the wave function of the Jastrow form. The known results for the systems with inverse-square interaction within harmonic well are reproduced in the limit τ → ∞ .
We find a set of exact eigenfunctions which provide the energy spectrum for the quantum N-body Calogero-Sutherland model for fermions or bosons with SU( nu ) spin degrees of freedom moving in a harmonic confinement potential. The eigenfunctions are explicitly constructed as a simple product of the Jastrow wavefunction for the ground state and the Hermite polynomials introduced to generate the excited states. The corresponding energy spectrum is given by a sum of the correlated ground-state energy and of the excited-state energy for SU( nu ) free particles in a harmonic well.
Abstract An elementary treatment of the quantum harmonic oscillator is proposed. No previous knowledge of linear differential equation theory or Fourier analysis are required, but rather only a few basics of elementary calculus. The pivotal role in our analysis is played by the sole particle localization constraint, which implies square integrability of stationary-state wavefunctions. The oscillator ground-state characterization is then achieved in a way that could be grasped, in principle, even by first-year undergraduates. A very elementary approach to build up and to characterize all higher-level energy eigenstates completes our analysis.