Multi-Particle Quasi Exactly Solvable Difference Equations
aa r X i v : . [ n li n . S I] A ug Yukawa Institute Kyoto
DPSU-07-3YITP-07-44August 2007
Multi-Particle Quasi Exactly Solvable DifferenceEquations
Satoru Odake a and Ryu Sasaki b a Department of Physics, Shinshu University,Matsumoto 390-8621, Japan b Yukawa Institute for Theoretical Physics,Kyoto University, Kyoto 606-8502, Japan
Abstract
Several explicit examples of multi-particle quasi exactly solvable ‘discrete’ quantummechanical Hamiltonians are derived by deforming the well-known exactly solvablemulti-particle Hamiltonians, the Ruijsenaars-Schneider-van Diejen systems. These aredifference analogues of the quasi exactly solvable multi-particle systems, the quantumInozemtsev systems obtained by deforming the well-known exactly solvable Calogero-Sutherland systems. They have a finite number of exactly calculable eigenvalues andeigenfunctions. This paper is a multi-particle extension of the recent paper by one ofthe authors on deriving quasi exactly solvable difference equations of single degree offreedom.
Recently a recipe to obtain a quasi exactly solvable difference equation from an exactlysolvable difference equation is developed by one of the present authors [1]. In the presentpaper we apply the recipe to obtain multi-particle quasi exactly solvable difference equations.A quantum mechanical system is called Quasi Exactly Solvable (QES), if a finite number ofeigenvalues and the corresponding eigenfunctions can be determined exactly [2]. Since theumber of exactly solvable states can be chosen as large as wanted, a QES system could beused as a good alternative to an exactly solvable system for theoretical as well as practicalpurposes. Many examples of QES systems of single degree of freedom have been known forsome time, whereas the known examples of those with many degrees of freedom are ratherlimited in their structure [3, 4]. They are all obtained by deforming Calogero-Sutherlandsystems [5, 6, 7], the exactly solvable Hamiltonian dynamics based on root systems. Amongthem those based on the BC type root systems (rational and trigonometric) and on the A type root systems (trigonometric) can contain an arbitrary number of particles. Therefore theexamples of multi-particle QES are infinite in number. In the present paper we derive severalexamples of multi-particle QES difference equations by deforming Ruijsenaars-Schneider vanDiejen (RSvD) systems [8, 9, 10, 11], which are the difference equation analogues of Calogero-Sutherland systems. To be more precise, we derive multi-particle QES difference equationsby deforming RSvD systems based on BC (rational and trigonometric) and A (trigonometric)type root systems. Thus the examples of multi-particle QES difference equations are nowinfinite in number. The Hamiltonian of RSvD systems has two kinds of interaction terms,the ‘single particle interaction’ part and the ‘multi-particle interaction’ part. The latter iskept intact and the former, the ‘single particle interaction’ part is deformed according to therecipe given in the recent paper [1].This paper is organised as follows. In section 2 the Hamiltonian of the rational BC typetheory is derived and the finite dimensional invariant polynomial space is identified. Thetrigonometric BC type theory is explained in section 3. The Hamiltonian of the trigonometric A type theory and the finite dimensional invariant polynomial space are derived in section4. The final section is for a short summary and comments. BC type theory The first multi-particle QES Hamiltonian is a simple deformation of the rational Ruijsenaars-Schneider-van Diejen (RSvD) [8, 9] system based on the BC n root system. Here n is therank of the root system as well as the degree of freedom with coordinates and conjugatemomenta: x def = ( x , x , . . . , x n ) ∈ R n , p def = ( p , p , . . . , p n ) .
2s always, the momentum operator is realised as a differential operator p j = − i∂ j = − i∂/∂x j . In ‘discrete’ quantum mechanical Hamiltonians the momentum operators appear inexponentiated forms e ± p j = e ∓ i∂ j , in contrast to the ordinary quantum mechanics, in whichthe momentum operators appear as polynomials. Thus their action on the wavefunction isa finite shift in the imaginary direction: e ± p j ψ ( x ) = ψ ( x , . . . , x j − , x j ∓ i, x j +1 , . . . , x n ) , leading to difference Schr¨odinger equations. The quasi exactly solvable rational Hamiltonianhas the following general form [1, 12, 11]: H def = n X j =1 (cid:16)p V j e p j q V ∗ j + q V ∗ j e − p j p V j − V j − V ∗ j (cid:17) + α M , M ∈ N , (2.1)= n X j =1 (cid:16)p V j e − i∂ j q V ∗ j + q V ∗ j e i∂ j p V j − V j − V ∗ j (cid:17) + α M . (2.2)Here α M is a compensation term indexed by a natural number M , to be specified shortly in(2.5) and (2.7). The potential function V j has the following general form: V j ( x ) def = w ( x j ) n Y k =1 k = j Y ε = ± v ( x j + εx k ) , (2.3)in which the ‘multi-particle interaction’ part v is not deformed v ( y ) def = 1 − i gy , g > . (2.4)Whereas the ‘single particle interaction’ part w allows two types of deformation for QES,corresponding to the linear and quadratic polynomial deformations introduced in section 3.1of [1]: Type I : w ( y ) def = ( a + iy ) w ( y ) , α M ( x ) def = M n X j =1 x j , (2.5)Type II : w ( y ) def = ( a + iy )( a + iy ) w ( y ) , (2.6) α M ( x ) def = M (cid:16) M − X α =1 a α + 2( n − g (cid:17) n X j =1 x j , (2.7)with a common undeformed w ( y ) [12, 11] w ( y ) def = Q α =3 ( a α + iy )2 iy (2 iy + 1) , a α > . (2.8)3s a single particle dynamics, the above w corresponds to the deformation of the harmonicoscillator with the centrifugal potential. The corresponding eigenfunctions are the Wilsonpolynomials [12, 13].The main part of the Hamiltonian is factorised [1, 12, 11]: H = n X j =1 A † j A j + α M , (2.9) A j def = − i (cid:16) e − i ∂ j q V ∗ j − e i ∂ j p V j (cid:17) , A † j = i (cid:16)p V j e − i ∂ j − q V ∗ j e i ∂ j (cid:17) , (2.10)exhibiting the hermiticity (self-adjointness) of the Hamiltonian.The pseudo ground state wavefunction φ is defined as the one annihilated by all the A j operators: A j φ = 0 , j = 1 , . . . , n. (2.11)It is given byType I : φ ( x ) def = (cid:12)(cid:12)(cid:12)(cid:12) n Y j =1 Q α =2 Γ( a α + ix j )Γ(2 ix j ) · Y ≤ j
1) + V j ( x ) ∗ ( e i∂ j − (cid:17) + α M ( x ) , (2.15) H φ = E φ, φ ( x ) = φ ( x ) P M ( x ) ⇐⇒ ˜ H P M = E P M . (2.16)In the undeformed limit, i.e. w = w and α M = 0, the theory is the exactly solvablerational BC n RSvD [9] system with a Hamiltonian ˜ H . The corresponding φ becomes thetrue ground state wavefunction. The exact solvability means that ˜ H maps a Weyl-invariant4olynomial in { x j } into another of the same degree. The BC n Weyl-invariant polynomialsin { x j } are simply symmetric (under any permutation j ↔ k ) polynomials in { x j } . For laterconvenience, let us introduce a monomial symmetric polynomial m λ ( { y j } ) = m ( λ ,...,λ n ) ( y , . . . , y n ) def = X ( l ,...,l n ) y l · · · y l n n , (2.17)where the summation with respect to ( l , . . . , l n ) is taken over all distinct permutations of λ def = ( λ , . . . , λ n ).In the deformed theory, it is straightforward to demonstrate that ˜ H maps a symmetricpolynomial in { x j } of degree equal or less than M into another:˜ H V M ⊆ V M , dim V M = (cid:0) M + nn (cid:1) , (2.18) V M def = Span (cid:2) m ( l ,...,l n ) ( { x j } ) (cid:12)(cid:12) ≤ l j ≤ M , ≤ j ≤ n (cid:3) . (2.19)This establishes the quasi exact solvability. The proof that an invariant polynomial is mappedinto another goes almost parallel with the undeformed theory. One simply has to verify thevanishing of the residues of the simple poles at q j = ± q k , , ± i/
2. The other step is to showthat a symmetric polynomial in { x j } with degree m ( ≤ M −
1) is mapped to another withdegree m + 1, whereas a symmetric polynomial of degree M remains with the same degree.This part goes almost the same as in the single particle case shown in [1], since it is causedby the deformation of w , the single particle interaction part. The present examples arethe difference equation version of the QES theory called rational BC Inozemtsev systemsdiscussed in section 6 of Sasaki-Takasaki paper [3]. BC type theory The next example is a QES deformation of the trigonometric BC n RSvD system. Becauseof the periodicity of the trigonometric potential, we introduce a slightly different notationfor the dynamical variables: θ def = ( θ , θ , . . . , θ n ) ∈ R n , < θ j < π, x j = cos θ j , z j = e iθ j . (3.1)The dynamical variables are θ . We denote D j def = z j ddz j . Then q D j is a q -shift operator, q D j f ( z ) = f ( z , . . . , z j − , qz j , z j +1 , . . . , z n ) , < q <
1. The quasi exactly solvable trigonometric Hamiltonian has the followinggeneral form [1, 12, 11]: H def = n X j =1 (cid:16)p V j q D j q V ∗ j + q V ∗ j q − D j p V j − V j − V ∗ j (cid:17) + α M . (3.2)The compensation term α M is given in (3.5). The potential function V j consists of the ‘singleparticle interaction’ part w and the ‘multi-particle interaction’ part v : V j ( z ) def = w ( z j ) n Y k =1 k = j Y ε = ± v ( z j z εk ) , (3.3) v ( y ) def = 1 − a y − y , w ( y ) def = (1 − a y ) w ( y ) , w ( y ) def = Q α =2 (1 − a α y )(1 − y )(1 − qy ) , (3.4) α M ( z ) def = ( q M − q − a n − a a a a a n X j =1 (cid:0) z j + z − j (cid:1) . (3.5)As before, the ‘multi-particle interaction’ part v is not deformed but the ‘single particleinteraction’ part w is multiplicatively deformed by a linear term from w . As a single particledynamics, the above w corresponds to the deformation of the P¨oschl-Teller potential. Thecorresponding eigenfunctions are the Askey-Wilson polynomials [12, 13].The main part of the Hamiltonian is factorised [1, 12, 11]: H = n X j =1 A † j A j + α M , (3.6) A j def = i (cid:16) q D j q V ∗ j − q − D j p V j (cid:17) , A † j = − i (cid:16)p V j q D j − q V ∗ j q − D j (cid:17) , (3.7)and the pseudo ground state wavefunction φ φ ( z ) def = (cid:12)(cid:12)(cid:12)(cid:12) n Y j =1 ( z j ; q ) ∞ Q α =1 ( a α z j ; q ) ∞ · Y ≤ j
1) + V j ( z ) ∗ ( q − D j − (cid:17) + α M ( z ) , (3.11) H φ = E φ, φ ( z ) = φ ( z ) P M ( z ) ⇐⇒ ˜ H P M = E P M . (3.12)In the undeformed limit, i.e. w = w and α M = 0, the theory is the exactly solvabletrigonometric BC n RSvD [9] system with a Hamiltonian ˜ H . The corresponding φ becomesthe true ground state wavefunction. The exact solvability means that ˜ H maps a Weyl-invariant polynomial in { x j = cos θ j = ( z j + z − j ) } into another of the same degree. The BC n Weyl-invariant polynomials in { x j } are simply symmetric (under any permutation j ↔ k ) polynomials in { x j } or in { z j + z − j } . The eigenfunctions of ˜ H are the BC typeJack polynomials [14].In the deformed theory, it is straightforward to demonstrate that ˜ H maps a Weyl-invariantpolynomial in { x j } of degree equal or less than M into another:˜ H V M ⊆ V M , dim V M = (cid:0) M + nn (cid:1) , (3.13) V M def = Span (cid:2) m ( l ,...,l n ) ( { z j + z − j } ) (cid:12)(cid:12) ≤ l j ≤ M , ≤ j ≤ n (cid:3) . (3.14)This establishes the quasi exact solvability. The proof that an invariant polynomial is mappedinto another goes almost parallel with the undeformed theory. One simply has to verify thevanishing of the residues of the simple poles at z j = z ± k , ± , ± q / , ± q − / . The other stepis to show that a symmetric polynomial in { x j } with degree m ( ≤ M −
1) is mapped toanother with degree m + 1, whereas a symmetric polynomial of degree M remains with thesame degree. This part goes almost the same as in the single particle case shown in [1],since it is caused by the deformation of w , the single particle interaction part. The presentexample is the difference equation version of the QES theory called trigonometric BC typeInozemtsev system discussed in section 7 of Sasaki-Takasaki paper [3].7 Trigonometric A type theory The QES deformation of the trigonometric A n − RS [8] system goes almost parallel with theprevious example, or even simpler. For the A -type theory, it is customary to consider A n − and to embed all the roots in R n . This is accompanied by the introduction of one more degreeof freedom, θ n and p n . The genuine A n − theory corresponds to the relative coordinates andtheir momenta, and the extra degree of freedom is the center of mass coordinate and itsmomentum. The Hamiltonian takes the general form (3.2) with the potential function V j V j ( z ) def = w ( z j ) n Y k =1 k = j v ( z j z − k ) , (4.1) v ( y ) def = 1 − a y − y , w ( y ) def = 1 − a y, (4.2) α M ( z ) def = ( q M − a n − a n X j =1 (cid:0) z j + z − j (cid:1) . (4.3)The undeformed theory, the trigonometric A n − RS system [8] has w ≡ α M = 0. Themain part of the Hamiltonian is factorised as in (3.6) and (3.7). The pseudo ground state wavefunction φ annihilated by all A j reads φ ( z ) def = (cid:12)(cid:12)(cid:12)(cid:12) n Y j =1 a z j ; q ) ∞ · Y ≤ j
1. The similarity transformedHamiltonian ˜ H in terms of φ has the same form as (3.11), (3.12).In the undeformed limit, i.e. w ≡ α M = 0, the theory is the exactly solvabletrigonometric A n − RS system with a Hamiltonian ˜ H . The eigenfunctions of ˜ H are thewell-known Jack polynomials in { z j } [14]. Since all the coefficients of the eigenvalue equation˜ H ϕ = E ϕ are real, the Jack polynomials in { z − j } are also eigenfunctions. In other words˜ H maps a symmetric polynomial in { z j } into another of the same degree.The eigenfunctions of the deformed Hamiltonian ˜ H are still symmetric polynomials in { z j } and { z − j } , but truncated to the maximal power of M for each variable, due to thesingle particle interaction term w (4.2) and the compensation term (4.3). This form of the8ompensation term is necessary for the hermiticity of the Hamiltonian. In other words ˜ H has the invariant subspace˜ H V M ⊆ V M , dim V M = (cid:0) M + nn (cid:1) , (4.6) V M def = Span (cid:2) m ( l ,...,l n ) ( { z j } ) (cid:12)(cid:12) −M ≤ l j ≤ M , ≤ j ≤ n (cid:3) . (4.7)This establishes the quasi exact solvability. The proof that a symmetric polynomial ismapped into another goes almost parallel with the undeformed theory. One simply hasto verify the vanishing of the residues of the simple poles at z j = z k . The other step is toshow that a symmetric polynomial in { z j } with degree m ( −M +1 ≤ m ≤ M −
1) is mappedto another with degree m ±
1, whereas a symmetric polynomial of degree ±M remains withthe same degree. This part goes almost the same as in the single particle case shown in [1],since it is caused by the deformation of w , the single particle interaction part. The presentexample is the difference equation version of the QES theory called trigonometric A typeInozemtsev system discussed in section 8 of Sasaki-Takasaki paper [3]. Quasi exactly solvable multi-particle difference equations are derived by deforming rationaland trigonometric BC type RSvD systems as well as the trigonometric A type RS systems.The method of multi-particle deformations is a simple extension of the single particle caserecently developed by one of the authors [1]. These examples are the difference equationanalogues of the quasi exactly solvable multi-particle quantum mechanical systems derivedby Sasaki-Takasaki [3].A few comments are in order. As in the single particle cases, the ranges of the parameterscan be loosened without losing QES. For example, the six positive parameters in Type II insection 2 (2.6), a , . . . , a could be replaced by three complex conjugate pairs with positivereal parts. Among the five real parameters a ,. . . , a in (3.10), four could be replaced bytwo complex conjugate pairs b , b ∗ , b , b ∗ with modulus less than unity, | b | < | b | < q → c , and set q = e − /c , a = q g , a = 1 − q − a/ , θ = 2 θ new . Moreover set H new = ( a a a a q − ) − / a − ( n − H a , a , a , a ) = ( q g , q g +1 / , − q g ′ , − q g ′ +1 / ) for the BC type theory, H new = a − ( n − / H for the A type theory (see [11]). Then in the c → ∞ limit, the Hamiltonian c H new reducesto the trigonometric Inozemtsev systems discussed in section 7 and 8 of [3].The corresponding statement for the rational theory is somehow complicated and itrequires a double limit. For the parameters in section 2, let us set ( a , a , a , a ) = ( cω p ω a + a ω ω ( ω ω − , − cω p ω a + a ω ω ( ω ω − , g , g + ) and x j = cx new j . Moreover set ˜ H new =4( a a a ) − ˜ H and a = c /ω for the type I theory, ˜ H new = 4( a a a a ) − ˜ H and a = a =2 c /ω for the type II theory (see [11]). Then a double limit lim ω →∞ (cid:0) lim c →∞ c ˜ H new (cid:1) gives theHamiltonian of the rational Inozemtsev system discussed in section 6 of [3].It is interesting to note that the weight function φ ( x ) for the polynomial eigenfunctions { P M ( x ) } is the zero mode (stationary distribution) of the corresponding deformed Fokker-Planck equation [15]. Acknowledgements
This work is supported in part by Grants-in-Aid for Scientific Research from the Ministryof Education, Culture, Sports, Science and Technology, No.18340061 and No.19540179.
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