2013
DOI: 10.3842/sigma.2013.003
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From Quantum ANto E8Trigonometric Model: Space-of-Orbits View

Abstract: A number of affine-Weyl-invariant integrable and exactly-solvable quantum models with trigonometric potentials is considered in the space of invariants (the space of orbits). These models are completely-integrable and admit extra particular integrals. All of them are characterized by (i) a number of polynomial eigenfunctions and quadratic in quantum numbers eigenvalues for exactly-solvable cases, (ii) a factorization property for eigenfunctions, (iii) a rational form of the potential and the polynomial entries… Show more

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Cited by 10 publications
(17 citation statements)
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“…It is a common invariant subspace for (C.3). It is shown that any quantum A d , BC d , D d rational, trigonometric, elliptic Calogero-Moser-Sutherland Hamiltonian has the finite-dimensional invariant subspace P (d) n , see [57], [11], [86], [87], [24], [65] 21 . For the rational and trigonometric cases the invariant subspace exists with any integer n = 0, 1, .…”
Section: Case Imentioning
confidence: 99%
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“…It is a common invariant subspace for (C.3). It is shown that any quantum A d , BC d , D d rational, trigonometric, elliptic Calogero-Moser-Sutherland Hamiltonian has the finite-dimensional invariant subspace P (d) n , see [57], [11], [86], [87], [24], [65] 21 . For the rational and trigonometric cases the invariant subspace exists with any integer n = 0, 1, .…”
Section: Case Imentioning
confidence: 99%
“…They form the infinite flag. The hidden algebra is not gl(3) anymore: it is infinite dimensional, 10-generated algebra of differential operators in two variables with generalized Gauss decomposition property (for discussion see the reviews [86,87]). For any quantum F 4 , E 6,7,8 rational, trigonometric Calogero-Moser-Sutherland Hamiltonian there exist infinitely-many finitedimensional invariant subspaces in polynomials of a special form [8], this is summarized in [9].…”
Section: Case Imentioning
confidence: 99%
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“…It is worth noting that the variables (8.1), (8.2) being A 2 Weyl invariants were obtained making the averaging over some orbits in A 2 root space, see e.g. [14]. It is an interesting open question whether (8) can be obtained as a result of averaging over some orbits, in particular, orbits generated by fundamental weights.…”
mentioning
confidence: 99%
“…Writing Θ in separable form Θ = A(α)B(β)Φ(φ), we see that the spectral problem for the operator (10) (see also (9)) separates as…”
Section: Prolate Spheroidal Coordinatesmentioning
confidence: 99%