sl ( 2 , R ) symmetry and solvable multiboson systems

Abstract

The one-mode and the two-mode multiboson systems with sl(2,R) symmetry are investigated.Hamiltonians of these systems are integrated using the theory of orthogonal polynomials. The coherent state representation for these systems is constructed.Comment: 23 pages, 1 figur

Quantum ball as an example of quantum Kähler manifold

“…where b ∓ n = ∓ 1 2 n 1 (n + n 2 + · · · + n N + N + µ), a ∓ n = N + µ + 1 + n 2 + · · · + n N 2 + n. see [14]. Thus there exists a Hilbert space isomorphism Φ µ,n 2 ,...,n N : …”

“…Thus the symplectic form ω µ and hermitian structure H given by (15) and (14) respectively, are invariant with respect to the action (1) of SU (1, N).…”

“…(M x ψ)(x) = xψ (x), see[14]. Let us remark here that the map E ∓n 2 ,...,n N : B(R) → L(H n 2 ,...,n N ) of the σ -algebra of Borel sets into the lattice of the orthogonal projectors L(H n 2 ,...,n N ) is the spectral measure of the operatorM ∓ H n 2 ...n N , i.e.…”

Exactly solvable models for multiatomic molecular Bose–Einstein condensates

“…The eigenvalues (60) are independent of the spectral parameter u and of the parameter ω, that are arbitrary.…”

“…Inspired by some of these ideas I present, in the present paper, the construction of two complete family of Bethe-ansatz solvable models for both homo-nuclear and hetero-nuclear molecular BECs obtained through a combination of three Lax operators constructed using special realizations of the su(2) Lie algebra and of the Heisenberg-Weyl Lie algebra, as well as a multibosonic representation of the sl(2) Lie algebra, discussed recently in [60]. Notice that the models obtained through this construction do not have spatial degrees of freedom.…”