From Quantum AN to E₈ Trigonometric Model: Space-of-Orbits View

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...

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Дата:	2013
Автор:	Turbiner, A.V.
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Опубліковано:	Інститут математики НАН України 2013
Назва видання:	Symmetry, Integrability and Geometry: Methods and Applications
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/149207
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Цитувати:	From Quantum AN to E₈ Trigonometric Model: Space-of-Orbits View / A.V. Turbiner // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 24 назв. — англ.

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spelling	irk-123456789-1492072019-02-20T01:24:08Z From Quantum AN to E₈ Trigonometric Model: Space-of-Orbits View Turbiner, A.V. 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 of the metric in the Laplace-Beltrami operator in terms of affine-Weyl (exponential) invariants (the same holds for rational models when polynomial invariants are used instead of exponential ones), they admit (iv) an algebraic form of the gauge-rotated Hamiltonian in the exponential invariants (in the space of orbits) and (v) a hidden algebraic structure. A hidden algebraic structure for (A–B–C–D)-models, both rational and trigonometric, is related to the universal enveloping algebra Ugln. For the exceptional (G–F–E)-models, new, infinite-dimensional, finitely-generated algebras of differential operators occur. Special attention is given to the one-dimensional model with BC₁≡(Z2)⊕T symmetry. In particular, the BC₁ origin of the so-called TTW model is revealed. This has led to a new quasi-exactly solvable model on the plane with the hidden algebra sl(2)⊕sl(2). 2013 Article From Quantum AN to E₈ Trigonometric Model: Space-of-Orbits View / A.V. Turbiner // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 24 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 35P99; 47A15; 47A67; 47A75 DOI: http://dx.doi.org/10.3842/SIGMA.2013.003 http://dspace.nbuv.gov.ua/handle/123456789/149207 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	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 of the metric in the Laplace-Beltrami operator in terms of affine-Weyl (exponential) invariants (the same holds for rational models when polynomial invariants are used instead of exponential ones), they admit (iv) an algebraic form of the gauge-rotated Hamiltonian in the exponential invariants (in the space of orbits) and (v) a hidden algebraic structure. A hidden algebraic structure for (A–B–C–D)-models, both rational and trigonometric, is related to the universal enveloping algebra Ugln. For the exceptional (G–F–E)-models, new, infinite-dimensional, finitely-generated algebras of differential operators occur. Special attention is given to the one-dimensional model with BC₁≡(Z2)⊕T symmetry. In particular, the BC₁ origin of the so-called TTW model is revealed. This has led to a new quasi-exactly solvable model on the plane with the hidden algebra sl(2)⊕sl(2).
format	Article
author	Turbiner, A.V.
spellingShingle	Turbiner, A.V. From Quantum AN to E₈ Trigonometric Model: Space-of-Orbits View Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Turbiner, A.V.
author_sort	Turbiner, A.V.
title	From Quantum AN to E₈ Trigonometric Model: Space-of-Orbits View
title_short	From Quantum AN to E₈ Trigonometric Model: Space-of-Orbits View
title_full	From Quantum AN to E₈ Trigonometric Model: Space-of-Orbits View
title_fullStr	From Quantum AN to E₈ Trigonometric Model: Space-of-Orbits View
title_full_unstemmed	From Quantum AN to E₈ Trigonometric Model: Space-of-Orbits View
title_sort	from quantum an to e₈ trigonometric model: space-of-orbits view
publisher	Інститут математики НАН України
publishDate	2013
url	http://dspace.nbuv.gov.ua/handle/123456789/149207
citation_txt	From Quantum AN to E₈ Trigonometric Model: Space-of-Orbits View / A.V. Turbiner // Symmetry, Integrability and Geometry: Methods and Applications. — 2013. — Т. 9. — Бібліогр.: 24 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT turbinerav fromquantumantoe8trigonometricmodelspaceoforbitsview
first_indexed	2025-07-12T21:04:48Z
last_indexed	2025-07-12T21:04:48Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 9 (2013), 003, 25 pages From Quantum AN (Sutherland) to E8 Trigonometric Model: Space-of-Orbits View? Alexander V. TURBINER Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Apartado Postal 70-543, 04510 México, D.F., Mexico E-mail: turbiner@nucleares.unam.mx Received September 21, 2012, in final form January 11, 2013; Published online January 17, 2013 http://dx.doi.org/10.3842/SIGMA.2013.003 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 of the met- ric in the Laplace–Beltrami operator in terms of affine-Weyl (exponential) invariants (the same holds for rational models when polynomial invariants are used instead of exponential ones), they admit (iv) an algebraic form of the gauge-rotated Hamiltonian in the expo- nential invariants (in the space of orbits) and (v) a hidden algebraic structure. A hidden algebraic structure for (A−B−C−D)-models, both rational and trigonometric, is related to the universal enveloping algebra Ugln . For the exceptional (G−F−E)-models, new, infinite- dimensional, finitely-generated algebras of differential operators occur. Special attention is given to the one-dimensional model with BC1 ≡ (Z2)⊕T symmetry. In particular, the BC1 origin of the so-called TTW model is revealed. This has led to a new quasi-exactly solvable model on the plane with the hidden algebra sl(2)⊕ sl(2). Key words: (quasi)-exact-solvability; space of orbits; trigonometric models; algebraic forms; Coxeter (Weyl) invariants; hidden algebra 2010 Mathematics Subject Classification: 35P99; 47A15; 47A67; 47A75 1 Introduction In this article we attempt to overview our constructive knowledge of (quasi)-exactly-solvable potentials having the form of a meromorphic function in trigonometric variables. Any model with such a potential is characterized by a discrete symmetry group, and possesses an (in)finite set of polynomial eigenfunctions in a certain trigonometric variables. In the case of exactly- solvable potentials an infinite discrete spectra is quadratic in the quantum numbers. All of these models are characterized by the appearance of a hidden (Lie) algebraic structure. They do not admit a separation of variables, they are completely-integrable and possess a commutative algebra of integrals. So far, no super-integrable models with trigonometric potentials are known, although all of them admit at least one particular integral [22]. A similar overview of the rational models (with a potential in the form of a meromorphic function in Cartesian coordinates) was given in [21]. Unlike the trigonometric models the rational models do admit a separation out of radial coordinate and hence, the emergence of integral of the second order leading to super-integrability. For exactly-solvable rational models their ?This paper is a contribution to the Special Issue “Superintegrability, Exact Solvability, and Special Functions”. The full collection is available at http://www.emis.de/journals/SIGMA/SESSF2012.html mailto:turbiner@nucleares.unam.mx http://dx.doi.org/10.3842/SIGMA.2013.003 http://www.emis.de/journals/SIGMA/SESSF2012.html 2 A.V. Turbiner eigenvalues depend on quantum numbers linearly, thus, their spectrum is a linear superposition of equidistant spectra. Any spinless quantum system is characterized by the Hamiltonian H = −∆ + V (x), x ∈ Rd. (1.1) The main problem of quantum mechanics is to find the spectrum the Hamiltonian looking for the Schrödinger equation HΨ(x) = EΨ(x), Ψ(x) ∈ L2 ( Rd ) . in the Hilbert space. Since the Hamiltonian is a differential operator it can be represented as infinite-dimensional matrix. Thus, the solving the Schrödinger equation is equivalent to diagonalizing the infinite-dimensional matrix. It is a transcendental problem: the characteristic polynomial is of infinite order and it has infinitely-many roots. In general, we do not know how to make such a diagonalization explicitly. One can try to describe or construct quantum system for which a transcendental nature of (1.1) degenerates (completely or partially) to algebraic one: the roots of the characteristic polynomial (energies), some or all, can be found explicitly (algebraically). Usually, in such a situation one can indicate an analytic form of (some or all) eigenfunctions. Such systems do exist and we call them solvable. If all energies are known they are called exactly-solvable (ES), if only some number of them is known we call them quasi- exactly-solvable (QES) [23]. Surprisingly, almost all such models the present author is familiar with, are provided by integrable systems emerging from the Hamiltonian reduction method [13] with real, continuous coupling constants. Sometimes, these models are called the Calogero– Moser–Sutherland models. Every Hamiltonian has a discrete symmetry – it is symmetric with respect to affine Weyl group. Usually, the multi-dimensional Hamiltonians of the trigonometric models are of the form H = 1 2 N∑ k=1 [ − ∂2 ∂y2 k ] + β2 8 ∑ α∈R+ ν\|α\|(ν\|α\| − 1) \|α\|2 sin2 β 2 (α · y) , (1.2) in the exactly-solvable case, where R+ is a set of positive roots in the root space ∆ of dimen- sion N , β is a parameter and µ\|α\| are coupling constants which depend on the root length. For roots of the same length the constants ν\|α\| are equal. Thus, the potential in (1.2) is a superpo- sition of the Weyl-invariant functions, each defined as a sum over roots of the same length. The configuration space is the Weyl alcove. The ground state wave function has a form Ψ0(y) = ∏ α∈R+ ∣∣∣∣sin β2 (α · y) ∣∣∣∣ν\|α\| . (1.3) The ground state energy has a form E0 = β2ε0(ν) and is known explicitly. Let us take the Hamiltonian in Rd which is symmetric with respect to the (maximal) discrete group G. One can construct invariants of G using a procedure of averaging some function over orbit(s). The d linearly independent invariants span a linear space called the space of orbits. These invariants are generating elements of the algebra of invariants. The main idea of this paper is to study the Hamiltonian in a space of orbits (space of invariants). Technically, it implies a change of variables from the original coordinates to the invariants. Conceptually, it means factoring out the discrete symmetry of the problem. It reveals a “primary” operator of the system which being dressed by the discrete symmetry becomes the Hamiltonian. We consider some models from the list of ones known so far. From Quantum AN (Sutherland) to E8 Trigonometric Model: Space-of-Orbits View 3 2 Solvable models 2.1 Generalities Many years ago, as the state-of-the-art, Sutherland found a many-body exactly-solvable and integrable Hamiltonian with trigonometric potential [18]. A few years later the Hamiltonian re- duction method was introduced (for review and references see e.g. Olshanetsky–Perelomov [13]). In this method an extended family of integrable and exactly-solvable Hamiltonians with trigono- metric potentials, associated with affine Weyl (Coxeter) symmetry, was found. The Sutherland model appeared as one of its representatives, the AN trigonometric model. The idea of the Hamiltonian reduction method is beautiful: • Take a simple group G. • Define the Laplace–Beltrami (invariant) operator on its symmetric space (free motion). • Radial part of the Laplace–Beltrami operator is the Olshanetsky–Perelomov Hamiltonian relevant from physical point of view. The emerging Hamiltonian is affine Weyl-symmetric, it can be associated with root system, it is integrable with integrals given by the invariant operators of order higher than two with a property of solvability. Trigonometric case. This case appears when the coordinates of the symmetric space are introduced in such a way that a negative-curvature surface occurs. Emerging the Calogero– Moser–Sutherland–Olshanetsky–Perelomov Hamiltonian in the Cartesian coordinates has the form (1.2) with the ground state given by (1.3). In the Hamiltonian reduction, the parame- ters ν\|α\| of the Hamiltonian take a set of discrete values, however, they can also be generalized to any real value without loosing the property of integrability as well as solvability with the only constraint being the existence of L2-solutions of the corresponding Schrödinger equation. The configuration space for (1.2) is the Weyl alcove. The Hamiltonian (1.2) is completely-integrable: there exists a commutative algebra of inte- grals (including the Hamiltonian) of dimension which is equal to the dimension of the configura- tion space (for integrals, see Oshima [14] with explicit forms of those). The Hamiltonian (1.2) is invariant with respect to the affine Weyl (Coxeter) group transformation, which is the discrete symmetry group of the corresponding root space, see e.g. [13]. The Hamiltonian (1.2) has a hidden (Lie)-algebraic structure. In order to reveal it (see [1, 2, 3, 4, 12, 15, 16]) we need to • Gauge away the ground state eigenfunction making a similarity transformation (Ψ0)−1(H− E0)Ψ0 = h, then • If the state-of-art variables are introduced for trigonometric models AN , BCN , G2 and F4 (see [1, 4, 15, 16], respectively), the Hamiltonian h becomes algebraic, however, • It can be checked, which, in fact, looks evident, that parameterizing the space of orbits of the Weyl (Coxeter) group by taking the Weyl (Coxeter) fundamental trigonometric invariants, τ (Ω) a (y;β) = ∑ w∈Ωa eiβ(w,y), (2.1) where Ωa is an orbit generated by fundamental weight wa, a = 1, 2, . . . , N (N is the rank of the root system); ~y is N -dimensional auxiliary vector which defines the Cartesian coordinates, as coordinates we arrive at the conclusion that the state-of-the-art variables [1, 4, 15, 16] coincide with (2.1). It should be emphasized that this fact was not clear to the authors of articles [1, 4, 15, 16] including the present author. 4 A.V. Turbiner From physical point of view, the expression (2.1) is a Weyl-invariant non-linear superpo- sition of plane waves with momenta proportional to β. The fundamental trigonometric invariants τ(β) taken as coordinates always lead to the gauge- rotated trigonometric Hamiltonian h in a form of algebraic differential operator with polynomial coefficients. It is proved by demonstration. It is worth emphasizing a surprising fact that the period(s) of the invariants τ(β) is half of the period(s) of the Hamiltonian (1.2) and the ground state function (1.3). It seems correct (this can be proved by demonstration) that the original Hamiltonian H (1.2) written in terms of the fundamental trigonometric invariants τ(β) takes the form H(τ) = −∆g + V (τ), (2.2) where ∆g = 1 √ g ∂τi √ ggij(τ)∂τj , is the Laplace–Beltrami operator with a metric gij(τ) with polynomial in τ matrix elements, hence with polynomial in τ coefficient functions in front of the second derivatives, and with the property that the coefficient functions in front of the first derivatives are also polynomials in τ ; V (τ) is a rational function, see below e.g. (2.7), (2.21). The form (2.2) can be called the rational form of the trigonometric model. It is evident that the similar rational form (2.2) appears for rational models when polynomial Weyl invariants are used as new coordinates to parameterize the space of orbits. In turn, the gauge-rotated Hamiltonian h in τ -variables takes the form h(τ) = −∆g + ∑ α∈R+ ν\|α\| ∑ a=1,...,N C \|α\|a (τ)∂τa , where Ca(τ) are polynomials in τ , see below e.g. (2.9), (2.27). The same representation is valid for the rational models. 2.2 A1/BC1 case or trigonometric Pöschl–Teller potential The BC1 trigonometric Hamiltonian reads1 HBC1(x) = − d2 dx2 + g2β 2 sin2 βx + g3β 2 4 sin2 βx 2 , (2.3) where β, g2, g3 are parameters. Symmetry: (Z2) ⊕ T (reflections x → −x, translation x → x + 2π/β). As for the configuration space, it can be taken the interval [0, πβ ]. If g2 = 0 the interval can be extended to [0, 2π β ]. At g3 = 0 (or g2 = 0) the Hamiltonian (2.3) degenerates to the A1 trigonometric Hamiltonian, which describes the relative motion of two particle system on a line. The ground state for (2.3) reads Ψ0 = ∣∣ sin(βx) ∣∣ν2 ∣∣∣∣sin(β2x )∣∣∣∣ν3 , E0 = − ( ν2 + ν3 2 )2 β2, (2.4) cf. (1.3), where ν2, ν3 are found from the relations g2 = ν2(ν2 − 1) > −1 4 , g3 = ν3(ν3 + 2ν2 − 1) > −1 4 . 1Common factor 1 2 is omitted. From Quantum AN (Sutherland) to E8 Trigonometric Model: Space-of-Orbits View 5 Note that if the parameters in (2.3) are related g2 = g3 2 (g32 − 1), the ground state energy (2.4) reaches its maximal value, E0 = 0. Any eigenfunction has a form Ψ0ϕ, where ϕ is a polynomial in the BC1 fundamental trigono- metric invariant τ(β) = cos(βx) (see (2.1)). Hence, the ground state function Ψ0 plays a role of a multiplicative factor. The BC1 trigonometric Hamiltonian (2.3) is easily related to the trigonometric Pöschl–Teller (PT) Hamiltonian HPT = − d2 dx2 + ( α2 − 1 4 ) β2 4 sin2 βx 2 + ( γ2 − 1 4 ) β2 4 cos2 βx 2 , (2.5) where α2 − 1 4 = g2 + g3, γ2 − 1 4 = g2. Replacing in (2.5) β → iβ, we arrive at the general hyperbolic Pöschl–Teller Hamiltonian H(h) PT = − d2 dx2 + ( α2 − 1 4 ) β2 4 sinh2 βx 2 − ( γ2 − 1 4 ) β2 4 cosh2 βx 2 , while the one-soliton Hamiltonian appears at α2 = 1 4 . In the case of the BC1 trigonometric Hamiltonian under the replacement β → iβ the BC1 Hyperbolic Hamiltonian occurs. Let us introduce a new variable τ = cos(βx) (2.6) (which is the 2π β -periodic, BC1-Weyl invariant) in the BC1 Hamiltonian (2.3). It appears that HBC1(τ) = −∆g + g2 2(1 + τ) + (g2 + g3) 2(1− τ) , (2.7) with amazingly simple meromorphic potential, where ∆g = ( τ2 − 1 ) d2 dτ2 + τ d dτ is the flat Laplace–Beltrami operator with metric g11 = ( τ2−1 ) . Overall multiplicative factor β2 in (2.7) is dropped off. It can be called a rational form of the BC1 trigonometric Hamiltonian. The eigenvalue problem for (2.7) is considered on the interval [−1, 1]. It can be easily seen that the rational form for the BC1 hyperbolic Hamiltonian is exactly the same as for the BC1 trigonometric Hamiltonian (!) and is given by (2.7). However, the domain for theBC1 hyperbolic Hamiltonian (2.7) is [1,∞). In the hyperbolic case taking τ = coshβx (cf. (2.6)) we obtain the same Hamiltonian (2.7). The ground state eigenfunction (2.4) in τ coordinate (2.6) becomes Ψ0(τ) = (1 + τ) ν2 2 (1− τ) ν2+ν3 2 . (2.8) At ν2 = 1 and ν3 = 0 it coincides to the Jacobian. Now let us make a gauge rotation hBC1 = 1 β2 Ψ−1 0 (HBC1 − E0)Ψ0, with Ψ0 given by (2.4) and write the result in the variable τ . After a simple calculations it reads hBC1(τ) = ( τ2 − 1 ) d2 dτ2 + [(2ν2 + ν3 + 1)τ + ν3] d dτ , (2.9) 6 A.V. Turbiner which is the algebraic form of the BC1 Hamiltonian (2.3). Its eigenvalues are εp = p2 + (2ν2 + ν3)p, p = 0, 1, 2, . . . , (2.10) being quadratic in quantum number p, while the eigenfunctions are the Jacobi polynomials, ϕp = P (ν2+ν3− 1 2 ,ν2− 1 2 ) p (τ). Eventually, the explicit form of an eigenfunction of the Hamiltonian (2.3) is Ψ(BC1) p = P (ν2+ν3− 1 2 ,ν2− 1 2 ) p (cos(βx)) ∣∣ sin(βx) ∣∣ν2 ∣∣∣∣sin(β2x )∣∣∣∣ν3 , p = 0, 1, 2, . . . . It can be easily checked that the gauge-rotated Hamiltonian hBC1(τ) has infinitely many finite-dimensional invariant subspaces Pn = 〈τp\|0 ≤ p1 ≤ n〉, n = 0, 1, 2, . . . , (2.11) hence, the infinite flag P, P0 ⊂ P1 ⊂ P2 ⊂ · · · ⊂ Pn ⊂ · · · ⊂ P, with the characteristic vector ~f = (1) (see below), is preserved by hBC1 . Thus, the eigenfunctions of hBC1 are elements of the flag P. Any subspace Pn contains (n + 1) eigenfunctions which is equal to dimPn. Take the algebra gl2 in (n+1)-dimensional representation realized by the first order differen- tial operators J− = d dτ , J0 n = τ d dτ − n, T 0 = 1, J+ n = τ2 d dτ − nt = τJ0 n, (2.12) where n = 0, 1, . . . and T 0 is the central element. Its finite-dimensional representation space is the space of polynomials Pn (2.11). Hence, the finite-dimensional invariant subspaces of the Hamiltonian hBC1 coincide with the finite-dimensional representation spaces of gl2 (2.12) for n = 0, 1, 2, . . .. It immediately implies that the algebra gl2 is the hidden algebra of the BC1 trigonometric Hamiltonian – it can be written in terms of gl2 generators (2.12) hBC1 = J0J0 − J−J− + (2ν2 + ν3 + 1)J0 + ν3J −, (2.13) where J0 ≡ J0 0 , J− ≡ J−0 . Thus, the Hamiltonian hBC1 is an element of the universal enveloping algebra Ugl2 . Among the generators of the algebra gl2 (2.12) there is the Euler-Cartan operator, J0 n = τ d dτ − n, which has zero grading; it maps a monomial in τ to itself. It defines the highest weight vector. This generator allows us to construct a particular integral – π-integral of zero grading of the (n+ 1)th order (see [22]) i (n) par(τ): its commutator with hBC1 vanishes on a subspace. If i(n) par(τ) = n∏ j=0 ( J0 n + j ) , (2.14) then [ hBC1(τ), i(n) par(τ) ] : Pn 7→ 0. From Quantum AN (Sutherland) to E8 Trigonometric Model: Space-of-Orbits View 7 Making the gauge rotation of the π-integral (2.14) with Ψ−1 0 (τ) given by (2.4) and changing variables τ back to the Cartesian coordinate we arrive at the quantum π-integral acting in the Hilbert space, I(n) par,BC1 (x) = Ψ0(τ)i(n) par(τ)Ψ−1 0 (τ) ∣∣∣ τ→x . Under such a gauge transformation the triangular space of polynomials Pn becomes the space Vn = Ψ0Pn. The Hamiltonian HBC1(x) commutes with I(n) par,BC1 (x) over this space[ HBC1(x), I(n) par,BC1 (x) ] : V(N−1) n 7→ 0. Any eigenfunction Ψ ∈ Vn is zero mode of the π-integral I(n) par,BC1 (x). It is worth noting a connection of the BC1 trigonometric model with the so-called Tremblay– Turbiner–Winternitz (TTW) model [19] and, in particular, with the I2(k) rational model (see e.g. [21]). In order to see it let us combine the BC1 trigonometric Hamiltonian HBC1(φ) (2.3) as the angular part and the radial part of two-dimensional spherical-symmetrical harmonic oscillator Hamiltonian as the radial part forming the 2D Hamiltonian HTTW(r, φ;ω, ν2, ν3, β) = −∂2 r − 1 r ∂r + ω2r2 + HBC1(φ) r2 , (2.15) which is nothing but the Hamiltonian of the TTW model [19]. If β = k is integer, this Hamiltonian corresponds to the I2(k) rational model [13]. Since the both Hamiltonians are exactly-solvable, the TTW model is also exactly-solvable but with spectra of two-dimensional anisotropic (!) harmonic oscillator with frequency ratio 1 : β. Any eigenfunction of (2.15) has the form of a polynomial p(r2, cos(βφ)) in variables r2 and cos(βφ) multiplied by a ground state function Ψ (TTW) 0 (r, φ) = r(ν2+ν3)β ∣∣ sin(βφ) ∣∣ν2 ∣∣∣∣sin(β2φ )∣∣∣∣ν3 e−ωr22 , (2.16) namely, Ψ(TTW)(r, φ) = p ( r2, cos(βφ) ) Ψ (TTW) 0 (r, φ). If in the construction (2.15) instead of two-dimensional radial harmonic oscillator, the radial Hamiltonian of the sextic QES 2D central potential (see e.g. [23]) is taken, the quasi-exactly- solvable extension of the TTW model occurs [19] H(qes) TTW(r, n;φ;ω, ν2, ν3, β, a) = −∂2 r − 1 r ∂r + a2r6 + 2aωr4 + [ ω2 − 2a(2n+ 2 + β(ν2 + ν3)) ] r2 + HBC1(φ) r2 , (2.17) cf. (2.15), here n is non-negative integer and a > 0 is a parameter. In this Hamiltonian a finite number of eigenstates can be found explicitly (algebraically). Their eigenfunctions have the form of a polynomial p(r2, cos(βφ)) of degree n in r2 multiplied by a factor Ψ (qes,TTW) 0 = r(ν2+ν3)β ∣∣ sin(βφ) ∣∣ν2 ∣∣∣∣sin(β2φ )∣∣∣∣ν3 e−ωr22 −ar44 , (2.18) cf. (2.16), namely, Ψ (qes,TTW) alg = p ( r2, cos(βφ) ) Ψ (qes,TTW) 0 . The factor (2.18) is the ground state eigenfunction of the Hamiltonian (2.17) at n = 0. If β is equal to non-negative integer k, a polynomial p(r2, cos(βφ)) belongs to the space P(1,k) with the characteristic vector ~f = (1, k), see below. 8 A.V. Turbiner 2.3 Quasi-exactly-solvable BC1 case (or QES trigonometric Pöschl–Teller potential) The Hamiltonian hBC1(τ) (2.9) is gl(2)-Lie-algebraic operator (2.13) which has infinitely-many finite-dimensional invariant subspaces in polynomials (2.11). By adding to hBC1(τ) (2.9) the operator δh(qes)(τ) = 2b ( τ2 − 1 ) d dτ − 2bnτ + 2b ( n+ ν2 + ν3 + 1 2 ) , where b is a parameter and n is non-negative integer, as a result we get the operator h (qes) BC1 (τ) = hBC1 + δh(qes), (2.19) which has a single finite-dimensional invariant subspace Pn = 〈τp \| 0 ≤ p ≤ n〉, of the dimension (n + 1). Hence, this operator is quasi-exactly-solvable - it can be written in terms of gl2 generators in (n+ 1)-dimensional representation (2.12), h (qes) BC1 = J0 nJ 0 n − J−J− − 2bJ+ n + (2n+ 2ν2 + ν3 + 1)J0 n + (ν3 − 2b)J− + n(n+ 2ν2 + ν3 + 1). Making the gauge rotation of (2.19) with Ψ̃0 = e− ν2+ν3 2 log(1−τ)− ν2 2 log(1+τ)ebτ and the change of variable τ = cos(βx) we arrive at the BC1-trigonometric QES Hamiltonian [23] H(qes) BC1 (x) = − d2 dx2 + ν2(ν2 − 1)β2 sin2 βx + ν3(ν3 + 2ν2 − 1)β2 4 sin2 βx 2 + b2β2sin2 βx + 2bβ2(2n+ 2ν2 + ν3 + 1) sin2 βx 2 , (2.20) cf. (2.3), where b, ν2, ν3, β are parameters, n is non-negative integer. In τ -variable (2.6) the BC1-trigonometric QES Hamiltonian appears in rational form H(qes) BC1 (τ) = −∆g + g2( 1−τ2 ) + g3 2(1−τ) + b2 ( 1−τ2 ) + b(2n+ 2ν2 + ν3 + 1)(1−τ), (2.21) cf. (2.7), where ∆g = (τ2 − 1) d2 dτ2 + τ d dτ is the flat Laplace–Beltrami operator with metric g11 = (τ2 − 1). Overall multiplicative factor β2 in (2.20) is dropped off. In the Hamiltonian (2.20) the (n+ 1) eigenfunctions are of a form Pn(cos(βx)) ∣∣ sin(βx) ∣∣ν2 ∣∣∣∣sin(β2x )∣∣∣∣ν3 e−b cos(βx), where Pn(τ) is a polynomial of degree n, they can be found by algebraic means. It is evident that i (n) par(τ) (2.14) remains the particular integral – π-integral of the BC1-trigonometric QES Hamiltonian (2.19) (see [22])[ h (qes) BC1 (τ), i(n) par(τ) ] : Pn 7→ 0. From Quantum AN (Sutherland) to E8 Trigonometric Model: Space-of-Orbits View 9 Interestingly, the BC1-trigonometric QES Hamiltonian (2.20) degenerates to the so-called Magnus–Winkler (MW) Hamiltonian or, in other words, to the QES Lame Hamiltonian (see e.g. [23]) H(qes) BC1 = − d2 dx2 + b2β2sin2 βx+ 2bβ2(2n+ ν + 1) sin2 βx 2 , where ν = 0, 1. For ν = 0 and given n there exist two families of eigenfunctions ϕ (0,+) n,i = Pn(cos(βx))e−b cos(βx), i = 0, 1, . . . , n, ϕ (0,−) n−1,i = Pn−1(cos(βx)) sin(βx)e−b cos(βx), i = 0, 1, . . . , n− 1, which correspond to periodic (anti-periodic) boundary conditions, correspondingly. These eigen- functions describe lower (upper) edges of Brillouin zones, respectively. Polynomial factors in ϕ (0,+) n,i and ϕ (0,−) n−1,i are eigenfunctions of h (qes,0,+) BC1 = J0 nJ 0 n − J−J− − 2bJ+ n + (2n+ 1)J0 n − 2bJ− + n(n+ 1), h (qes,0,−) BC1 = J0 n−1J 0 n−1 − J−J− − 2bJ+ n−1 + (2n+ 1)J0 n−1 − 2bJ− + n(n+ 2), respectively (see (2.12)). For ν = 1 and given n there also exist two families of eigenfunctions ϕ (1,−) n,i = Pn(cos(βx)) sin ( β 2 x ) e−b cos(βx), i = 0, 1, . . . , n, ϕ (1,+) n,i = Pn(cos(βx)) cos ( β 2 x ) e−b cos(βx), i = 0, 1, . . . , n, which correspond to (anti)-periodic boundary conditions, correspondingly. These eigenfunc- tions describe upper (lower) edges of Brillouin zones, respectively. Polynomial factors in ϕ (1,−) n,i and ϕ (1,+) n,i are eigenfunctions of h (qes,1,−) BC1 = J0 nJ 0 n − J−J− − 2bJ+ n + 2(n+ 1)J0 n + (1− 2b)J− + n(n+ 2), h (qes,1,+) BC1 = J0 nJ 0 n − J−J− − 2bJ+ n + 2(n+ 1)J0 n − (1 + 2b)J− + n(n+ 2), respectively (see (2.12)). If in a construction (2.15) to obtain the TTW model we replace the BC1-trigonometric Hamiltonian HBC1(φ) (2.3) by the BC1-trigonometric QES Hamiltonian H(qes) BC1 (φ) (2.19) H(qes) TTW(r, φ;ω, ν2, ν3, β) = −∂2 r − 1 r ∂r + ω2r2 + H(qes) BC1 (φ) r2 , a new quasi-exactly-solvable extension of the TTW model is obtained H̃(qes) TTW(r;φ,m;ω, ν2, ν3, β, b) = −∆(2) + ω2r2 + ν2(ν2 − 1)β2 r2 sin2 βφ + ν3(ν3 + 2ν2 − 1)β2 4r2 sin2 βφ 2 + b2β2sin2 βφ r2 + 2bβ2(2m+ 2ν2 + ν3 + 1) sin2 βφ 2 r2 , cf. (2.15), where ∆(2) is 2D Laplacian, b, ν2, ν3, β are parameters, m is non-negative integer. 10 A.V. Turbiner .. . . x x x x 1 2 3 N Figure 1. N -body Sutherland model. If in the construction (2.15) instead of two-dimensional radial harmonic oscillator, the radial Hamiltonian of the sextic QES 2D radial potential [23] is taken and the BC1-trigonometric Hamiltonian HBC1(φ) (2.3) is replaced by the BC1-trigonometric QES Hamiltonian H(qes) BC1 (φ) (2.19) the most general quasi-exactly-solvable extension of the TTW model occurs Ĥ(qes) TTW(r, n;φ,m;ω, ν2, ν3, β, a, b) = −∆(2) + a2r6 + 2aωr4 + [ ω2 − 2a(2n+ 2 + β(ν2 + ν3)) ] r2 + ν2(ν2 − 1)β2 r2 sin2 βφ + ν3(ν3 + 2ν2 − 1)β2 4r2 sin2 βφ 2 + b2β2sin2 βφ r2 + 2bβ2(2m+ 2ν2 + ν3 + 1) sin2 βφ 2 r2 , (2.22) where n, m is non-negative integer and a > 0, b are parameters. In this Hamiltonian a finite number of eigenstates can be found explicitly (algebraically). Their eigenfunctions have the form of a polynomial p ( r2, cos(βφ) ) of degree n in r2 and of degree m in cos(βx) multiplied by a factor Ψ̂ (qes,TTW) 0 (r, φ) = r(ν2+ν3)β ∣∣ sin(βφ) ∣∣ν2 ∣∣∣∣sin(β2φ )∣∣∣∣ν3 e−ωr22 −ar44 −b cos(βφ), (2.23) cf. (2.16), namely, Ψ̂ (qes,TTW) alg (r, φ) = p ( r2, cos(βφ) ) Ψ̂ (qes,TTW) 0 (r, φ). The factor (2.23) becomes the ground state eigenfunction of the Hamiltonian (2.22) at n=m=0. 2.4 Case AN−1 This is the celebrated Sutherland model (AN−1 trigonometric model) which was found in [18]. It describes N identical particles on a circle (see Fig. 1) with singular pairwise interaction ∝ 1 h2 where h is the horde. The Hamiltonian is HSuth = −1 2 N∑ k=1 ∂2 ∂x2 k + gβ2 4 N∑ k<l 1 sin2 (β 2 (xk − xl) ) , (2.24) where g is the coupling constant and β is a parameter. The symmetry of the system is SN⊕T⊕Z2 (permutations xi → xj , translation xi → xi + 2π/β and all xi → −xi). The ground state of the Hamiltonian (2.24) reads Ψ0(x) = ∏ i<j ∣∣∣∣sin2 ( β 2 (xi − xj) )∣∣∣∣ν , g = ν(ν − 1) ≥ −1 4 (2.25) From Quantum AN (Sutherland) to E8 Trigonometric Model: Space-of-Orbits View 11 (cf. (1.3)). Let us make the gauge rotation hSuth = 2 β2 Ψ−1 0 (HSuth − E0)Ψ0, where E0 is the ground state energy. Then introduce center-of-mass variables Y = ∑ xi, yi = xi − 1 N Y, i = 1, . . . , N, here N∑ i=1 yi = 0, and then new permutationally-symmetric, translationally-invariant, periodic relative variables [16] (x1, x2, . . . xN )→ ( Y, τn(x) = σn ( eiβy(x) ) ∣∣n = 1, 2, 3, . . . , (N − 1) ) , (2.26) where σk(x) = ∑ i1<i2<···<ik xi1xi2 · · ·xik , σk(−x) = (−)kσk(x), are elementary symmetric polynomials, and τ0 = τN (x) = 1, τk(x) = 0, k < 0 or k > N. The ground state function (2.25) in τ -variables takes a form of a polynomial in some power, e.g. Ψ (A2) 0 (x) = ( 4τ3 1 + 4τ3 2 − 18τ1τ2 − τ2 1 τ 2 2 + 27 ) ν 2 . After the center-of-mass separation, the gauge rotated Hamiltonian takes the algebraic form [16] hSuth = N−1∑ i,j=1 Aij(τ) ∂2 ∂τi∂τj + N−1∑ i=1 Bi(τ) ∂ ∂τi , (2.27) where Aij = (N − i)j N τiτj + ∑ l≥max(1,j−i) (j − i− 2l)τi+lτj−l, Bi = ( 1 N + ν ) i(N − i)τi, Eigenvalues of the gauge-rotated Hamiltonian (2.27) are Nε{p} = νN N−1∑ i=1 i(N − i)pi + N−1∑ i,j=1 (N − i)jpipj , being quadratic in quantum numbers {p1, p2, . . . , p(N−1)} where p1, p2, . . . , p(N−1) = 0, 1, 2, . . .. It is easy to check that the gauge-rotated Hamiltonian hSuth has infinitely many finite- dimensional invariant subspaces P(N−1) n = 〈 τ1 p1τ2 p2 · · · τ(N−1) pN−1 \| 0 ≤ ∑ pi ≤ n 〉 . (2.28) where n = 0, 1, 2, . . .. As a function of n the spaces P(N−1) n form the infinite flag (see below). 12 A.V. Turbiner 2.4.1 The gld+1-algebra acting by 1st order differential operators in Rd It can be checked by the direct calculation that the gld+1 algebra realized by the first-order differential operators acting in Rd in the representation given by the Young tableaux as a row (n, 0, 0, . . . , 0︸︷︷︸ d−1 ) has a form J −i = ∂ ∂τi , i = 1, 2, . . . , d, Jij0 = τi ∂ ∂τj , i, j = 1, 2, . . . , d, J 0 = d∑ i=1 τi ∂ ∂τi − n, J + i = τiJ 0 = τi  d∑ j=1 τj ∂ ∂τj − n  , i = 1, 2, . . . , d, (2.29) where n is an arbitrary number. The total number of generators is (d + 1)2. If n takes the integer values, n = 0, 1, 2, . . ., the finite-dimensional irreps occur P(d) n = 〈 τ1 p1τ2 p2 · · · τdpd \| 0 ≤ ∑ pi ≤ n 〉 (cf. (2.28)). It is a common invariant subspace for (2.29). The spaces P(d) n at n = 0, 1, 2, . . . can be ordered P(d) 0 ⊂ P(d) 1 ⊂ P(d) 2 ⊂ · · · ⊂ P(d) n ⊂ · · · ⊂ P(d). (2.30) Such a nested construction is called infinite flag (filtration) P(d). It is worth noting that the flag P(d) is made out of finite-dimensional irreducible representation spaces P(d) n of the algebra gld+1 taken in realization (2.29). It is evident that any operator made out of generators (2.29) has finite-dimensional invariant subspace which is finite-dimensional irreducible representation space. 2.4.2 Algebraic properties of the Sutherland model It seems evident that the Hamiltonian (2.27) has to have a representation as a second order polynomial in generators (2.29) at d = N − 1 acting in RN−1, hSuth = Pol2 ( J −i ,Jij 0 ) , where the raising generators J + i are absent. Thus, gl(N) (or, strictly speaking, its maximal affine subalgebra) is the hidden algebra of the N -body Sutherland model. Hence, hSuth is an element of the universal enveloping algebra Ugl(N). The eigenfunctions of the N -body Sutherland model are elements of the flag of polynomials P(N−1). Each subspace P(N−1) n is represented by the Newton polytope (pyramid). It contains CN−1 n+N−1 eigenfunctions, which is equal to the volume of the Newton polytope. They are orthogonal with respect to Ψ2 0, see (2.25). The Hamiltonian (2.24) is completely-integrable: there exists a commutative algebra of in- tegrals (including the Hamiltonian and the momentum of the center-of-mass motion) of dimen- sion N which is equal to the dimension of the configuration space (for integrals, see Oshima [14] with explicit forms of those). Each integral Ik has a form polynomial in momentum of degree k ≤ N . Making gauge rotation with Ψ2 0, separating center-of-mass motion and changing vari- able to (2.26) any integral appears in a form differential operator with polynomial coefficients. Evidently, it preserves the flag of polynomials (2.30) and can be written as a non-linear combi- nation of the generators (2.29) at d = N −1 from its affine subalgebra. The explicit formulae of integrals in (2.29) are unknown. The spectra of the integral which is a polynomial in momentum From Quantum AN (Sutherland) to E8 Trigonometric Model: Space-of-Orbits View 13 of degree k is given by a polynomial in quantum numbers of the degree k. All eigenfunctions of the integrals are common. Among the generators of the hidden algebra there is the Euler–Cartan operator, J 0 n = N−1∑ i=1 τi ∂ ∂τi − n, see (2.29), which has zero grading and plays a role of constant acting as identity operator on a monomial in τ . It defines the highest weight vector. This generator allows us to construct the particular integral – π-integral of zero grading (see [22]) i(n) par(τ) = n∏ j=0 ( J 0 n + j ) (2.31) such that[ hSuth(τ), i(n) par(τ) ] : P(N−1) n 7→ 0. Making the gauge rotation of the π-integral (2.31) with Ψ−1 0 (τ) given by (2.25) and changing variables τ (see (2.26)) back to the Cartesian coordinates we arrive at the quantum π-integral, I(n) par,Suth(x) = Ψ0(τ)i(n) par(τ)Ψ−1 0 (τ) ∣∣∣ τ→x . It is a differential operator of the (n+ 1)th order. Under such a gauge transformation the triangular space of polynomials P(N−1) n becomes the space V(N−1) n = Ψ0P(N−1) n . The Hamiltonian HSuth(x) commutes with I(n) par,Suth(x) over this space[ HSuth(x), I(n) par,Suth(x) ] : V(N−1) n 7→ 0. Any eigenfunction Ψ ∈ V(N−1) n is zero mode of the π-integral I(n) par,Suth(x). 2.5 Case BCN The BCN -Trigonometric model is defined by the Hamiltonian, HBCN = −1 2 N∑ i=1 ∂2 ∂xi2 + gβ2 4 N∑ i<j [ 1 sin2 (β 2 (xi − xj) ) + 1 sin2 (β 2 (xi + xj) )] + g2β 2 2 N∑ i=1 1 sin2 βxi + g3β 2 8 N∑ i=1 1 sin2 βxi 2 , (2.32) where β, g, g2, g3 are parameters. Symmetry: SN ⊕ (Z2)⊗N ⊕ T (permutations xi → xj , reflections xi → −xi, translation xi → xi + 2π/β). BCN root space contains roots of the three lengths: 1, √ 2, 2. The BCN fundamental weights coincide to the CN fundamental weights. The ground state function for (2.32) reads Ψ0 = ∏ i<j ∣∣∣∣sin(β2 (xi−xj) )∣∣∣∣ν ∣∣∣∣sin(β2 (xi+xj) )∣∣∣∣ν  N∏ i=1 ∣∣ sin(βxi) ∣∣ν2 ∣∣∣∣sin(β2xi )∣∣∣∣ν3 , (2.33) 14 A.V. Turbiner cf. (1.3), where ν, ν2, ν3 are found from the relations g = ν(ν − 1) > −1 4 , g2 = ν2(ν2 − 1) > −1 4 , g3 = ν3(ν3 + 2ν2 − 1) > −1 4 . Any eigenfunction has a form Ψ0ϕ, where ϕ is a polynomial in the CN fundamental trigonometric invariants τ(β) (2.1). Hence, Ψ0 plays a role of multiplicative factor. The BCN Hamiltonian (2.32) degenerates to the BN Hamiltonian at g2 = 0, to the CN Hamiltonian at g3 = 0 and to the DN Hamiltonian at g2 = g3 = 0. For the BN Hamiltonian there exist two families of eigenfunctions with multiplicative factors Ψ (1) 0,BN = ∏ i<j ∣∣∣∣sin(β2 (xi − xj) )∣∣∣∣ν ∣∣∣∣sin(β2 (xi + xj) )∣∣∣∣ν [∣∣∣∣sin(β2xi )∣∣∣∣ν3] , and Ψ (2) 0,BN = ∏ i<j ∣∣∣∣sin(β2 (xi − xj) )∣∣∣∣ν ∣∣∣∣sin(β2 (xi + xj) )∣∣∣∣ν [ N∏ i=1 ∣∣ sin(βxi) ∣∣][∣∣∣∣sin(β2xi )∣∣∣∣ν3], respectively. For the DN Hamiltonian there exist three families of eigenfunctions with multi- plicative factors Ψ (1) 0,DN = ∏ i<j ∣∣∣∣sin(β2 (xi − xj) )∣∣∣∣ν ∣∣∣∣sin(β2 (xi + xj) )∣∣∣∣ν  , Ψ (2) 0,DN = ∏ i<j ∣∣∣∣sin(β2 (xi − xj) )∣∣∣∣ν ∣∣∣∣sin(β2 (xi + xj) )∣∣∣∣ν [ N∏ i=1 ∣∣ sin(βxi) ∣∣] , Ψ (3) 0,DN = ∏ i<j ∣∣∣∣sin(β2 (xi − xj) )∣∣∣∣ν ∣∣∣∣sin(β2 (xi + xj) )∣∣∣∣ν [∣∣∣∣sin(β2xi )∣∣∣∣] , respectively. Let us make a gauge rotation hBCN = 1 β2 (Ψ0)−1(HBCN − E0)Ψ0, and then change variables [4] (x1, x2, . . . , xN )→ ( τk = σk(cosβx) ∣∣ k = 1, 2, . . . , N ) , (2.34) where σk is the elementary symmetric polynomial, τ0 = 1 and τk = 0 for k < 0 and k > N . It can be checked that τk are CN trigonometric invariants with period 2π β . We arrive at [4] hBCN = N∑ i,j=1 Aij(σ) ∂2 ∂σi∂σj + N∑ i=1 Bi(σ) ∂ ∂σi , (2.35) with coefficients Aij = −Nτi−1τj−1 + ∑ l≥0 [ (i− l)τi−lτj+l + (l + j − 1)τi−l−1τj+l−1 From Quantum AN (Sutherland) to E8 Trigonometric Model: Space-of-Orbits View 15 − (i− 2− l)τi−2−lτj+l − (l + j + 1)τi−l−1τj+l+1 ] , (2.36) Bi = [ 1 + ν(2N − i− 1)+ 2ν2 + ν3 ] iτi− ν3(i−N − 1)τi−1+ ν(N − i+ 1)(N − i+ 2)τi−2, cf. (2.9). This is an algebraic form of the BCN trigonometric Hamiltonian. For polynomial eigenfunctions we find the eigenvalues are ε{p} = N∑ i=1 [ ν(2N − i− 1) + 2ν2 + ν3 ] ipi + N∑ i,j=1 ipipj , cf. (2.10), hence, the spectrum is quadratic in quantum numbers pi = 0, 1, . . ., where i = 1, 2, . . . , N . The Hamiltonian hBCN has infinitely many finite-dimensional invariant subspaces of the form P(N) n , see (2.28), where n = 0, 1, 2, . . .. They naturally form the flag P(N), see (2.30). The Hamiltonian can be immediately rewritten in terms of generators (2.29) at d = N as a polynomial of the second degree, hBCN = Pol2 ( J −i ,Jij 0 ) , where the raising generators J + i are absent. Hence, gl(N + 1) is the hidden algebra of the BCN trigonometric model, the same algebra as for the AN -rational model. The eigenfunctions of the BCN trigonometric model are elements of the flag of polynomials P(N). Each subspace P(N) n contains CNn+N eigenfunctions (volume of the Newton polytope (pyramid) P(N) n ). They are orthogonal with respect to Ψ2 0, see (2.33). The rational form (2.2) of the BCN trigonometric Hamiltonian (2.32) can be derived making the gauge rotation of the algebraic form (2.35) with inverse of the ground state function in τ -variables, (Ψ0(τ))−1, HBCN (τ) = −∆g + VBCN (τ), where ∆g is the Laplace–Beltrami operator with a metric gij(τ) = Aij (see (2.36)) and VBCN (τ) is a potential. The explicit expression for VBC1(τ) is presented in (2.7) while the ground state eigenfunction Ψ (BC1) 0 (τ) is given by (2.8). The configuration space in τ coordinate is the interval, τ ∈ [−1, 1] (trigonometric case) or half-line, τ ∈ [1,∞) (hyperbolic case). As for BC2 case, VBC2(τ) = g 1− τ2 τ2 1 − 4τ2 + g2 4 2− τ1 1 + τ1 + τ2 + 1 4 2(g2 + g3) + g2τ1 − g3τ2 1− τ1 + τ2 , (2.37) and Ψ (BC2) 0 (τ) = ( τ2 1 − 4τ2 ) ν 2 (1 + τ1 + τ2) ν2 2 (1− τ1 + τ2) ν2+ν3 2 , and the configuration space is illustrated by Fig. 2. As for BC3 VBC3(τ) = g τ4 1 − τ3 1 τ3 − 6τ2 1 τ2 + 9τ1τ2τ3 + 9τ2 2 − τ3 2 − 27τ2 3 τ2 1 τ 2 2 − 4τ3 1 τ3 − 4τ2 2 − 27τ2 3 + 18τ1τ2τ3 + g2 2 3 + 2τ1 + τ2 1 + τ1 + τ2 + τ3 + g2 + 4g3 4 3− 2τ1 + τ2 1− τ1 + τ2 − τ3 , (2.38) and Ψ (BC3) 0 (τ) = ( τ2 1 τ 2 2 − 4τ3 1 τ3 − 4τ2 2 − 27τ2 3 + 18τ1τ2τ3 ) ν 2 × (1 + τ1 + τ2 + τ3) ν2 2 (1− τ1 + τ2 − τ3) ν2+ν3 2 . 16 A.V. Turbiner Figure 2. An illustration of the configuration space for BC2 trigonometric model in τ -variables (light brown area) and for BC2 hyperbolic model (light blue area on the right). The Hamiltonian (2.32) is completely-integrable: there exists a commutative algebra of in- tegrals (including the Hamiltonian) of dimension N which is equal to the dimension of the configuration space (for integrals, see Oshima [14] with explicit forms of those). Each integ- ral Ik has a form polynomial in momentum of degree 2k ≤ 2N . Making gauge rotation with Ψ2 0 and changing variable to (2.26) any integral appears in a form differential operator with poly- nomial coefficients. Evidently, it preserves the flag of polynomials (2.30) and can be written as a non-linear combination of the generators (2.29) at d = N from its affine subalgebra. The explicit formulae of integrals in generators (2.29) are unknown. The spectra of the integral which is a polynomial in momentum of degree 2k is given by a polynomial in quantum numbers of the degree 2k. All eigenfunctions of the integrals are common. It is evident that for the BCN trigonometric model there exists a particular integral – π- integral of zero grading (see [22]) i(n) par(τ) = n∏ j=0 ( J 0 n + j ) (cf. (2.31)), such that[ hBCN (τ), i(n) par(τ) ] : P(N) n 7→ 0. Making the gauge rotation of the π-integral (2.31) with Ψ−1 0 (τ) given by (2.33) and changing variables τ (see (2.34)) back to the Cartesian coordinates we arrive at the quantum π-integral, I(n) par,BCN (x) = Ψ0(τ)i(n) par(τ)Ψ−1 0 (τ) ∣∣ τ→x. It is a differential operator of the (n+ 1)th order. Under such a gauge transformation the triangular space of polynomials P(N) n becomes the space V(N) n = Ψ0P(N) n . The Hamiltonian HBCN (x) commutes with I(n) par,BCN (x) over this space[ HBCN (x), I(n) par,BCN (x) ] : V(N) n 7→ 0. Any eigenfunction Ψ ∈ V(N) n is zero mode of the π-integral I(n) par,BCN (x). From Quantum AN (Sutherland) to E8 Trigonometric Model: Space-of-Orbits View 17 Now we are in a position to draw an intermediate conclusion about AN and BCN trigono- metric models. • Both AN - and BCN -trigonometric (and rational) models possess algebraic forms associated with preservation of the same flag of polynomials P(N). The flag is invariant with respect to linear transformations in space of orbits τ 7→ τ + A. It preserves the algebraic form of Hamiltonian. • Their Hamiltonians (as well as higher integrals) can be written in the Lie-algebraic form h = Pol2(J (b ⊂ glN+1)), where Pol2 is a polynomial of 2nd degree in generators J of the maximal affine subalgebra of the algebra b of the algebra glN+1 in realization (2.29). Hence, glN+1 is their hidden algebra. From this viewpoint all four models are different faces of a single model. • Supersymmetric AN - and BCN -rational (and trigonometric) models possess algebraic forms, preserve the same flag of (super)polynomials and their hidden algebra is the su- peralgebra gl(N + 1\|N) (see [4]). In a connection to flags of polynomials we introduce a notion ‘characteristic vector’. Let us consider a flag made out of “triangular” linear space of polynomials P(d) n,~f = 〈xp11 x p2 2 · · ·x pd d \| 0 ≤ f1p1 + f2p2 + · · ·+ fdpd ≤ n〉, where the “grades” f ’s are positive integer numbers and n = 0, 1, 2, . . .. In lattice space P(d) n,~f defines a Newton pyramid. Definition 1. Characteristic vector is a vector with components fi: ~f = (f1, f2, . . . , fd). From geometrical point of view ~f is normal vector to the base of the Newton pyramid. The characteristic vector for flag P(d) is ~f0 = (1, 1, . . . , 1)︸︷︷︸ d . 2.6 Case G2 Take the Hamiltonian HG2 = −1 2 3∑ k=1 ∂2 ∂x2 k + gβ2 4 3∑ k<l 1 sin2(β2 (xk − xl)) + g1β 2 4 3∑ k<lk, l 6=m 1 sin2(β2 (xk + xl − 2xm)) , where g, g1 and β are parameters. It describes a trigonometric generalization of the rational Wolfes model of three-body interacting system or, in the Hamiltonian reduction nomenclature, the G2-trigonometric model [13]. The symmetry of the model is dihedral group D6 ⊕ T . The ground state function is Ψ0 = 3∏ i<j ∣∣∣∣sin β2 (xi − xj) ∣∣∣∣ν 3∏ k<lk, l 6=m ∣∣∣∣sin β2 (xi + xj − 2xk) ∣∣∣∣µ 18 A.V. Turbiner with ν, µ > −1 2 as solutions of g = ν(ν − 1) > −1 4 , g1 = 3µ(µ− 1) > −3 4 . Making the gauge rotation hG2 = (Ψ0)−1(HG2 − E)Ψ0, and changing variables [15] Y = ∑ xi, yi = xi − 1 3 Y, i = 1, 2, 3, (x1, x2, x3)→ ( Y, τ1, τ2 ) , where τ1 = 2[cos(β(y1 − y2)) + cos(β(2y1 + y2)) + cos(β(y1 + 2y2))], τ2 = 2[cos(3βy1) + cos(3βy2) + cos(3β(y1 + y2))] are G2 trigonometric invariants, and separating the center-of-mass coordinate we arrive at [15] hG2 = − ( 4 + τ1 + τ2 3 − τ2 1 3 ) ∂2 τ1τ1 + ( 12 + 4τ2 + τ1τ2 − 2τ2 1 ) ∂2 τ1τ2 + ( 9τ1 + 3τ2 + 3τ1τ2 + τ2 2 − τ3 1 ) ∂2 τ2τ2 + [ 2ν + 1 + 3µ+ 2ν 3 τ1 ] ∂τ1 + [ 6µ+ (1 + 2µ+ ν)τ2 + 2ντ1 ] ∂τ2 , (2.39) which is the algebraic form of the G2 trigonometric Hamiltonian. The eigenvalues of hG2 are ε{p} = p2 1 3 + p1p2 + p2 2 + (µ+ ν)p1 + (2µ+ ν)p2 quadratic in quantum numbers p1, p2 = 0, 1, 2, . . .. The Hamiltonian hG2 has infinitely many finite-dimensional invariant subspaces P(2) n,(1,2) = 〈τ1 p1τ2 p2 \| 0 ≤ p1 + 2p2 ≤ n〉, n = 0, 1, 2, . . . , (2.40) hence the flag P(2) (1,2) with the characteristic vector ~f = (1, 2) is preserved by hG2 . The eigen- functions of hG2 are are elements of the flag P(2) (1,2). Each space (P(2) n,(1,2) P (2) n−1,(1,2)) contains ∼n eigenfunctions which is equal to length of the Newton line Ln = 〈τ1 p1τ2 p2 \|p1 + 2p2 = n〉. A natural question to ask whether does an algebra of differential operators exist for which P(2) n,(1,2) is the space of (irreducible) representation. We call this algebra g(2) [15]. 2.7 Algebra g(2) Let us consider the Lie algebra spanned by seven generators J1 = ∂t, J2 n = t∂t − n 3 , J3 n = 2u∂u − n 3 , J4 n = t2∂t + 2tu∂u − nt, Ri = ti∂u, i = 0, 1, 2, R(2) ≡ (R0, R1, R2). (2.41) It is non-semi-simple algebra gl(2,R) nR(2) (S. Lie [11, p. 767–773] at n = 0 and A. González- Lopéz et al. [9] at n 6= 0 (case 24)). If the parameter n in (2.41) is a non-negative integer, it has (2.40) P(2) n = ( tpuq \| 0 ≤ (p+ 2q) ≤ n ) , From Quantum AN (Sutherland) to E8 Trigonometric Model: Space-of-Orbits View 19 Figure 3. Triangular diagram relating the subalgebras L, U and g`2. P2(g`2) is a polynomial of the 2nd degree in g`2 generators. It is a generalization of the Gauss decomposition for semi-simple algebras. as common (reducible) invariant subspace. By adding three operators T0 = u∂2 t , T1 = u∂tJ (n) 0 , T2 = uJ (n) 0 ( J (n) 0 + 1 ) = uJ (n) 0 J (n−1) 0 , (2.42) where J (n) 0 = t∂t + 2u∂u − n, to gl(2,R)nR(2) (see (2.41)), the action on P(2) n,(1,2) gets irreducible. Multiple commutators of J4 n with T (2) 0 generate new operators acting on P(2) n,(1,2), Ti ≡ [J4, [J4, [. . . J4, T0] . . .]︸︷︷︸ i = u∂2−i t J (n) 0 ( J (n) 0 + 1 ) · · · (J (n) 0 + i− 1) = u∂2−i t i−1∏ j=0 J (n−j) 0 , i = 0, 1, 2, all of them are differential operators of degree 2. These new generators have a property of nilpotency, Ti = 0, i > 2, and commutativity: [Ti, Tj ] = 0, i, j = 0, 1, 2, U (2) ≡ (T0, T1, T2). The generators (2.41) plus (2.42) span a linear space with a property of decomposition: g(2) . = R(2) o (gl2 ⊕ J0) n U (2) (see Fig. 3). It is worth mentioning a property of conjugation R(2) ⇔ T (2): ∂τ2 ↔ τ2J (n) 0 ( J (n) 0 + 1 ) , τ1∂τ2 ↔ τ2∂τ1J (n) 0 , τ1 2∂τ2 ↔ τ2∂ 2 τ1 . where J (n) 0 = τ1∂τ1 + 2τ2∂τ2 − n. Eventually, infinite-dimensional, eleven-generated algebra (by (2.41) and J0 plus (2.42), so that the eight generators are the 1st order and three generators are of the 2nd order differential operators) occurs. The Hamiltonian hG2 can be rewritten in terms of the generators (2.41), (2.42) with the absence of the highest weight generator J4 n, hG2 = − ( 4J1 + J2 − 2J3 − 12R0 + 2R2 ) J1 + 1 6 ( 2J2 + 3J3 ) J2 + ( J3 + 3 2 R1 ) J3 + (9R0 −R2)R1 − 1 3 T0 + 2νJ1 + 3µ+2ν 3 J2 + 2µ+ν−1 2 J3 + 6µR0 + ( 2ν − 3 2 ) R1 20 A.V. Turbiner (see [15]), where J2,3 ≡ J2,3 0 . Hence, gl(2,R)nR(2) is the hidden algebra of the G2 trigonometric model. The G2 trigonometric Hamiltonian admits the integral in a form of the 6th order differential operator [14]. After gauge rotation with Ψ0 in variables τ1,2 the integral has to take the alge- braic form which is not known explicitly. This integral preserves the same flag P(2) (1,2) as the Hamiltonian (2.39). It can be rewritten in term of generators of the algebra g(2). In addition to it, there exists π-integral of zero grading (see [22]) i(n) par(τ) = n∏ j=0 ( J (n) 0 + j ) = n∏ j=0 J (n−j) 0 (cf. (2.31)), such that[ hG2(τ), i(n) par(τ) ] : P(2) n,(1,2) 7→ 0. Making the gauge rotation of the π-integral (2.31) with Ψ−1 0 (τ) given by (2.33) and changing variables τ (see (2.34)) back to the Cartesian coordinates we arrive at the quantum π-integral, I(n) par,G2 (x) = Ψ0(τ)i(n) par(τ)Ψ−1 0 (τ) ∣∣ τ→x. It is a differential operator of the (n+ 1)th order. Under such a gauge transformation the triangular space of polynomials P(2) n,(1,2) becomes the space V(N) n = Ψ0P(2) n,(1,2). The Hamiltonian HG2(x) commutes with I(n) par,G2 (x) over this space[ HG2(x), I(n) par,G2 (x) ] : V(N) n 7→ 0. Any eigenfunction Ψ ∈ V(N) n is zero mode of the π-integral I(n) par,G2 (x). Summarizing let us mention that in addition to the flag P(2) (1,2) the G2 trigonometric Hamilto- nian preserves two more flags: P(3,5) and P(5,9), where their characteristic vectors (3, 5) and (5, 9) coincide to the Weyl vector and co-vector, respectively. 2.8 Cases F4 and E6,7 These three cases are described in some details in [2, 12] and in [3, p. 1416], respectively. 2.9 Case E8 (in brief) In this Section a brief description of E8 trigonometric case is given, all details can be found in [3]. The E8 trigonometric Hamiltonian has a form (1.2), HE8 ( β 2 ) = −1 2 ∆(8) + gβ2 4 8∑ j<i=1 [ 1 sin2 β 2 (xi + xj) + 1 sin2 β 2 (xi − xj) ] + gβ2 4 ∑ {νj} 1[ sin2 β 4 ( x8 + 7∑ j=1 (−1)νjxj )] , (2.43) From Quantum AN (Sutherland) to E8 Trigonometric Model: Space-of-Orbits View 21 and it acts in R8. The second summation being one over septuples {νj} where each νj = 0, 1 and 7∑ j=1 νj is even. Here g = ν(ν−1) > −1/4 is the coupling constant and β is a parameter. The configuration space is the principal E8 Weyl alcove. Symmetry of the E8 trigonometric model is given by the affine E8 Weyl group of the order 696 729 600. The ground state function Ψ0 is given by (1.3). Making a gauge rotation of the Hamiltonian hE8 = 1 β2 (Ψ0)−1(HE8 − E0)Ψ0, where E0 = 310β2ν2 is the ground state energy, and introducing the new variables τ1,...,8(β), which are the fundamental trigonometric invariants with respect to the E8 Weyl group, we arrive at the E8 trigonometric Hamiltonian in the algebraic form hE8 = 4∑ i,j=1 Aij(τ) ∂2 ∂τi∂τj + 4∑ j=1 Bj(τ, ν) ∂ ∂τj , (2.44) where Aij(τ), Bj(τ ; ν) are polynomials in τ with integer coefficients and Bj(τ ; ν) depend on ν linearly (see [3, Appendix A]). It is easy to check that the algebraic operator hE8 has infinitely-many finite-dimensional invariant subspaces P(2,2,3,3,4,4,5,6) n = 〈τn1 1 τn2 2 τn3 3 τn4 4 τn5 5 τn6 6 τn7 7 τn8 7 \| 0 ≤ 2n1 + 2n2 + 3n3 + 3n4 + 4n5 + 4n6 + 5n7 + 6n8 ≤ n〉, n ∈ N, all of them have with the same characteristic vector ~f = (2, 2, 3, 3, 4, 4, 5, 6), they form the infinite flag. The spectrum of the Hamiltonian hE8 (2.44) is quadratic in quantum numbers [3, 10]. Eigenfunctions φn,{p} of hE8 are elements of P(2,2,3,3,4,4,5,6) n . The number of eigenfunctions in P(2,2,3,3,4,4,5,6) n is equal to the dimension of P(2,2,3,3,4,4,5,6) n . The space P(2,2,3,3,4,4,5,6) n is a finite-dimensional representation space of a Lie algebra of differential operators which we call the e(8) algebra [6]. It is infinite-dimensional but finitely generated algebra of differential operators, with 968 generating elements in a form of differential operators of the orders 1st (54), 2nd (24), 3rd (18), 4rd (18), 5rd (28), 6rd (5) plus one of zeroth order (constant). They span 100 + 100 Abelian (conjugated) subalgebras of lowering and raising generators2 L and U and one algebra B of the Cartan type of dimension 15 plus one central element. Among the generators of B there is the Euler–Cartan operator J (n) 0 = 2τ1∂τ1 + 2τ2∂τ2 + 3τ3∂τ3 + 3τ4∂τ4 + 4τ5∂τ5 + 4τ6∂τ6 + 5τ7∂τ7 + 6τ8∂τ8 − n. (2.45) Taking the algebra B and a pair of conjugated Abelian algebras one can show that the commu- tation relations lead to the diagram of Fig. 4. Depending on what pair L, U the degree p takes the following values: 2, 3, 4, 5, 6, 7, 8, 9, 10. The E8 trigonometric model is completely-integrable – there exist seven algebraically in- dependent mutually commuting differential operators of finite order that commute with the Hamiltonian (2.43) [10, 13]. We are not aware on the existence of their explicit forms. It seems evident that any of these integrals after the gauge rotation with the ground state function Ψ0 the space of orbits should take an algebraic form of a differential operator with polynomial 2It implies that these commutative subalgebras can be divided into pairs. In every pair the elements of different subalgebras are related via a certain operation of conjugation similar to one described for g(2) on p. 18. 22 A.V. Turbiner Figure 4. Triangular diagram relating the subalgebras L, U and B. Pp(B) is a polynomial of the pth degree in B generators. It is a generalization of the Gauss decomposition for semi-simple algebras. coefficient functions. Any integral as well as the Hamiltonian is an element of the algebra e(8). In addition to “global” integrals, there exists π-integral of zero grading (see [22]) i(n) par(τ) = n∏ j=0 ( J (n) 0 + j ) = n∏ j=0 J (n−j) 0 , where J (n) 0 is given by (2.45) (cf. (2.31)) such that[ hE8(τ), i(n) par(τ) ] : P(2,2,3,3,4,4,5,6) n 7→ 0. It is worth mentioning that the operator (2.44) has a certain property of degeneracy: it also preserves the infinite flag of the spaces of polynomials with the characteristic vector ~f = (29, 46, 57, 68, 84, 91, 110, 135). This vector coincides to the E8 Weyl (co)vector. Hence, the eigenfunctions of hE8(τ) are the elements of this flag as well. It implies the existence of another π-integral ĩ (n) par(τ) with J (n) 0 given by J (n) 0 = 29τ1∂τ1 + 46τ2∂τ2 + 57τ3∂τ3 + 68τ4∂τ4 + 84τ5∂τ5 + 91τ6∂τ6 + 110τ7∂τ7 + 135τ8∂τ8− n, such that[ hE8(τ), ĩ(n) par(τ) ] : P(29,46,57,68,84,91,110,135) n 7→ 0. 3 Conclusions • For trigonometric Hamiltonians for all classical AN , BCN , BN , CN , DN and for ex- ceptional root spaces G2, F4, E6,7,8, similar to the rational Hamiltonians including non- crystallographic H3,4, I2(k) (see [21]), there exists an algebraic form after gauging away the ground state eigenfunction, and changing variables from Cartesian to fundamental trigonometric Weyl invariants (see [1, 2, 3, 4, 12, 15, 16]). Their eigenfunctions are poly- nomials in these variables. They are orthogonal with respect to the squared ground state eigenfunction. Coefficient functions in front of the second derivatives of these gauge-rotated Hamiltonians which are polynomials in fundamental trigonometric Weyl invariants define a metric A of flat space in the space of orbits. We will call this metric the V.I. Arnold metric, he was the first to calculate a similar metric in the case of polynomial Weyl invariants. This metric has a property that in the Laplace–Beltrami operator the coefficient functions in front of the first derivatives are polynomials in fundamental trigonometric invariants. This property is similar to one which occurs in the case of rational models. The (rational) Arnold metric for the space of orbits parameterized by polynomial Weyl invariants can be considered as an appropriate degeneration of the (trigonometric) Arnold metric for the space of orbits parameterized by fundamental trigonometric Weyl invariants. From Quantum AN (Sutherland) to E8 Trigonometric Model: Space-of-Orbits View 23 Figure 5. Triangular diagram relating the subalgebras L, U and B. Pp(B) is a polynomial of the pth degree in B generators. It is a generalization of the Gauss decomposition for semi-simple algebras where p = 1. Table 1. Minimal characteristic vectors for rational (non)crystallographic and trigonometric crystallo- graphic systems (see [3]). For latter case the Weyl vector and co-vector as possible characteristic vectors occur. Characteristic vectors for H3, H4, I2(k) are from [7, 8, 19], respectively. Model Rational Trigonometric minimal integer Weyl integer co-Weyl AN (1, 1, . . . , 1)︸︷︷︸ N (1, 1, . . . , 1)︸︷︷︸ N BCN (1, 1, . . . , 1)︸︷︷︸ N (1, 1, . . . , 1)︸︷︷︸ N G2 (1,2) (1,2) (3,5) (5, 9) F4 (1,2,2,3) (1,2,2,3) (8,11,15,21) (11,16,21,30) E6 (1,1,2,2,2,3) (1,1,2,2,2,3) (8,8,11,15,15,21) (8,8,11,15,15,21) E7 (1,2,2,2,3,3,4) (1,2,2,2,3,3,4) (27, 34, 49, 52, 66, 75, 96) (27, 34, 49, 52, 66, 75, 96) E8 (1,3,5,5,7,7,9,11) (2,2,3,3,4,4,5,6) (29,46,57,68,84,91,110,135) (29,46,57,68,84,91,110,135) H3 (1,2,3) — H4 (1,5,8,12) — I2(k) (1, k) — • Any trigonometric Hamiltonian is characterized by a hidden algebra. These hidden al- gebras are Ugl(N+1) for the case of classical AN , BCN , BN , CN , DN and new infinite- dimensional but finite-generated algebras of differential operators for all other cases. All these algebras have finite-dimensional invariant subspace(s) in polynomials. Rational Hamiltonians are characterized by the same hidden algebra with a single exception of the E8 case. • The generating elements of any such hidden algebra can be grouped into an even number of (conjugated) Abelian algebras Li, Ui and one Lie algebra B. They obey a (generalized) Gauss decomposition rule (see Fig. 5). A study and a description of all these algebras is in progress and will be given elsewhere. • Any algebraic Hamiltonian h of a trigonometric model preserves one or several flags of invariant subspaces with characteristic vectors given by the highest root vector, the Weyl vector and the Weyl co-vector (see Table 1). With the single exception of the E8 case the flags for rational and trigonometric models coincide. • The original Weyl-invariant periodic Hamiltonian (1.1) written in the fundamental trigono- metric invariants (2.1) corresponds to a particle moving in flat space with (trigonometric) Arnold metric A in a rational potential, H(τ) = −∆A + ∑̀ k gkVk(τ), 24 A.V. Turbiner where ∆A is the Laplace–Beltrami operator, gk, k = 1, . . . , ` are coupling constants, ` is the number of different root lengths in the root space. Vk(τ) are rational functions. So far, we are unaware about the explicit form of the functions Vk(τ) for all root systems except for some particular cases (see (2.7), (2.21), (2.37), (2.38)). • The existence of an algebraic form of the Hamiltonian h of a trigonometric model al- lows us to construct integrable discrete systems in the space of orbits with the same hidden algebra structure, having a property of isospectrality, on uniform, exponential and mixed uniform-exponential lattices following the strategy presented in [17] (uniform lat- tice) and [5] (exponential lattice). • The space of orbits formalism allowed us to show that both rational and trigonometric models for any root system are essentially algebraic: the (appropriately) gauge-rotated Hamiltonians are algebraic operators, their invariant subspaces are spaces of polynomials. A natural question to ask is: How the elliptic Calogero–Moser systems look like in a space of orbits formalism; are they algebraic just like rational and trigonometric systems? 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Phys., Springer, New York, 2000, 473–484, solv-int/9710004. http://dx.doi.org/10.1142/S0217732309031387 http://arxiv.org/abs/0904.0484 http://dx.doi.org/10.1016/0370-1573(83)90018-2 http://dx.doi.org/10.3842/SIGMA.2007.061 http://arxiv.org/abs/math-ph/0502028 http://dx.doi.org/10.1142/S0217751X98001815 http://dx.doi.org/10.1142/S0217751X98001815 http://arxiv.org/abs/hep-th/9606092 http://dx.doi.org/10.1142/S0217732395002374 http://arxiv.org/abs/hep-th/9506105 http://dx.doi.org/10.1142/S0217732395001927 http://dx.doi.org/10.1142/S0217732395003318 http://arxiv.org/abs/funct-an/9501001 http://dx.doi.org/10.1103/PhysRevA.4.2019 http://dx.doi.org/10.1088/1751-8113/42/24/242001 http://arxiv.org/abs/0904.0738 http://dx.doi.org/10.3842/SIGMA.2011.071 http://arxiv.org/abs/1106.5017 http://dx.doi.org/10.1088/1751-8113/46/2/025203 http://arxiv.org/abs/1206.2907 http://dx.doi.org/10.1007/BF01466727 http://arxiv.org/abs/solv-int/9710004 1 Introduction 2 Solvable models 2.1 Generalities 2.2 A1/BC1 case or trigonometric Pöschl-Teller potential 2.3 Quasi-exactly-solvable BC1 case (or QES trigonometric Pöschl-Teller potential) 2.4 Case AN-1 2.4.1 The gld+1-algebra acting by 1st order differential operators in Rd 2.4.2 Algebraic properties of the Sutherland model 2.5 Case BCN 2.6 Case G2 2.7 Algebra g(2) 2.8 Cases F4 and E6,7 2.9 Case E8 (in brief) 3 Conclusions References

From Quantum AN to E₈ Trigonometric Model: Space-of-Orbits View

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