International Conference on

Secondary calculus and cohomological Physics Moscow, August 1997

REDUCED SYSTEM OF BETHE EQUATIONS FOR THREE-BOSON INTERACTION

V.A. Andreev

P.N. Lebedev Physical Institute, Optical Department, Leninsky Prospect 53, 117924 MOSCOW, Russia

O.A. Ivanova

M. V. Lomonosov Moscow State University, Department of Mechanics and Mathematics, 119899, MOSCOW, Russia

Abstract. The reduced system of Bethe equations is found for three-boson interaction. The invariants and surfaces in the space of Bethe parameters, connected with this system, are constructed. It is shown that at these surfaces the eigenvectors can be presented in power form. Connection between Bethe ansatz and polynomial extensions of Lie algebras is briefly discussed. The computational procedures corresponding to the original Bethe equations and the method of polynomial algebras are compared.

1. Introduction

Bethe ansatz plays an important role in the theory of integrable quantum models [1]-[4]. Initially, it has been applied to the lattice models such as isotropic and anisotropic spin chain [1, 2] and Hubbard model [5, 6]. It was also applied to the quantum versions of sine-Gordon [7, 8] and nonlinear Schrödinger equations [9].

The other class of models was introduced by Gaudin [10]. The possibility of functional Bethe ansatz for them was shown by Sklyamin [11]. These models have been investigated in many works [12]-[15]. The algebraic Bethe ansatz was applied by Jurĉo to the Gaudin’s models related to nonlinear quantum optics [16]. They describe such processes as second harmonic generation, three-boson interaction, the Dike model [17] and some cases of fourboson interaction.

The dynamics of three-boson interaction was investigated in our work [18]. It turned out, that in this case the Bethe ansatz formalism has some specific properties distinguishihg the three-boson interaction from other models. The point is that the eigenvectors of Hamiltonian in this approach have the form

Key words and phrases: Bethe ansatz, eigenvector, invariant, surface, polynomial algebra

|λ_{1}, …, λ_{N}⟩ = (N!)^{−1/2}B(λ_{1}) ⋯ B(λ_{N})|0⟩.  (1)

Here |0⟩ is a pseudovacuum vector and B(λ_{i}) are some specific operators depending on the parameters λ_{i}. In most models vectors (1), corresponding to different sets {λ_{1}, …, λ_{N}}, {λ′_{1}, …, λ′_{N}}, are linearly independent. This condition together with the demand of (1) to be the eigenvector for Hamiltonian gives us the equations for the parameters {λ_{i}, i = 1, …, N}. And in fact, in the spin chain models, in quantum sine-Gordon and nonlinear Schrödinger equations these vectors are linearly independent [19]-[21], but in case of threeboson interaction it is not so. In this model eigenvectors (1) with different {λ_{i}, i = 1, …, N} can be linearly dependent or even coincide.

Therefore the admissible parameters defining an eigenvector (1) occupy some surface S in space C^{N}. Equations of these surfaces have been found in Ref.[22]. In this work we show how the existence of such dependencies can change the form of equations for the Bethe ansatz parameters. In more exact terms, the structure of equations for the Bethe ansatz parameters is found to be more closely connected with the algebraic structure of the model. The Hamiltonians of Gaudin’s models have the form of the sum of products of operators of two algebras. The original Bethe equations are connected only with the form of these algebras. But it turns out that if one considers the concrete representations of these algebras, the structure of corresponding equations can be simplified. We call them reduced Bethe equations.

We define the coordinate system at the parametric surfaces S and with it’s help compare the computational procedures corresponding to the original Bethe equations, the reduced Bethe equations and the method of polynomial algebras [23, 24]. The algebraic sense of the parameters λ_{i} is also clarified.

2. Algebraic Bethe ansatz scheme

According to the usual Bethe ansatz scheme in terms of Gaudin models [16] one must consider the 2 × 2 matrix L(λ) with operator matrix elements

L(λ) = ( \begin{matrix}A(λ) & B(λ) \\ C(λ) & D(λ) \end{matrix} ) .  (2)

This matrix must be a solution of the classical Yang-Baxter equation

[L(λ) ⨂ I, I ⨂ L(μ)] + [r(λ − μ), L(λ) ⨂ I + I ⨂ L(μ)] = 0.  (3)

Here I = ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) and r(λ) is 4 × 4 matrix

r(λ) = ( \begin{matrix}f(λ) & 0 & 0 & 0 \\ 0 & 0 & g(λ) & 0 \\ 0 & g(λ) & 0 & 0 \\ 0 & 0 & 0 & f(λ) \end{matrix} ) .  (4)

It follows from equation (3) that elements of matrix (2) must satisfy the following commutation relations

[A(λ), B(μ)] = g(λ − μ)B(λ) − f(λ − μ)B(μ),  (5)
[A(λ), C(μ)] = f(λ − μ)C(μ) − g(λ − μ)C(λ),
[D(λ), B(μ)] = f(λ − μ)B(μ) − g(λ − μ)B(λ),
[D(λ), C(μ)] = g(λ − μ)C(λ) − f(λ − μ)C(μ),
[B(λ), C(μ)] = g(λ − μ)(A(λ) − A(μ) − D(λ) + D(μ)),
[A(λ), A(μ)] = [B(λ), B(μ)] = [C(λ), C(μ)] = [D(λ), D(μ)] =
= [A(λ), D(μ)] = 0.

The Hamiltonian of a three-boson interaction has the form

H = H_{0} + gH_{I} = ω_{1}a_{1}^{+}a_{1} + ω_{2}a_{2}^{+}a_{2} + ω_{3}a_{3}^{+}a_{3} + g(a_{1}^{+}a_{2}a_{3} + a_{3}^{+}a_{2}^{+}a_{1}),  (6)
ω_{1} = ω_{2} + ω_{3}.

In this case the elements of matrix L(λ) can be taken as

A(λ) = −\frac{1}{2}λ − \frac{1}{λ}S_{3},  B(λ) = a_{3}^{+} + \frac{1}{λ}S_{+},  (7)

C(λ) = a_{3} + \frac{1}{λ}S_{−}, D(λ) = \frac{1}{2}λ + \frac{1}{λ}S_{3} = −A(λ).

Here the 2-boson realization of SU(2) algebra is used

S_{+} = a_{1}^{+}a_{2}, S_{−} = a_{2}^{+}a_{1}, S_{3} = \frac{1}{2}(a_{1}^{+}a_{1} − a_{2}^{+}a_{2}).  (8)

It is easy to verify that if  f(λ) = g(λ) = \frac{1}{λ}  (9)

operators (7) satisfy commutation relations (5). The eigenfunctions of Hamiltonian (6) can be constructed in the form (1) with the pseudovacuum vector

|0⟩_{M} = |0, M, 0⟩.  (10)

This vector satisfies the equations

C(λ)|0⟩_{M} = 0, A(λ)|0⟩_{M} = a(λ)|0⟩_{M}, D(λ)|0⟩_{M} = d(λ)|0⟩_{M}
a(λ) = −d(λ) = \frac{1}{2λ}(M − λ^{2}).  (11)

Now we find the equations for the parameters {λ_{i}, i = 1, …, N}. For this, one must consider the trace of the matrix L^{2}(λ).

T (λ) = \frac{1}{2}T rL^{2}(λ).  (12)

This operator has the important property

[T (λ), T (μ)] = 0,  (13)
[T (λ), B(μ)] = g(λ − μ)(A(μ)B(λ) − B(λ)D(μ))+  (14)
+f(λ − μ)(B(μ)D(μ) − A(λ)B(μ)).

The pseudovacuum vector (10) is the eigenvector for operator (12)

T (λ)|0⟩_{M} = t_{M,0}(λ)|0⟩_{M},  (15)

t_{M,0}(λ) = \frac{1}{2}(a^{2}(λ) + d^{2}(λ) + a′(λ) − d′(λ)).

In case of three-boson interaction, i.e., for (7),(8),(11) we have

T (λ) = \frac{1}{4}λ^{2} + (a_{3}^{+}a_{3} + \frac{1}{2}) + S_{3} + \frac{1}{λ^{2}} (S_{3}^{2} + \frac{1}{2}(S_{+}S_{−} + S_{−}S_{+}))+

+ \frac{1}{λ}(a_{3}S_{+} + a_{3}^{+}S_{−}),  (16)

t_{M,0}(λ) = \frac{1}{4}λ^{2} − \frac{M}{2} + \frac{1}{2} + \frac{1}{λ^{2}} \frac{M}{2} (\frac{M}{2} + 1) .

It is obvious that the operator T (λ) commutes both with the whole Hamiltonian (6) and with its interacting part H_{I}

[T (λ), H] = [T (λ), H_{I}] = 0  (17)
and  H_{I} = res T (λ),  λ → 0.  (18)

We are lookihg for the eigenvectors of the operator T (λ) in the form (1)

Ψ_{M,N}(λ_{1}, …, λ_{N}) ≡ |λ_{1}, …, λ_{N}⟩_{M} = (N!)^{− \frac{1}{2}} B(λ_{1}) ⋯ B(λ_{N})|0⟩_{M}.  (19)

Acting on it by operator (12) and using the commutation relations (5) and (14) one can find the equation [16]

T (λ)|λ_{1}, …, λ_{N}⟩_{M} =  (20)

= {∑_{i=1}^{N} f^{2}(λ − λ_{i}) − g^{2}(λ − λ_{i}) + ∑_{j≠i} f(λ − λ_{i})f(λ − λ_{j}) + t_{M,o}(λ)+

+ ∑_{i=1}^{N} f(λ − λ_{i})(d(λ) − a(λ))} |λ_{1}, …, λ_{N}⟩_{M}+

+ ∑_{i=1}^{N} g(λ − λ_{i}){a(λ_{i}) − d(λ_{i}) − 2 ∑_{j≠i} f(λ_{i} − λ_{j})}|λ_{1}, ..., λ_{i−1}, λ, λ_{i+1}, ..., λ_{N}⟩_{M}.

As we want the vector (19) to be the eigenvector of operator (12), it follows from the last equation that the additional terms in the right-hand part of (20), containing vectors

Ψ_{M,N}^{i} = |λ_{1}, …, λ_{i−1}, λ_{0}, λ_{i+1}, …, λ_{N}⟩,  (21)

must be equal to zero. If all these vectors are linearly independent and ortogonal to vector (19) we have the system

a(λ_{i}) − d(λ_{i}) − 2 ∑_{j≠i} f(λ_{i} − λ_{j}) = 0,  i = 1, …, N.  (22)

In the case of (9),(11) this system takes the form

λ_{i} + ∑_{j≠i} \frac{2}{λ_{i} − λ_{j}} = \frac{M}{λ_{i}} ,  i = 1, …, N.  (23)

This very system (23) is usually regarded as the system of Bethe equations for the threeboson interaction, see for example [16]. It is clear from (20) that the conditions (23) are necessary and sufficient for vector (19) to be the eigenvector of the operator (16) only if all vectors (21)) are linearly independent, but in case of three-boson interaction these vectors are linearly independent only if N ≤ M. But if N > M vectors (19),(21) are linearly dependent. As the result of this fact the conditions (23) can be weakened and replaced by the another system. Now we find these equations.

3. Reduced system of Bethe equations

The explicit form of vector (19) was found for the three-boson interaction in [18].

Ψ_{M,N}(λ_{1}, …, λ_{N}) =  (24)

= ∑_{k=0}^{min(N,M)}(\frac{k!M!(N − k)!}{(M − k)!N!})^{\frac{1}{2}} ( ∑_{i_{1}<···<i_{k}}(λ_{i_{1}} ⋯ λ_{i_{k}})^{−1})|k, M − k, N − k⟩.

We see that for every N > M this sum contains (M + 1) members, therefore among (N + 1) vectors (19),(21) only (M + 1) ones are linearly independent. Let us construct the set of linearly independent vectors. As the first member of this set we take vector (24), the other M vectors have the form

Ψ_{l} ≡ |0, M, N⟩ − (\frac{(M − l)!N!}{l!M!(N − l)!})^{\frac{1}{2}} ( ∑_{i_{1}<···<i_{l}}(λ_{i_{1}} … λ_{i_{l}})^{−1})^{−1}|l, M − l, N − l⟩.

l = 1, …, M.  (25)

All vectors (25) are linearly independent and ortogonal to the vector (24). Let us calculate the scalar product

c_{l}^{j} = ⟨Ψ_{l} | Ψ_{M,N}^{j} ⟩ = 1 − ( ∑_{i_{1}<···<i_{l}}^{j} (λ_{i_{1}} … λ_{i_{l}})^{−1})( ∑_{i_{1}<···<i_{l}}(λ_{i_{1}} … λ_{i_{l}})^{−1})^{−1}.  (26)

In the sum ∑^{J} parameter λ_{j} is replaced by the parameter λ.

Substituting in (20) vectors (25) instead of vectors (21) one gets the system of equations

∑_{i=1}^{N} g(λ − λ_{i}){a(λ_{i}) − d(λ_{i}) − 2 ∑_{j≠i} f(λ_{i} − λ_{j})}c_{l}^{i} = 0, l = 1, …, M.  (27)

Taking into account formulas (9), and (11), system (27) can be transformed into the system

∑_{i=1}^{N}(λ_{i})^{−l}(λ_{i} − \frac{M}{λ_{i}} + 2 ∑_{j≠i} \frac{1}{λ_{i} − λ_{j}} ) = 0,  l = 1, …, M.  (28)

Thus, if M < N one can use the reduced system (28) instead of usual system (23). System (23) contains N equations and N unknown parameters λ_{i}. Its solutions are points in N-dimensional complex space. System (28) contains M equations for N unknown parameters, its solutions are (N − M)-dimensional complex surfaces in N-dimensional space of parameters. Each surface consists of M connected components. All points from some connected component give the same vector (24), so

Ψ_{M,N}(λ_{1}, …, λ_{N}) ≡ Ψ_{M,N}(μ_{1}, …, μ_{N}),

if {λ_{i}} and {μ_{i}} belong to the same connected component. Equations (23) give us the specific points from each component. For M = 1, N = 1, 2, …, 14 these points have been evaluated in [18]. Thus we see that the usual Bethe equations (23) give the correct set of parameters, but this set is not unique. Note that equations (28), in contrast to (23), permit solutions containing equal λ_{i} = λ_{j}.

4. Spectrum formula and invariants of the surfaces

The eigenvalue of the operator T (λ) was found in [16].

t_{M,N}(λ) = ∑_{i=1}^{N} [f^{2}(λ − λ_{i}) − g^{2}(λ − λ_{i}) + f(λ − λ_{i})(d(λ) − a(λ)) +

+ ∑_{i≠j} f(λ − λ_{i})f(λ − λ_{j})] + t_{M,0}(λ).  (29)

In the case of (9),(11),(15) it takes the form

t_{M,N}(λ) = ∑_{i=1}^{N}\{\frac{λ^{2} − M}{λ(λ − λ_{i})} + ∑_{j≠i} \frac{1}{(λ − λ_{i})(λ − λ_{j})}\}+

+ \frac{1}{4}λ^{2} − \frac{M}{2} + \frac{1}{2} + \frac{1}{λ^{2}} \frac{M}{2} (\frac{M}{2} + 1).  (30)

This formula is valid if N ≤ M, but if N > M it is valid only for those {λ_{i}, i = 1, …, N} which satisfy equations (23); but if parameters {λ_{i}, i = 1, …, N} satisfy only the weaker system (28) the formula (30) is not valid. We must add to it some terms, because now vectors Ψ_{M,N} and Ψ_{M,N}^{i} are linearly dependent. Such addition, Δ_{M,N}, to eigenvalue (30) has the form

Δ_{M,N}(λ) = C_{M,N}^{−2} ∑_{i=1}^{N} ⟨Ψ_{M,N}^{i} | Ψ_{M,N} ⟩F_{i}(λ).  (31)

Here

C_{M,N}^{2} = ∑_{k=0}^{M}(\frac{k!(N − k)!M!}{(M − k)!N!})( ∑_{i_{1}<···<i_{k}}(λ_{i_{1}} … λ_{i_{k}})^{−1})^{2}

— is the square of the norm of vector (23) and

F_{i}(λ) = \frac{1}{λ − λ_{i}} (\frac{M}{λ_{i}} − λ_{i} − 2 ∑_{j≠i} \frac{1}{λ_{i} − λ_{j}} ).

Using the fact

⟨Ψ_{M,N}^{i} | Ψ_{M,N} ⟩ =

= ∑_{k=0}^{M}(\frac{k!(N − k)!M!}{(M − k)!N!})( ∑_{i_{1}<···<i_{k}}^{i} (λ_{i_{1}} … λ_{i_{k}})^{−1})( ∑_{i_{1}<···<i_{k}}(λ_{i_{1}} … λ_{i_{k}})^{−1})

we get

Δ_{M,N}(λ) = ∑_{i=1}^{N} F_{i}(λ).  (32)

Therefore, for arbitrary set of parameters {λ_{i}, i = 1, …, N}, satisfing the system (28), the eigenvalue of the operator T (λ) has the form

t_{M,N}(λ) = ∑_{i=1}^{N} \{\frac{λ^{2} − M}{λ(λ − λ_{i})} + ∑_{j≠i} \frac{1}{(λ − λ_{i})(λ − λ_{j})} + \frac{M − λ_{i}^{2}}{λ_{i}(λ − λ_{i})}−

−2 ∑_{j≠i} \frac{1}{(λ − λ_{i})(λ_{i} − λ_{j})}\} + \frac{1}{4}λ^{2} − \frac{M}{2} + \frac{1}{2} + \frac{1}{λ^{2}} \frac{M}{2} (\frac{M}{2} + 1) =

= \frac{1}{λ} ∑_{i=1}^{N} \frac{M}{λ_{i}} + N + \frac{1}{4}λ^{2} − \frac{M}{2} + \frac{1}{2} + \frac{1}{λ^{2}} \frac{M}{2} (\frac{M}{2} + 1).  (33)

Comparing formulas (16) and (33) we have

E_{I} = ∑_{i=1}^{N} \frac{M}{λ_{i}} ,  H_{I}Ψ_{M,N} = E_{I}Ψ_{M,N}.  (34)

In the work [16] for E_{I} the following expression was found

E_{I} = ∑_{i=1}^{N} λ_{i}.  (35)

But this formula (35) is valid only for those {λ_{i}} which satisfy equations (23), while formula (33) is valid for all sets {λ_{i}}, belonging to surfaces (28). So we see that the quantity

S_{1} = ∑_{i=1}^{N} λ_{i}^{−1} = \frac{1}{M} E_{I}

is an invariant of the surface (to be more exact — of its connected component). The other invariants have the form

S_{k} = ∑_{i_{1}<···<i_{k}}(λ_{i_{1}} … λ_{i_{k}})^{−1} = (−1)^{k} \frac{N!}{(N − k)!} \frac{A_{M−k}}{A_{M}} , k = 1, …, M.  (36)

Here A_{l} — is a l × l -lower principal minor of the matrix A_{MN}

A_{MN} =∥ H_{I} − EI ∥=∥ h_{ij} − Eδ_{ij} ∥=∥ a_{ij} ∥,  i, j = 1, …, M + 1.  (37)
h_{ij} = δ_{i+1,j}h_{i} + δ_{i,j+1}h_{j}, h_{i} = ((M + 1 − i)(N + 1 − i))^{1/2}.
A_{l} =∥ a_{ij}^{l} ∥, i, j = 1, …, l, a_{ij}^{l} = a_{M+1−l+i,M+1−l+j}.  (38)

Equation det A_{MN} = 0 gives us the eigenvalues of vectors (24). Formulas (36) can be found by comparing (24) with the usual structure of eigenvectors of the matrix (37).

It was shown in [18] that the space D of the eigenvectors of Hamiltonian (6) is the direct sum of subspaces D_{M,N}

D = ⨁_{M,N=0}^{∞} D_{M,N}, D_{M.N} = {|K, M − K, N − K⟩, K = 0, 1, … min(M, N)} , (39)
dim D_{M,N} = min(M, N) + 1.

The vectors from these subspaces are the eigenvectors of two operators

K_{1} = S_{3}^{2} + \frac{1}{2}(S_{+}S_{−} + S_{−}S_{+}),  K_{2} = (a_{3}^{+}a_{3} + \frac{1}{2}) + S_{3}.  (40)

with the following eigenstates (33)

K_{1} |D_{M,N} = \frac{M}{2} (\frac{M}{2} + 1),  K_{2} |D_{M,N} = N − \frac{M}{2} + \frac{1}{2}.  (41)

In case N ≥ M D_{M,N} is the space of (M + 1)-dimensional representation of the algebra SU(2). If N < M, the space D_{M,N} forms only part of the space of the representation. Different vectors (24) from the space D_{M,N} have different energy E = ∑_{i=1}^{N} \frac{M}{λ_{i}}. The operators K_{1}, K_{2}, H_{I} are the coefficients attached to different degrees of the parameter λ in formula (16) for the operator T r(λ) (λ^{−2}, λ^{0}, λ^{−1} -correspondingly).

The original equations for the Bethe ansatz parameters (23) correspond to the case N ≤ M, and dim S = 0, therefore S is a set of points in the space C^{N}. The reduced Bethe equations (28) correspond to the case N > M and dim S = N − M > 0. The number N of equations of the original Bethe system (23) can be reduced now by the dimension of surface S

N − dim S = M.

M is the number of equations of reduced system (28). Hamiltonian H_{I} (6) can be presented by the generators of algebra SU(2){S_{3}, S_{+}, S_{−}} and Heisenberg’s algebra {a_{3}, a_{3}^{+}}

H_{I} = a_{3}S_{−} + a_{3}^{+}S_{+}.  (42)

They are the same for all N, M. The structure of reduced system (28) is more closely connected with concrete values of these numbers. In Table 1., Ref.[18], the spaces of the representations of algebra SU(2) are given by rows, the infinite columns correspond to the representation of Heisenberg’s algebra {a_{3}, a_{3}^{+}}, and the spaces D_{M,N} are the diagonals. The structure of these diagonals and of the reduced Bethe equations is defined by the intersection of the spaces of representation of these two algebras.

5. Asymptotics and generating functions for the eigenvectors

The fact that for N > M series (24) contains only (M + 1) members can be used for calculating the asymptotics of the vectors Ψ_{M,N} for large N. For this, let us consider the (M + 1)-dimensional irreducible representation of algebra SU(2). In the space of this representation one can find the eigenvectors of the operator S_{+} + S_{−}

(S_{+} + S_{−})Ψ_{M}^{m} = E_{M}^{m}Ψ_{M}^{m},  m = 0, 1, …, M.  (43)

The solutions of this equation have the form

Ψ_{M}^{m} = ∑_{k=0}^{M} C_{M,k}^{m}l_{k},  C_{M,0}^{m} = 1.  (44)

Here {l_{k}, k = 0, …, M} is a basis in the space of the representation, such that

S_{3}l_{k} = (k − \frac{M}{2} )l_{k}.

Then it follows from (37) that vector (24)

Ψ_{M,N}^{m} → ∑_{k=0}^{M} C_{M,k}^{m}|k, M − k, N⟩  (45)
and  E_{M,n}^{m} → \sqrt{N}E_{M}^{m}  if N → ∞.

As a basis {l_{k}, k = 0, …, M} we can use l_{k} = |k, M − k, 0⟩. Then

Ψ_{M}^{m} = ∑_{k=0}^{M} C_{M,k}^{m}|k, M − k, 0⟩

and equation (41) takes the form

Ψ_{M,N}^{m} → (N!)^{−1/2}(a_{3}^{+})^{N}Ψ_{M}^{m}  if N → ∞.  (46)

6. Coordinate systems on the parametric surfaces

It was found that the equations of the parametric surfaces S_{N−M} have the form [22]

∑_{i_{1}<···<i_{k}}(λ_{i_{1}} … λ_{i_{k}})^{−1} = I_{k} = (−1)^{k} \frac{N!}{(N − k)!} \frac{A_{M−k}}{A_{M}} , k = 1, …, M.  (47)

In order to fix a point on the surface S_{N−M} one must define other (N − M) equations

∑_{i_{1}<···<i_{M+l}}(λ_{i_{1}} … λ_{i_{M+l}})^{−1} = J_{l},  l = 1, …, N − M.  (48)

These numbers, J_{l}, can be considered as the coordinates of point λ on the surface S_{N−M}. Thus each point λ = λ_{1}, …, λ_{N} in N-dimensional complex space C^{N} is defined by two sets of numbers {I_{k}, k = 1, …, M} and {J_{l}, l = 1, …, N − M}. Members I_{k} of the first set are the values of invariants (47) of solutions (24), they define the surface S_{N−M}. Members J_{l} of the second set are not the invariants of the solutions and define the point on the surface. But in practice it is more convenient to consider the quantities

C_{n} = ∑_{i_{1}<···<i_{n}}(−1)^{n}(λ_{i_{1}} … λ_{i_{n}}) n = 1, …, N.  (49)

If N > M the quantities C_{n} are not the invariants of solutions (24), but with their help one can construct the polynomials

P_{N} = ∑_{i=0}^{N} x^{N−i}C_{i},  C_{0} = 1  (50)

whose roots are the parameters λ_{1} … λ_{N}. The quantities C_{n} can be directly found via matrix elements of the Hamiltonian H.

It was shown in [22], that in case of N ≤ M

C_{n} = \frac{1}{n!}A_{n},  n = 1, …, N,  (51)

where A_{n} is the lower major n-dimensional minor of (N + 1) × (N + 1) matrix A_{MN}.

If N > M, one must consider the extended (N + 1) × (N + 1) matrix Ã_{MN} instead of (M + 1) × (M + 1) matrix A_{MN} (37). This matrix has a bloc structure

Ã_{MN} = ∥{A_{MN} \atop 0} {0 \atop G_{MN}} ∥ .  (52)

The upper bloc coincides with the matrix A_{MN}. The lower (N −M)×(N −M)- dimensional bloc G_{MN} has such a structure, that numerical values of all quantities C_{n} can be calculated with the help of formula

C_{n} = \frac{1}{n!}Ã_{n},  (53)

where Ã_{n} is the lower major n-dimensional minor of (N + 1) × (N + 1) extended matrix Ã_{MN} (48). Since matrix Ã_{MN} has a bloc structure the minor Ã_{n} can be factorised for all n ≥ N − M

Ã_{n} = G · A_{n−N+M},  G = det ∥G_{MN}∥.  (54)

Now we can find the connection between I_{K}, J_{l} and minors of the matrix Ã_{MN}.

I_{K} = \frac{C_{N−K}}{C_{N}} = (−1)^{K} \frac{N!}{(N − K)!} \frac{Ã_{N−K}}{Ã_{N}} = (−1)^{K} \frac{N!}{(N − K)!} \frac{A_{M−K}}{A_{M}} .  (55)
K = 1, …, M.
J_{l} = \frac{C_{N−M−l}}{C_{N}} = (−1)^{M+l} \frac{N!}{(N − M − l)!} \frac{Ã_{N−M−l}}{Ã_{N}} , l = 1, …, N − M.  (56)

We see that the invariants I_{K} of solution (24) are the functions of the elements of the matrix A_{MN} only, which has clear physical sense. The coordinates J_{l} of the points on the parametric surface depend on the elements of the lower “unphysical” bloc G_{MN} of matrix Ã_{MN}. More precisely, quantities J_{l} are functions of the lower major minors of the matrix G_{MN}. There are N − M of such minors, including the determinant G

A_{J,1}, …, A_{J,N−M} = G.  (57)

numbers can be also considered as the coordinates of the point on the surface S_{N−M}. Defining any explicit form for the elements of the matrix G_{MN}, one can use them as the third set of coordinates on the surface S_{N−M}.

For example,

G_{MN} = ∥d_{ij} − Eδ_{ij}∥,  d_{ij} = (δ_{i,j+1} + δ_{i+1,j})r_{i} + δ_{ij}d.  (58)

Now parameters {d, r_{i}, i = 1, …, N − M − 1} can be considered as the coordinates of the point on the surface S_{N−M}.

It is necessary to note that the invariants I_{K} are not all independent. Each solution (24) is defined by three parameters M, N, E_{I}. Correspondingly, among the invariants I_{K} only three are independent; all others can be found will the help of recurrence condition

(N − K + 1)I_{K−1} + E_{I}I_{K} + \frac{h_{M−K+1}}{N − K} I_{K+1} = 0.  (59)

As we know, ther is one point marked out on the surface S_{N−M}. This point is defined by the original system (23). Let us find its coordinates in the system (59). It can be done by comparing the SU(2) and SU(1, 1) forms of the Bethe ansatz for three-boson interaction, or, more exactly, formulas (24),(37) with formulas (A.2) and (A.4) in [22]. Then it is easy to see that

d = 0, r_{l} = ((N − M − l)(−l)(M + 1 + l))^{1/2},  l = 1, …, N − M − 1  (60)

are the coordinates of this point.

7. The comparison of different computational procedures

Let us compare the computational procedures corresponding to the original and reduced systems for the Bethe ansatz parameters. In case of system (23) one must deal with the extended (N + 1) × (N + 1) matrix Ã_{MN} (52), which contains physical and unphysical blocs A_{MN} and G_{MN}, correspondingly. First, one must solve the equation

|A_{MN}| = 0  (61)

and find the eigenvalues E_{I} of the Hamiltonian H_{I}. Then it is necessary to evaluate the minors Ã_{n} of the matrix Ã_{MN} and to construct a polynomial

P_{N} = ∑_{n=0}^{N} \frac{x^{N−n}}{n!} Ã_{n}.  (62)

The roots of this polynomial are the desired quantities r_{n}.

In case of the reduced system (28) we must first evaluate the invariants I_{K} with the help of matrix A_{MN} and formula (47). Then we must divide N parameters λ_{n} into two parts

λ_{i}^{−1} = μ_{i}, i = 1, …, M.
λ_{M+j}^{−1} = ν_{j},  j = 1, …, N − M.  (63)

The values of ν_{j} may be taken arbitrarily. We fix them and evaluate the quantities

Q_{l} = ∑_{j_{1}<···<j_{l}}(ν_{j1} … ν_{jl}),  l = 1, …, N − M.  (64)

The parameters μ_{i} now satisfy the equations

∑_{i_{1}<···<i_{K}}(μ_{i_{1}} … μ_{i_{K}}) = I_{K} − Q_{K} − ∑_{l=1}^{K−1} I_{K−l}Q_{l} = L_{K}., K = 1, …, M.  (65)

Therefore, M parameters μ_{i} are the roots of the polynomial

P̃_{M} = ∑_{K=0}^{M} x^{K}L_{M−K}.  (66)

So, in case of the original Bethe equation (23) one must begin with solving the equation (61) of order (M + 1); then solve the equation P_{N} = 0 of order N. In case of the reduced Bethe equations one must solve first the same equation (61), and then the equation P̃_{M} = 0, whose order is M. If M << N the difference in computations may be quit distinct.

In the next section we consider another approach to the same problem, based on the infinite dimensional polynomial algebras. By this method to find eigenvalues and eigenstates of the Hamiltonian H it is sufficient to solve only equation (61) . Therefore, the knowledge of the Bethe parameters λ is necessary not only for constructing the eigenstates of the Hamiltonian H_{I}, but also for investigating the algebraic structure of the model. The knowledge of the parameters λ helps us to represent some operators from polynomial algebra SU_{d}^{(1,2)}(2) in terms of SU(2) and Heisenberg’s algebras.

8. Bethe ansatz and polynomial extensions of Lie algebras

The other approach to quantum optical problems based on the polynomial extensions of Lie algebras was put forth in [23, 24]. According to it, the Hamiltonian of the system must be presented in the form

H = ∑_{i=1}^{N} α_{i}Y_{i},  α_{i} = const,  (67)

where Y_{i} are operators from the enveloping algebra of some finite-dimensional algebra G. These operators satisfy the commutation relations

[Y_{i}, Y_{j}] = f_{ij}(Y_{k}),  i, j, k = 1, …, N,  (68)

where f_{i,j} are structural polynomials of finite order. Knowing these polynomials one can study the properties of Hamiltonian (52). In case of three-boson interaction Hamiltonian (6) can be presented in the form

H = C + g(Y_{+} + Y_{−}),  (69)

where

C = H_{0} = ω_{1}a_{1}^{+}a_{1} + ω_{2}a_{2}^{+}a_{2} + ω_{3}a_{3}^{+}a_{3}, Y_{+} = a_{1}a_{2}^{+}a_{3}^{+}, Y_{−} = a_{1}^{+}a_{2}a_{3}.  (70)

These operators together with the additional operator

Y_{0} = \frac{1}{3}(−a_{1}^{+}a_{1} + a_{2}^{+}a_{2} + a_{3}^{+}a_{3})  (71)

satisfy the commutation relations

[Y_{0}, Y_{±}] = ±Y_{±}; [C, Y_{±0}] = 0, [Y_{−}, Y_{+}] = Ψ_{1,2}(Y_{0} + 1) − Ψ_{1,2}(Y_{0}).  (72)

The structural polynomial Ψ_{1,2}(Y_{0}) defines the polynomial algebra SU_{d}^{(1,2)}(2) with commutation relations (72). It has the form [23]

Ψ_{1,2}(Y_{0}) = \frac{1}{4}(R_{0} − Y_{0} + 1)(2Y_{0} + R_{0} + R_{1})(2Y_{0} + R_{0} − R_{1}),  (73)

R_{0} = \frac{1}{3}(2a_{1}^{+}a_{1} + a_{2}^{+}a_{2} + a_{3}^{+}a_{3}), R_{1} = −a_{2}^{+}a_{2} + a_{3}^{+}a_{3}.

We try to find the eigenvectors of Hamiltonian (69) in a form

|E⟩ = ∑_{n=1}^{s} Q_{n}(E)(Y_{+})^{n}|s, k, 0⟩.  (74)

The coefficients Q_{n} satisfy the recurrence relation

Q_{n−1} − E_{I}Q_{n} + Ψ_{1,2}(l_{1} + n + 1)Q_{n+1} = 0.  (75)

Here E_{I} is an eigenvalue of the interaction Hamiltonian H_{I} and l_{1} is an eigenvalue of operator Y_{0}

H_{I}|E⟩ = (Y_{+} + Y_{−})|E⟩ = E_{I}|E⟩,

Y_{0}|s, k, 0⟩ = \frac{1}{3}(k − s)|s, k, 0⟩ = l_{1}|s, k, 0⟩.

Recurrence relations (75) are valid for all polynomial deformations SU_{d}^{(p,q)}(2) of algebra SU(2) with the specific polynomial Ψ_{p,q} in each case. In case of algebra (70) – (73) we have

Q_{n−1} − E_{I}Q_{n} + (s − n)(n + 1)(n + 1 − k)Q_{n+1} = 0,  (76)
Q_{−1} ≡ 0, Q_{0} ≡ 1.

The condition Q_{s+1} = 0 gives us the equation for the eigenvalues E_{I} of the operator H_{I}. This condition is equivalent to equation (61). We see that equation (61) and the recurrence relation give us the coefficients Q_{n} and eigenstate (74) without using the Bethe parameters λ. These parameters are necessary for transforming the vector |E⟩ from the form (59) into the form (1), or, more formally, for splitting the polynomial algebra SU_{q}^{(1,2)}(p, q) into the composition of algebra SU(2) and Heisenberg’s algebra {a_{3}, a_{3}^{+}}.

By comparing of formulas (24) and (74) for the eigenvectors of Hamiltonian (24) and (74) one can find the connection between polynomials Q_{n} and invariants S_{i} (36).

Q_{i} = (\frac{(M − i)!(N − M + i)!}{M!(N − M)!i!})^{1/2} \frac{S_{M−i}}{S_{M}} ,  {i = 0, …, M. \atop M < N}  (77)

Q_{i} = (\frac{(N − i)!i!(N − M)!}{N!(M − N + i)!})^{1/2} \frac{S_{N−i}}{S_{N}} , {i = 0, …, N. \atop M ≥ N}

In general, comparing the method of polynomial algebras with the method of Bethe ansatz one can conclude that according to the method of polynomial algebras eigenvectors (74) of Hamiltonian are generated by the operator having a form of a polynomial, while in the method of Bethe ansatz the generating operator (19) has the factorized form. Correspondingly, in the first case we are interested in the coefficients Q_{n} of this polynomial, and in the second one in the roots λ_{i} of the related algebraic equation. We shall investigate this connection in more detail elsewhere.

9. Conclusion

We have found that if N > M the system of Bethe equations (23) can be weakened. In fact, for each M only finite number of states are described by the usual system (23), but for infinite number of states reduced system (23) can be used. This system contains M equations for any N > M. With its help one can find their asymptotics for large N. Solutions of equation (28) satisfy the invariance conditions (36). The other quantum optical models, such as Dike model and four-boson interaction have similar properties, and we intend to investigate them subsequently. We also intend to investigate connection between equations (23),(28) and the polynomial algebras.

Acknowledgement

The research described in this publication is supported in part by Grant N 96-02-17987 of the Russian Foundation for Basic Research. reduced system of bethe equations for three-boson interaction 15

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