From Quantum $A_N$ (Calogero) to $H_4$ (Rational) Model

Symmetry , Integrabilit y and Geometry: Metho ds and Applications SIGMA 7 (2011), 071, 20 pages F rom Quan tum A N (Calogero ) to H 4 (Rational) Mo del ⋆ Alexa nder V. TURBINER Instituto de Ciencias Nucle ar es, Universidad Nacional A ut´ onoma de M´ exic o, Ap artado Postal 70-543, 04510 M´ exic o, D.F., Mexic o E-mail: turbiner@nucle ar es.unam.mx Received F ebruary 28, 20 11, in f ina l form July 12, 201 1; Published online July 18 , 2011 doi:10.38 42/SIGMA.20 11.071 Abstract. A br ief and inco mplete review of known in tegrable and (quasi)-exa ctly-solv able quantum mo dels with ra tio nal (meromor phic in Cartesian co ordinates) p otentials is given. All of them ar e characterized by ( i ) a discrete s ymmetry of the Hamiltonian, ( ii ) a num ber of p o lynomial eige nfunctions, ( ii i ) a factoriz a tion pr op erty for e ig enfunctions, and admit ( iv ) the separation of the r adial co ordinate a nd, hence, the existence of the 2nd order in tegral, ( v ) an algebraic form in inv ariants of a discr ete sy mmetr y g r oup (in s pace o f or bits). Key wor ds: (quasi)-exact-s olv abilit y; r ational mode ls ; algebra ic forms; Coxeter (W eyl) in- v ar iants, hidden alg ebra 2010 Mathematics Subje ct Classific ation: 35P99 ; 47A15; 4 7 A67; 47 A75 1 In tro duction In this pap er we will make an attempt to o v erview our constructiv e kno wledge ab out (quasi)- exactly-sol v able p otentia ls having a form of a meromorp h ic function in Cartesian co ordin ates. All these mo dels h a v e a discr ete group of symmetry , admit s eparation of v ariable(s), p ossess an (in)f in ite set of p olynomial eigenfunctions. They ha v e an inf inite discrete sp ectrum w hic h is linear in the quan tum num b ers. All of them are c haracterized b y the pr esence of a hidden (Lie) algebraic str u cture. Each of them is a type of isosp ectral deformation of the isotropic harmonic oscillato r. Let us consider the Hamiltonian = the Schr¨ odinger op erator H = − ∆ + V ( x ) , x ∈ R d . The main problem of quantum mec hanics is to solv e the S c hr¨ odinger equation H Ψ( x ) = E Ψ( x ) , Ψ( x ) ∈ L 2  R d  , f in ding th e sp ectrum (the energies E and eigenfunctions Ψ). Since the Hamiltonian is an inf inite- dimensional matrix, solving the Sc hr¨ odinger equation is equiv alen t to diagonalizing the inf inite- dimensional m atrix. It is a transcendental problem: the c haracteristic p olynomial is of inf inite order and it has inf initely-ma n y ro ots. Usually , we d o n ot kn o w how to make suc h a d iagonal- ization exactly (explicitly) but we can ask: Do mo dels exist f or which the r o ots ( ener gi e s ) , some of the them or al l, c an b e found expl icitly ( exactly )? Suc h mo d els do exist and w e call them solvable . If all energies are kno wn they are ca lled exactly-solvable (ES ), if only some num b er of th em is known w e call them quasi-exactly-solvable (QES ). Sur prisingly , all su c h m o dels th e ⋆ This pap er is a contribution to th e Sp ecial Issue “Sy mmetry , S ep aration, Sup er-integrabilit y and Sp ecial F unctions (S 4 )”. The full collection is av ailable at http://www.emis .de/journals/SIGMA/S4.h tml 2 A.V. T urbiner present author familiar with , are pro vided b y int egrable systems. The Hamiltonians of these mo dels are of the form H ES = − 1 2 ∆ + ω 2 r 2 + W (Ω) r 2 , in the exactly-solv able case and H QES = − 1 2 ∆ + ˜ ω 2 k r 2 + W (Ω) + Γ r 2 + ar 6 + br 4 , in the qu asi-exactly-solv able case, wh ere ω , ˜ ω k , Γ are p arameters, W (Ω) is a fun ction on unit sphere and r is the radial co ordinate. In b oth cases there exists the integral F = 1 2 L 2 + W (Ω) , (1) where L is the angular momentum op erator, due to the separation of v ariables in spherical co ordinates. No w w e consider some examples from the ones known so far. 2 Solv able mo dels 2.1 Case O ( N ) The Hamiltonian reads H O ( N ) = 1 2 N X i =1  − ∂ 2 ∂ x i 2 + ω 2 x i 2  + ν ( ν − 1) N P i =1 x i 2 , (2) or, in spherical co ordinates, H O ( N ) = − 1 2 r N ∂ ∂ r  r N ∂ ∂ r  + 1 2 ω 2 r 2 + F + ν ( ν − 1) r 2 , (3) F = 1 2 L 2 . (4) The Hamiltonian (2) is O ( N ) symmetric. It describ es a spher ical-symmetric h armonic oscillator with a generalized centrifugal p oten tial. Needless to sa y that the Hamiltonian H O ( N ) and F comm ute, [ H O ( N ) , F ] = 0 . Th us, F has common eigenfunctions w ith the Hamiltonian H O ( N ) . The sp ectrum can b e imme- diately found explicitly , and all eigenfunctions are of the t yp e P n  r 2  r ˜ ℓ Y { ℓ } (Ω) e − ω r 2 2 , where Y { ℓ } (Ω) is a N -dimen s ional spher ical harmonics, F Y { ℓ } (Ω) = γ Y { ℓ } (Ω). The Hamil- tonian (2) describ es an N -dimensional harmonic oscilla tor with generalized ce n trifugal term. Substituting in (3) the op erator F by its eigen v alue γ and gauging a w a y Ψ 0 = r ˜ ℓ e − ω r 2 2 w e arrive at the Laguerre op erator h O ( N ) ≡ (Ψ 0 ) − 1 ( H O ( N ) − E 0 )Ψ 0   r 2 = t = − 2 t∂ 2 t +  2 ω − 1 − N 2 − ˜ ℓ  ∂ t , (5) F rom Qu an tum A N (Calogero) to H 4 (Rational) Mo del 3 where E 0 is the lo w est energy and the parameter ˜ ℓ is chosen in su c h a wa y as to r emo v e singular term ∝ 1 r 2 in the p oten tia l in (3). (5) is th e algebraic form of the Hamiltonian (3 ). The gauge-rota ted Hamilto nian h O ( N ) (5) is sl (2)-Lie-alg ebraic (see b elo w), it h as inf initely-man y f in ite-dimensional inv arian t subspaces in p olynomials P n , n = 0 , 1 , . . . forming the inf inite f lag (see b elo w), its eigenfunctions P n ( r 2 = t ) are nothing but th e asso ciat ed Laguerre p olynomials. By adding to h O ( N ) (5) the op erator δ h (qes) = 4  at 2 − γ  ∂ ∂ t − 4 ak t + 2 ω k , (6) w e get the op erator h O ( N ) + δ h (qes) whic h has a single f inite-dimensional inv arian t sub space P k = h t p | 0 ≤ p ≤ k i , of the d imension ( k + 1). Hence, this op erator is quasi-exactly-solv able. Making th e change of v ariable t = r 2 and gauge rotation with ˜ Ψ 0 = t ˆ γ e − ω t 2 − at 2 4 w e arrive at the O ( N )-symmetric Q ES Hamiltonian [2] H O ( N ) = − 1 2 r N ∂ ∂ r  r N ∂ ∂ r  + a 2 r 6 + 2 aω r 4 + 1 2 ˜ ω 2 r 2 + F + Γ r 2 , (7) where ˆ γ , Γ, ˜ ω are parameters and γ is replaced by the op erator F . In (7) a f inite n umber of the eigenfunctions is of the form P k  r 2  r 2ˆ γ Y { ℓ } (Ω) e − ω r 2 2 − at 2 4 , they can b e foun d algebraically . It is worth noting th at at a = 0 th e op erator h O ( N ) + δ h (qes) remains exac tly-solv able, it preserv es the inf inite f lag of p olynomials P and the emerging Hamil- tonian has a form of (3). 2.2 Case ( Z 2 ) N The Hamiltonian reads H ( Z 2 ) N = 1 2 N X i =1  − ∂ 2 ∂ x 2 i + ω 2 x i 2  + 1 2 N X i =1 ν i ( ν i − 1) x i 2 , (8) or, in spherical co ordinates, H ( Z 2 ) N = − 1 2 r N ∂ ∂ r  r N ∂ ∂ r  + 1 2 ω 2 r 2 + F + W ( Z 2 ) N (Ω) r 2 , where W ( Z 2 ) N (Ω) = 1 2 N X i =1 ν i ( ν i − 1)  r x i  2 , and F is giv en by (4). T he Hamiltonian (8) is ( Z 2 ) N symmetric. It def ines the so-called S moro- dinsky–Win ternitz in tegrable system [1] whic h is in realit y the maximally-sup erinteg rable (there exist (2 N − 1) inte grals includin g the Hamiltonian) and exactly-solv able. Gauging aw ay in (8) the ground state, Ψ 0 = N Q i =1 ( x 2 i ) ν i 2 exp  − ω x 2 i 2  , and changing v ariables to t i = x 2 i w e arrive at the algebraic form. Also it admits QES extension. The system d escrib ed by the Hamiltonian (8) at ν i = ν is a p articular case of th e B C N -rational s y s tem (see b elo w). 4 A.V. T u rbiner Figure 1. N -b o dy Caloger o mo del. 2.3 Case A N − 1 This is the celebrated Calogero mo del ( A N − 1 rational mo del) whic h was found in [3]. It descri- b es N iden tical particles on a line (see Fig. 1) with singular p airwise in teractio n. The Hamiltonian is H Cal = 1 2 N X i =1  − ∂ 2 ∂ x 2 i + ω 2 x 2 i  + ν ( ν − 1) N X i>j 1 ( x i − x j ) 2 , (9) where the singular part of the p oten tial can b e written as N X i>j 1 ( x i − x j ) 2 = W A N − 1 (Ω) r 2 , W A N − 1 (Ω) = N X i =1  1 x i r − x j r  2 , (10) Here r is th e radial co ord inate in the space of relativ e co ordinates (see b elo w for a def inition) and W A N − 1 (Ω) is a function on the unit sph ere. Symmetry: S n (p ermutati ons x i → x j ) plu s Z 2 (all x i → − x i ). The ground sta te of the Hamiltonian (9) reads Ψ 0 ( x ) = Y i