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Physics of Particles and Nuclei

Integrability Aspects of the Current Algebra Representation and the Factorized Quantum Nonlinear Schrödinger Type Dynamical Systems

Nikolay N. Bogolubov, Dominik Prorok, Anatolij K. Prykarpatski

Published Jul 1, 2020• 1 citation
Publisher

Abstract

Abstract In this work we study integrability aspects of the current algebra representation and the factorized quantum nonlinear Schrödinger type dynamical systems, initiated before by G. Goldin with collaborators, in suitably renormalized Fock type Hilbert spaces. There is developed the current algebra representation scheme of reconstructing algebraically factorized quantum Hamiltonian and symmetry operators in the Fock type space. There is presented its application to constructing quantum factorized Hamiltonian systems and their symmetry operators in case of quantum integrable spatially many- and one-dimensional dynamical systems. As examples we have studied in detail the factorized structure of Hamiltonian operators, describing such quantum integrable spatially one-dimensional models as the Calogero–Moser–Sutherland and Nonlinear Schrödinger dynamical systems of spin-less Bose-particles.

DOI
10.1134/s1063779620040152

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Abstract In this work we study integrability aspects of the current algebra representation and the factorized quantum nonlinear Schrödinger type dynamical systems, initiated before by G. Goldin with collaborators, in suitably renormalized Fock type Hilbert spaces. There is developed the current algebra representation scheme of reconstructing algebraically factorized quantum Hamiltonian and symmetry operators in the Fock type space. There is presented its application to constructing quantum factorized Hamiltonian systems and their symmetry operators in case of quantum integrable spatially many- and one-dimensional dynamical systems. As examples we have studied in detail the factorized structure of Hamiltonian operators, describing such quantum integrable spatially one-dimensional models as the Calogero–Moser–Sutherland and Nonlinear Schrödinger dynamical systems of spin-less Bose-particles.
There is developed a current algebra representation scheme for reconstructing algebraically factorized quantum Hamiltonian and symmetry operators in the Fock type space and its application to quantum Hamiltonian and symmetry operators in case of quantum integrable spatially many- and one-dimensional dynamical systems. As examples, we have studied in detail the factorized structure of Hamiltonian operators, describing such quantum integrable spatially many- and one-dimensional models as generalized oscillatory, Calogero-Sutherland, Coulomb type and nonlinear Schrödinger dynamical systems of spinless bose-particles.
Quantum integrable particle systems with inverse square interactions in one-dimensional space are investigated by use of the quantum inverse scattering method. For the case of particles with $\mathrm{SU}(\ensuremath{\nu})$ spins, a hidden symmetry is the $\mathrm{SU}(\ensuremath{\nu})\ensuremath{\bigotimes}\mathrm{U}(1)\ensuremath{\simeq}\mathrm{U}(\ensuremath{\nu})$ current algebra. The system has a higher symmetry, the ${W}_{1+\ensuremath{\infty}}$ algebra. This WWW algebra structure unifies the Calogero type (1x2\frac{1}{{x}^{2}}x21​ interactions) and the Sutherland type (1sin2x\frac{1}{{sin}^{2}x}sin2x1​ interactions). The energy spectra of the systems are obtained in a simple manner based on the fermion representation of the algebra.
In this short review paper the detailed analysis of six two-dimensional quantum {\it superintegrable} systems in flat space is presented. It includes the Smorodinsky-Winternitz potentials I-II (the Holt potential), the Fokas-Lagerstrom model, the 3-body Calogero and Wolfes (equivalently, G2G_2G2​ rational, or I6I_6I6​) models, and the Tremblay-Turbiner-Winternitz (TTW) system with integer index kkk. It is shown that all of them are exactly-solvable, thus, confirming the Montreal conjecture (2001); they admit algebraic forms for the Hamiltonian and both integrals (all three can be written as differential operators with polynomial coefficients without a constant term), they have polynomial eigenfunctions with the invariants of the discrete symmetry group of invariance taken as variables, they have hidden (Lie) algebraic structure g(k)g^{(k)}g(k) with various kkk, and they possess a (finite order) polynomial algebras of integrals. Each model is characterized by infinitely-many finite-dimensional invariant subspaces, which form the infinite flag. Each subspace coincides with the finite-dimensional representation space of the algebra g(k)g^{(k)}g(k) for a certain kkk. In all presented cases the algebra of integrals is a 4-generated (H,I1,I2,I12≡[I1,I2])(H, I_1, I_2, I_{12}\equiv[I_1, I_2])(H,I1​,I2​,I12​≡[I1​,I2​]) infinite-dimensional algebra of ordered monomials of degrees 2,3,4,5, which is a subalgebra of the universal enveloping algebra of the hidden algebra.
We develop some calculation schemes to determine dynamics of a wide class of integrable quantum-optical models using their symmetry adapted reformulation in terms of polynomial Lie algebras supd(2)su_{pd}(2)supd​(2). These schemes, based on "diagonal" representations of model evolution operators (via diagonalizing Hamiltonians with the help of the supd(2)su_{pd}(2)supd​(2) defining relations), are implemented in the form adapted for numerical calculations. Their efficiency is demonstrated on the example of the second-harmonic-generation model.