On frequencies of small oscillations of some dynamical systems associated with root systems

A.M. Perelomov ^{∗}

Max-Planck-Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany

Abstract

In the paper by Calogero and author [CP 1978] the formula for frequencies of small oscillations of Sutherland system (A_{l} case) was found. In present note the generalization of this formula for the case of arbitrary root system is given.

1. Introduction. In this note, we consider the classical Hamiltonian systems associated to the root systems, with the potential v(q) = g^{2} sin^{−2} q. Such systems were introduced in [OP 1976] as the generalization of the Calogero-Sutherland systems [Ca 1971], [Su 1972].

It is known that these systems have the equilibrium configurations which are distinguished by many remarkable features (see papers [Ca 1977ab], [Pe 1978], [CP 1978], [CP 1979], [ABCOP 1979], [MOP 1981], [OP 1981], [KPS 1992], [BMS 2000], [CS 2002] and books [Pe 1990], [Ca 2001]). In the simplest case (A_{l} case with the potential v(q) = g^{2} sin^{−2} q), the explicit expression for frequencies of small oscillations near equilibrium was found in the paper [CP 1978]. For the case of the potential v(q) = q^{2} + g^{2}q^{−2}, these frequencies are known for the case of arbitrary root system and they are proportional to the degrees of basic invariants of the Weyl group of the corresponding root system [Pe 1978].

The main result of this note is the explicit formula for the frequencies of such oscillations for trigonometric case. Namely, these frequencies are proportional to the coefficients r_{j}(g) in the expansion of the sum of positive roots in the simple roots {α_{j}}:

∗On leave of absence from Institute for Theoretical and Experimental Physics, 117259 Moscow, Russia. Current e-mail address: perelomo@dftuz.unizar.es

2 ρ(g) = ∑_{α∈R^{+}} g_{α} α = ∑_{j=1}^{l} r_{j}(g) α_{j}.  (1)

This is the generalization of the old result by Calogero and author [CP 1978].^{1}

As a byproduct at g_{α} = 1, we discovered the identity for the root systems which we cannot find in the literature, namely,

∏_{j=1}^{l}(d_{j} − 1)d_{j} = z · ∏_{k=1}^{l} r_{k}.  (2)

Here {d_{j}} are the degrees of the basic invariants of the root systems, and z = z(G) is the dimension of the center of the simple compact simplyconnected Lie group G corresponding to the root system R.

2. General description. The systems under consideration were introduced in [OP 1976] and they are described by the Hamiltonian (for more details, see [OP 1981] and [Pe 1990])

H = \frac{1}{2} p^{2} + U(q),  p^{2} = (p, p) = ∑_{j=1}^{l} p_{j}^{2},  (3)

where p = (p_{1}, ..., p_{l}), and q = (q_{1}, ..., q_{l}) are the momentum vector and the coordinate one in the l-dimensional vector space V ∼ ℝ^{l} with the standard scalar product (·, ·). The potential U(q) is constructed by means of the certain system of vectors R = {α} = R^{+} ∪ R^{−} in V, so-called root system.

U = ∑_{α∈R^{+}} g_{α}^{2} v(q_{α}),  q_{α} = (α, q).  (4)

The constants g_{α} satisfy the condition g_{α} = g_{β} if (α, α) = (β, β). Such systems are completely integrable for v(q) of five types. Here we consider only the case v(q) = sin^{−2} q.

Note that these systems are the generalization of the Calogero-Sutherland systems [Ca 1971] and [Su 1972], for which R^{+} = {e_{i} − e_{j}, i < j} and {e_{j}}, j = 1, ..., l + 1 is the standard basis in ℝ^{l+1}.

1 For numerical calculations of such type quantities see [CS 2002].

3. Root systems. We give here the basic definitions. For more details, see [Hu 1972], [Ma 1987] and [OV 1990]. Let V be the l-dimensional real vector space with a standard scalar product (·, ·), (α, β) = ∑ α_{j}β_{j}, and let s_{α} be the reflection in the hyperplane through the origin orthogonal to the vector α

s_{α} q = q − (q, α^{∨}) α,  α^{∨} = \frac{2}{(α, α)} α.  (5)

Consider a finite set of nonzero vectors R = {α} generating V and satisfying the following conditions:

1. For any α ∈ R, the reflection s_{α} conserves R : s_{α}R = R.

2. For all α, β ∈ R, we have (α^{∨}, β) ∈ ℤ.

The set {s_{α}} generates the finite group W (R) – the Weyl group of R. The root system R is called reduced system if only vectors in R collinear to α are ±α. Let us choose the hyperplane which does not contain the root. Then we have R = R^{+} ⋃ R^{−}, and R^{+} is the set of positive roots. In R^{+} there is the basis {α_{1}, …, α_{l}} (the set of simple roots) such that α = ∑_{j} n_{j}α_{j}, n_{j} ≥ 0 for any α ∈ R^{+}. The root system R is called irreducible system if it can not be the union of two non-empty subsets R_{1} and R_{2} which are orthogonal each other.

Let {α_{1}, …, α_{l}} be the set of simple roots in R, R^{+} be the set of positive roots, and {λ_{j}} be a dual basis or the weight basis: (λ_{j}, α_{k}) = δ_{jk}.

Let Q be the root lattice, and Q^{+} be the cone of positive roots

Q = {β: β = ∑_{j=1}^{l} m_{j}α_{j}, m_{j} ∈ ℤ }; Q^{+} = {γ: γ = ∑_{j=1}^{l} n_{j}α_{j}, n_{j} ∈ ℕ} .

(6)

Let P be the weight lattice, and P^{+} be the cone of dominant weights

P = {λ: λ = ∑_{j=1}^{l} m_{j}λ_{j}, m_{j} ∈ ℤ}; P^{+} = {μ: μ = ∑_{j=1}^{l} n_{j}λ_{j}, n_{j} ∈ ℕ} .

(7)

Following [Ma 1987], we define a partial order on P as follows: λ ≥ μ if and only if λ − μ ∈ Q^{+} (or (λ, λ_{j}) ≥ (μ, λ_{j}) for all j = 1, …, l). The set of linear combinations over ℝ of the functions f_{λ}(q) = exp {2i(λ, q)}, λ ∈ P, q ∈ V may be considered as the group algebra A over ℝ of the free Abelian group P. For any λ ∈ P, let us denote e^{λ} ∼ f_{λ}(q) as the corresponding element of A, so that e^{λ}e^{μ} = e^{λ+μ}, (e^{λ})^{−1} = e^{−λ}, and e^{0} = 1, the identity element of A. Then e^{λ}, λ ∈ P form the ℝ-basis of A.

The Weyl group W (R) acts on P and hence also on A : s(e^{λ}) = e^{sλ} for s ∈ W and λ ∈ P. We denote as A^{W} the subalgebra of W -invariant elements of A. It is known that A^{W} is the algebra free generated by basic invariants of degrees d_{1}, ..., d_{l}. We introduce also a g-deformed vector

ρ(g) = \frac{1}{2} ∑_{α∈R^{+}} g_{α} α.

Let δ be the highest root, i.e. the positive root such that δ ≥ α for all α ∈ R^{+}.

4. Results. Let us consider two classical systems characterized by the Hamiltonians

H_{m} = \frac{1}{2} p^{2} + U_{m}(q),  m = 1, 2  (8)

with

U_{1}(q) = − ∑_{α∈R^{+}} g_{α} log | sin(q_{α})|,  (9)
U_{2}(q) = ∑_{α∈R^{+}} g_{α}^{2} sin^{−2}(q_{α}), q_{α} = (α, q), g_{α} = g_{β} if (α, α) = (β, β).(10)

The configuration space of such system is the Weyl alcove

{q | (q, α_{j}) > 0, j = 1, …, l, (q, δ) < π},  (11)

where {α_{j}} is the set of the simple roots, δ is the highest root, and as it well known the potential energies U_{1} and U_{2} have an isolated minimum at the point q̄ inside the alcove^{2}. Let us denote this minimum as U_{1}^{(0)} and U_{2}^{(0)}, U_{m}^{(0)} = U_{m}(q̄). Then near this point we have

U_{m}(q) ∼ U_{m}^{(0)} + \frac{1}{2} a_{jk}^{(m)} ξ_{j}ξ_{k},  ξ_{j} = q_{j} − q̄_{j}.  (12)

The matrices a^{(m)} are positive definite and as it is known ([Pe 1978]),

a^{(2)} = c (a^{(1)})^{2} ,  c = const.  (13)

2 Let us note that the first system is the generalization of the so-called ”Dyson system“, [Dy 1962a], [Dy 1962b].

So, we need to know just one of them. The frequencies of small oscillations ω_{j}^{(m)} near the equilibrium configuration are square roots of eigenvalues of

these matrices and we have ω_{j}^{(2)} = c (ω_{j}^{(1)})^{2}. The eigenvalues of these matrices were known before only for A_{l} case (see [CP 1978]). Here we give the explicit expression for them for general case.

Theorem. Eigenvalues of the matrix a^{(1)} are equal to the quantities {2 r_{j}(g)}, j = 1, …, l, where {r_{j}(g)} are the coefficients in the expansion

2 ρ(g) = ∑_{α∈R^{+}} g_{α} α = ∑_{j=1}^{l} r_{j}(g) α_{j}.  (14)

Proof. Note that the function U_{1}(q) is equal to − log |Ψ_{0}^{κ}|, where Ψ_{0}^{κ} is the solution of the Schrödinger equation ([OP 1983])

(\frac{1}{2} p̂^{2} + U_{2}(q)) Ψ_{0}^{κ} = E_{0}(κ) Ψ_{0}^{κ},  g_{α}^{2} = κ_{α}(κ_{α} − 1), p̂_{j} = −i \frac{∂}{∂q_{j}} , (15)

Ψ_{0}^{κ} = ∏_{α∈R^{+}}(sin q_{α})^{κ_{α}}.  (16)

The function Ψ_{0}^{κ}(q) is positive inside the Weyl alcove and has a maximum at point q = q̄. Let κ_{α} = s κ̃_{α} and let us consider the limit at s → ∞. Then the function

Φκ̃ ; 0(ξ) = lim_{s→∞} \frac{Ψ_{0}^{κ} (q̄ + ξ/\sqrt{s})}{Ψ_{0}^{κ}(q̄)},  κ_{α} = s κ̃_{α}  (17)

takes the Gaussian form^{3}

Φκ̃ ; 0(ξ) = exp (− \frac{1}{2} a_{jk} ξ_{j}ξ_{k}) .  (18)

lim_{s→∞} s^{(−n/2)}P_{n}^{(α,α)}(x/\sqrt{s}) = (2^{n}n!)^{−1}H_{n}(x),

3 In other words at the limit κ_{α} → ∞ (s → ∞) our quantum system become the oscillatory one for which the correspondence between classical and quantum systems is well known. Note that at this limit wave functions go to oscillatory ones. For example for the case of one degree of freedom we have where P_{n}^{(α,β)}(x) and H_{n}(x) are Jacobi and Hermite polynomials correspondingly. For multidimensional case analogous formulae give the multivariable Hermite polynomials.

From other side, at this limit the Schrödinger equation

(\frac{1}{2} p̂^{2} + U_{2}(q)) Ψ_{m} = E_{m} Ψ_{m},  m = (m_{1}, …, m_{l})  (19)

takes the form

(− Δ_{ξ} + \frac{1}{2} ∑_{j,k=1}^{l} b_{j,k} ξ_{j}ξ_{k}) Φ_{m} = Ẽ_{m} Φ_{m},  (20)

where

b = a^{2},  Ẽ_{m} = Ẽ_{0} + ∑_{j=1}^{l} ω_{j} m_{j},  (21)

and ω_{j} are the eigenvalues of the matrix a. Note that the spectrum of the quantum problem

HΨ_{m} = E_{m} Ψ_{m}.

is given by the formula

E_{m} = ∑_{i,j=1}^{l}(λ_{i}, λ_{j}) m_{i} m_{j} + 2 ∑_{j=1}^{l}(λ_{j}, ρ(κ)) m_{j} + E_{0}(κ).  (22)

And at s → ∞ and we obtain

Ẽ_{m} = lim_{s→∞} (s^{−1} E_{m}(κ)) = Ẽ_{0} + 2 ∑_{j=1}^{l}(λ_{j}, ρ(κ̃)) m_{j}.  (23)

Comparing (21) and (23), we obtain the main formula

(ω_{j}^{(1)})^{2} = 2 r_{j}(κ̃) = 2 (λ_{j}, ρ(κ̃)).  (24)

So, the classical result is obtained from the quantum result at s → ∞.

In conclusion, we give the explicit expression for the maximum value of the function Ψ_{0}^{κ}(q). This formula was given by I. Macdonald as an conjecture [Ma 1982] that was proved later by E. Opdam [Op 1989]. For the simplest case (κ_{α} = 1 for all α), it has the form

|Ψ_{0}^{1}(q̄)|^{2} = \{ ∏_{α∈R^{+}}(sin q_{α})\}^{2} = \frac{|W |}{2^{|R|}} ∏_{j=1}^{l} (\frac{d_{j}}{d_{j} − 1})^{d_{j}−1} ,  (25)

where |W | is the order of W, and |R| is the number of roots.

Acknowledgements. I am grateful to Prof. R. Sasaki who raised the question on small oscillations, to the Physics Department of Oviedo University and Max-Planck-Institut für Mathematik, Bonn for the hospitality.

References

[ABCOP 1979] Ahmed S., Bruschi M., Calogero F., Olshanetsky M.A. and Perelomov A.M., Properties of the zeros of the classical polynomials and of Bessel functions, Nuovo Cim. 49 173-199

[BMS 2000] Bordner A.J., Manton N.S. and Sasaki R., Calogero-Moser models V: supersymmetry, and quantum Lax pairs, Prog. Theor. Phys.

103 463-487; hep-th/9910033

[CP 1978] Calogero F. and Perelomov A.M., Properties of certain matrices related to the equilibrium configuration of one-dimensional many-body problems with pair potentials V_{1} = − log | sin x| and V_{2} = 1/ sin^{2} x, Commun. Math. Phys. 59 (1978) 109-116

[CP 1979] Calogero F. and Perelomov A.M., Some diophantine relations involving circular functions of rational angles, Lin. Alg. Appl. 25, 91-94

[CS 2002] Corrigan E. and Sasaki R., Quantum vs classical integrability in Calogero-Moser systems, hep-th/0204039

[Ca 1971] Calogero F., Solution of the one-dimensional N-body problem with quadratic and/or inversely quadratic pair potentials, J. Math. Phys. 12 419-436

[Ca 1977a] Calogero F., On the zeros of the classical polynomials, Lett. Nuovo Cim. 19 505-508

[Ca 1977b] Calogero F., Equilibrium configuration of the one-dimensional n-body problem with quadratic and inversely quadratic pair potentials, Lett. Nuovo Cim. 20 251-253

[Ca 2001] Calogero F., Classical Many-Body Problems in One-Two- and Three-dimensional Space Amenable to Exact Treatments (Solvable and/or Linearizable), Springer-Verlag [Dy 1962a] Dyson F.J., Statistical theory of the energy levels of complex

systems, I, II, III, J. Math. Phys. 3, 140-156, 157-165, 166-175

[Dy 1962b] Dyson F.J., A Brownian motion model for the eigenvalues of a random matrix, J. Math. Phys. 3, 1191-1198

[Hu 1972] Humphreys J.E., Introduction to Lie Algebras and Representation Theory, New York, Springer

[KPS 1992] Kogan Ya.I., Perelomov A.M. and Semenoff G., Charge distribution in two-dimensional electrostatics, Phys. Rev. B45, No.20, 12084-12087

[MOP 1981] Mikhailov A.V., Olshanetsky M.A. and Perelomov A.M., Twodimensional generalized Toda lattice, Commun. Math. Phys. 79, 473-488

[Ma 1982] Macdonald I., Some conjectures for root systems, SIAM J. Math. Anal. 13, 988-1007

[Ma 1987] Macdonald I., Orthogonal polynomials associated with root systems, Preprint

[Ma 1995] Macdonald I., Symmetric Functions and Hall Polynomials, Oxford Univ. Press

[OP 1976] Olshanetsky M.A. and Perelomov A.M., Completely integrable Hamiltonian systems connected with semisimple Lie algebras, Invent. Math. 37, 93-108

[OP 1981] Olshanetsky M.A. and Perelomov A.M., Classical integrable finite-dimensional systems related to Lie algebras, Phys. Reps. 71 314-400

[OP 1983] Olshanetsky M.A. and Perelomov A.M., Quantum Integrable Systems Related to Lie Algebras, Phys. Reps. 94, 313-404

[OV 1990] Onishchik A.L. and Vinberg E.B., Lie Groups and Algebraic Groups, Springer-Verlag, Berlin-Heidelberg

[Op 1989] Opdam E., Some applications of hypergeometric shift operators, Invent. Math. 98, 1-18 [Pe 1978] Perelomov A.M., Equilibrium configurations and small oscilla-

tions of some dynamical systems, Ann. Inst. H. Poincaré A28, 407-415

[Pe 1990] Perelomov A.M., Integrable Systems of Classical Mechanics and Lie Algebras. I, Birkhauser

[Su 1972] Sutherland B., Exact results for a quantum many-body problem in one-dimension. II, Phys. Rev. A5 1372-1376