olshanetsky m.a., perelomov a.m. quantum integrable systems related to lie algebras

Systems of type VI’ Generalized periodic Toda lattices 366 Some quantum integrable finite-dimensional systems related to Lie algebras are considered.. Introduction The purpose of this p

QUANTUM INTEGRABLE SYSTEMS RELATED

TO LIE ALGEBRAS M.A OLSHANETSKY and A.M PERELOMOV

Institute of Theoretical and Experimental Physics, Moscow, 117259, USSR

Received 15 November 1982

Contents:

1 Examples Systems with one degree of freedom 318 15 Systems with v(q) = 5(q) Bethe Ansatz 369

3* Abstract quantum systems, related to root systems 331 16.1 Factorization of the ground-state wave function 372

4 The proof of complete integrability of the systems 335 16.2 Green’s functions on symmetric spaces 376

5** Complete integrability in the abstract case 337 Appendices

6* Wave functions 340 A Groups generated by reflections and their root systems 378

8* Systems of type II (v(q) = sinh”? q) 350 C Laplace operators and spherical functions 394 9* Systems of type III (v(q) = sin“? 4) 352 D Connection between Hamiltonians and Laplace operators 399

10* Systems of type IV (v(q) = ?(4)) 354 E Proof of Propositions of section 5 401

12* Systems of type VI (v(q) = exp 4) 363

13* Systems of type VI’ (Generalized periodic Toda lattices) 366

Some quantum integrable finite-dimensional systems related to Lie algebras are considered This review continues the previous review of the same authors [83] devoted to the classical aspects of these systems The dynamics of some of these systems is closely related to free motion in

symmetric spaces Using this connection with the theory of symmetric spaces some results such as the forms of spectra, wave functions, S-matrices, quantum imegrals of motion are derived In speci’ cases the considered systems describe the one- ~dimensional n-body systems interacting pairwise

7

Weierstrass function, so that the first three cases are merely subcases of the fourth The system characterized by the Toda nearest-neighbour

potential exp(q; ~ gj+1) is moreover considered

This review presents from a general and universal point of view results obtained mainly over the past fifteen years Besides, it contains some new results both of physical and mathematical interest

Single orders for this issue

accompanied by check

| PHYSICS REPORTS (Review Section of Physics Letters) 94, No 6 (1983) 313-404

QUANTUM INTEGRABLE SYSTEMS RELATED TO LIE ALGEBRAS

M.A OLSHANETSKY and A.M PERELOMOV Institute of Theoretical and Experimental Physics, Moscow, 117259, USSR

This article should be considered as the follower of a previous one by the same authors It discusses Quantum systems while the earlier article, Physics Reports 71, No 5 (1981) 313-400 limited itself to Classical systems While both are self contained these two articles gain at being considered together

0 Introduction The purpose of this paper is to survey the quantum systems described by a Hamiltonian of the type

H=5 > pi+UQu -4n), j=l By = ~i 88g, (0.1)

where the potential U(qi, q,) is a function of a special type, so that there exist n commuting integrals of motion The corresponding classical systems were considered in detail in our previous paper [83]

It is worthwhile noticing that some of these quantum systems have been investigated earlier than their classical counterparts

For the first time in 1961-74 a number of exact results were obtained on the type of wave functions, spectrum and the nature of scattering for one-dimensional many-body problems, i.e for systems with potential

and a function v(q) of type (0.5) was considered in [10, 59}

The quantum periodic and unperiodic Toda lattices were also considered, reading

Lexp é VI (for U(0.8))

(Here Y(é) is the Weierstrass function.) The delta potential (0.3) is related to the quantum inverse scattering method in one-dimensional quantum field theory Numerous papers are devoted to this approach See, for example, the reviews [13, 55, 98-100] We partially touch on this subject as far as it concerns finite-dimensional systems

There exists a number of results about the complete system of quantum commuting observables (quantum integrals of motion), wave functions, spectra and so on The interest for models with known

đC SOI1U ONS S aSSoclafted A mn ine At la Ty C duantim propiems A | nrec and more Dd

with a realistic interaction acting between them (Coulomb or nuclear), do not enable one to obtain any

^ Avr 4 Ae © -T- ort awe Arno 7) - lì a

in the more realistic cases Besides, these models may be useful to estimate the accuracy of different

approximate me OdUSs wWort re ng 16 ne c d T 0 ne VSTCI OF gered here

related to a number of interesting systems with an infinite number of degrees of freedom For example, the Toda lattice is related to the duality equation in the Yang-Mills theory [31], the rational solutions to the Korteweg-de Vries, the Kadomtsev—Petviashvily and the Benjamin—Ono equations are described by means of classical solutions of the systems of types I, II and III (see [83]) There exists a number of other examples

In order to take into consideration systems of type (0.8), and also a number of others, it is necessary vectors R = {a} in configuration space with some special properties

space The potential (0.8) is described by means of the subsystem {2e, — e.— €3, —2e2+ €,+ €3, —2e3+

et C;, Oj — Ớy; A k=1, 2, 3; j<k} The nonperiodic Toda lattice (0.9) is related to the subsystem

= {e; — G41;/=1, n—1}, and for the periodic Toda lattice it is necessary to add to R, the vector

€, — €; AS we shall see further, the subsystems R, have a certain group-theoretical significance

It turns out that the root systems are related canonically to simple Lie algebras or to some homogeneous spaces of Lie groups (symmetric spaces), or to infinite dimensional Lie algebras (the so- called Kac-Moody algebras)

Therefore, besides the evident discrete symmetries of the potential under consideration, as for example the invariance of the potentials of type (0.1) under transformations of the permutation group, there exist as a rule additional hidden symmetries, generated by the Lie algebras mentioned above All types of potentials, except the periodic Toda lattice and potentials of type IV (0.9), have such types of symmetries

The key point of the relation between the theory of symmetric spaces (SS) and quantum systems of types I-III, V and VI is the existence of a simple transformation of the Hamiltonians under consideration to Laplace—Beltrami operators on SS These operators are uniquely determined by means

of a Riemannian metric on SS

Using the theory of SS developed by Gelfand, Harish-Chandra, Beresin, Karpelevich, Gindikin and Kostant [4, 18, 20, 21, 28, 29, 24] one can obtain a number of results for the quantum systems On the other hand, a number of formulae obtained in the papers mentioned above, and generalized in this paper for an arbitrary root system, are new mathematical results

Apparently this approach does not apply to systems of type IV and to the periodic Toda lattice Thus there are less results for these systems However, it turns out that the periodic Toda lattice is related to Kac-Moody algebras [32] The relation between systems of type IV and Kac-Moody algebras, or any other infinite algebras, is not yet clear

The paper is organized as follows The paper begins with self-contained examples of one-dimensional systems In section 2 we present a

a venera are ma An oO ne erry Pe Aig are nrere c er kỈ n1 s ñ = "Tom k2 uid Ð k2 ANnrenr H Ð oan `

more rigorous level using an algebraic approach In sections 4 and 5 we deal with the integrals of

6 We describe the properties of the various systems in sections 7-15

In appendices A-C basic facts about root systems and symmetric spaces are derived The proofs of some statements formulated in section 5 are given in appendices D and E Sections denoted by one asterisk use partly the material presented in appendices A-C and those denoted by two asterisks are based entirely on this material Appendices A and B are reproduced from [83], so as to make this paper self-contained

Everywhere in this paper the particle mass and the Planck constant are set equal to unity

We had long discussions on different parts of this paper with F Calogero tO whom we are © deeply

° Or we °®OnSsrde C ° ne exa 2Ì€S—O œt\;cfe oy * O =deo c€—O C©edOo , W + = JOSSESS

explicit solutions The description of general systems of such type is given in the next section

L The motion of one particle in the poterial g”q

In this case

H=i?p?+g?4?= 3 đ!dq?+ 824”, (1.1)

where it is convenient to write g” as

We are interested in the solutions

As, = Exh ; Wi (0) =0 5 FE, = $k? ` (1.3)

The following results can be easily demonstrated:

1 For g?>-—š, the operator H is selfadjoint

2 The spectrum is continuous and covers the semi-axis 0s Ei, <m,

3 The wave function of the zero-energy state is of the form

4 After substituting ¢ = Wod:, the operator H is transformed into the operator — B/2, where

d2

U4 W xu and we shall normalize the function x(q by the conditior p, (0) = 1 _equatio

6 Let us give the explicit expression for the function ¢(q)

i (q) = 2°77 P(e + a(kqy oS -1nlka) (1.9) where J,,(x) is the standard Bessel function

As a consequence we obtain the expansion of ¢,(q) as a series,

After averaging we obtain the integral representation for the function

¢,(q) = | (cosh q— sinh g øñ)"“*'* đụ (8) (1.18)

6 The explicit expression for the function ¢,(q) has the form

x(q) = F((u + ik)/2, (w — ik)/2, w +2; —sinh? q), (1.19)

where F(a, b, c; x) is the hypergeometric function

7 The asymptotic behaviour of the function ¢,(q) as q >®% has the form

and the following results are valid:

1 For g? >—3, the operator H is selfadjoint

2 The spectrum of H is discrete:

Bội(4)= ~l + 2p) di(q)- (1.27)

For =(n- 1)/2 the operator B is the radial part of the Laplace-Beltrami operator on the n- dimensional sphere

§" = {x ER"*'|x = (cos g, Asin q), n° = 1}

The function ¢,(q) is called zonal spherical function

5 The free motion of a particle on the sphere §” is described by the functions

which are the analogs of plane waves in Euclidean space

The integral representation for the function ¢,(q) has the form

|wP=1

6 The function ¢;(q) can be expressed through the Gegenbauer polynomials C/:

IV The motion in the potential g* P(q)

The Weierstrass function PA(q)* is doubly-periodic in the complex plane g and, hence, depends on

two parameters (semiperiods) w and w’ Let us note that the systems considered in sections 1, 2 and 3

2 special cases of this system They may be obtained as imiting ases when one or both of the periods

of the function A(q) go to infinity

It is known that for any A there are two solutions to this equation, depending on two parameters k and 5,

=e o(q-byo(q), z=e”"“ơ(q+bơ(4) (1.34)

where a(q) is the Weierstrass sigma-function, and the quantities b and k can be found from the equations

Here ¢ is equal to o’/o These solutions are linearly independent, if b is not a semiperiod (b# «a, w’)

However, the solutions are singular at g = 0 and at g = 2w (2w is a real period of the function A(q) which can be taken equal to 77) But the sum of these solutions,

(4) = (4) + Yo(q) = e“ o(q — b)/a(q) +e" o(g + by/a(q), (1.36)

is the solution to (1.32) which is regular at q = 0

The requirement that this solution be regular also at g = 7 yields the equation e" ơ(m_— b)+e"*" ơ(m + b)=0,

The quantity 5,, is imaginary (b,, = i8,,) and we can write instead of (1.39)

Note that as a limiting case we can obtain one of the previous systems If, for example w = 7/2 and w’

V The motion in the potential g7q~* + 3w7q’

The Schrédinger equation in this case has the form

2

The spectrum of this equation is discrete The normalized ground-state wave function has the form

Yo(q) = Nog” exp(—20q”) (1.46)

The energy is given by the expression

For the half-integer values 4 = (n — 1)/2 this equation coincides with the radial part of the equation for

The explicit solutions to (1.49) can be expressed through Laguerre polynomials,

Xo = cosh 3q +327 e?”,

X, = ez, there follows that the pair (g, z) defines a global coordinate system on H* These are the so-called

horospheric coordinates Parabolas (q = const., —œ < z <%) are horospheres in Lobachevsky geometry

for which H is one of the possible realizations The reduction of the Laplace operator in RỶ to H” has

The equation determining the function f,(q) may be evaluated after inserting (1.56) in (1.52):

2(p)= cA)TCi(p~ À))TCi(p+À)) | (1.64)

The function %(p) is the solution to the equation

vanishing as p> +

VI' The motion in the potential v(q) = g” cosh q

We have the Schrédinger equation

with boundary conditions W,(x)—> 0 at x>+oœ

The solution to this equation is expressed in terms of Mathieu functions and is not given here

We present here only the expression for the spectrum of this problem in the quasiclassical approximation According to the Bohr-Sommerfeld quantization rules, we have

n-dimensional space $ (p; = —i /4q;) The potential U(q) is constructed by a certain system of vectors {a} in the space §, the so-called root system In this section we shall discuss some particular examples

1 The simplest and at the same time the most important example is the one-dimensional system of n pairwise interacting particles In this case

by ia in the potential of type II II or i one puts a= 0, one gets the systems of type III or I respectively

Note that the results reviewed i in this paper are trivially extended to the larger class of functions v(é)

_ obtamned i b đ 1DDTOD co Ate 1 a afe argo men ataia Op đ ate PPT vs đ 4 h Cy 7,

introduction of different types of Particles For instance, the results apply toa model with two types of

a+ alae An_a 2e ANS ` h 2 2 ¬ a 3 3 he r2 /^3f1/x13 4 = r^

Jd s § ũ s § USCS bả Ud; L4 U 1 — Ệ u 1O UY 9 ¥pC

The system contains n, particles of one type and n.=n-—n, particles of the other type Particles of

——————— điRerent type are attracted pairwise via the potential —g*a* cosh~* a(q; ~ 4 ) Same type of particles are -

instead repulsed pairwise via the potential g*a’ sinh”* a(q; — q,) The analogous situation for classical systems was considered in [95, 84] We shall not exploit this possibility below

Due to the singularity of the potential U(q) (2.1) at q, = q; the ordering of the particles in the course

of the motion cannot be changed, and we put q; > q, if j < k The configuration space A for the systems

I, I] and V in the c.m.s is therefore the inner part of the polyhedral angle:

4

àe z| Me and for the systems III and IV it has the form of the simplex

J=1

The configuration spaces A and A, for n =3 are respectively the interior of the angle 7/3, and the regular triangle:

41 4o

Hence we have an n-body problem on a line in the first-case and on a circle in the second case In the case of the system of type III the radius of the circle is equal to 1/2, the angle coordinates of the particles are equal to 2aq, and the potential is equal to g?a7/r’, where r is the distance between particles

on the plane Similarly for the system of type II one can consider the motion along a hyperbola with potential g’a’/r’ instead of the motion along a straight line with potential g’a” sinh”*a(q; — q,.)

We shall assign to the systems of types I, III and V the systems of types Ip, II) and Vo which are the limit

of those as g?> 0 One gets thereby immediately billiards problems in the polyhedral angle A (2.4), in the simplex A, (2.5) and in A (2.4) in the field of a harmonic potential

The next example is the system of n particles with the nearest-neighbour interaction given by the

potentials

or U(a)= > P0G-4i), Gui=a), (WT) 27)

—————— The potentials (2.2), (2.6) and (2.7) are called A„-; type*

2 The next example corresponds to the potentials, denoted as BC,,* which may be written as

Ũ() = g” Ð, [0(4 ~ 4) + 0( + 4)] + gì = 0(q.) + g3 > v(2q.), (2.9)

k<j

where v(€) is one of the five types (2.3)

The BC,, generalization of the Toda lattice takes the form

(nonperiodic case) and

(periodic case)

In the formulae (2.10) and (2.11) the function v(é) has the exponential form (2.8)

One can consider the BC, systems I-V as those of 2n+1 particles on the line (2.2) If their coordinates and momenta satisfy at the initial time the symmetry conditions

and these conditions will be conserved in time under the equations of motion and the potential (2.2) will take the form (2.9) The same is true for the potentials (2.10) and (2.11)

The configuration space for the potentials of types I, []_and V is of the form

and for the potentials VI (2.10) and VI’ (2.11) it is the whole space 0

3 In particular the systems under consideration describe the interaction between three particles with coordinates q;, q2, g3 in a special form For the systems of types I-V (2.3) the potentials have the form

U(q)= g [oíq.— q›) + Đ(đì — 43) + Đ(đ›— 4) + + qi:+4qs)+ — + 0(—24; + g; +

he case or the potential a pe n gated in Ji ana in nese stems are denoted as ¬

systems The corresponding generalized Toda potentials have the form

and

U(q) = 0(qa— q›)+ v(-243+ Git G2) + v(-2q1+ Gat qs) — (VỤ) (2.16) Note that the condition ¥?_, g; = 0 allows one to consider the systems G2 as systems with two degrees of freedom

4, Let us consider the most general Hamiltonian (2.1) for the system of type V with two degrees of freedom In polar coordinates q = (r, ¢) on a plane © the potential has the form

2 m/2—1 2

Un 4)=Š "> sin nˆ(ø+2k m7 > sin ?(+@k+0)+5#, (if m is even),

Un (q) = g-* > sin *(¢ +2k m) + P, (if m is odd) (2.18)

It is worthwhile to stress that for m = 3, 4 and 6 these systems are equivalent to those of type V (or I if

w = (0) considered before Namely there is the following correspondence U3(q)~ (2.2) n=3,

Us(q) ~ (2.14)

The systems (2.17) and (2.18) are called systems of type Iz(m)

It is interesting to consider in (2.17), (2.18) the limit g?>0, gi0 In this case the Schrédinger equation is reduced to the billiards problem in the field w*r’/2 in the angle z/m

the previous section The necessary information for the reader is contained in appendix A

generalized Toda lattices (systems of types VI, VI’) were defined in [63]

1 Let R= {a} be the system of roots in n-dimensional space 9, R, be the subsystem of positive roots, g = (q:,. 4G,) and p= (py, .p,) be the vectors of the coordinates and momenta, respectively, and q = (a,q) First of all let us consider the systems of types I-V, which are described by the Hamiltonian

Here the coupling constants gz are the same for the equivalent roots, i.e those roots that are connected with each other by transformations of the Coxeter group W, and the function v(q) has one of the five forms (2.3) It is worth stressing that the noncrystallographic cases I,(n), H3 and Hy, are also considered for the potentials of types I and V The additional quadratic term in the potential of type V has the form

and for the systems of types III and IV it is the Weyl alcove

where d depends on the real period of the function ø(£) and @ is the maximal root in R, Thus, the potentials U(q) (3.2) for the systems of types III and IV are invariant under the affine Weyl group W, which corresponds to the systems of table A3 in appendix A including BA2, from table A4 The last root system is equivalent to the unique nonreduced root system BC, It would be interesting to investigate the potentials I-V for the whole table A4, but we do not know any positive results about them now

We shall consider only irreducible root systems As it follows from (3.2), a reducible root system corresponds toa Hamiltonian system which is a union of noninteracting subsystems

Ve shall discuss also the billiards problem p2 0) m the Wevl chambers (the svstem o ne in

the Weyl alcoves (the systems of type Mlb), a and the billiards Problem in the Weyl chambers with an

^ Mm Sf arn wrall Aannan « re OP “4

AGG iond ta4n 1onic-poter di YO} ip alg g8 A GÌ k C PT d

case as well

generalized nonperiodic Case IS connected with the algebras of zero › height For simplicity we single out

ne = spe He Cian pe M ne genera eq pe 1OdIC ca e ’ cOrr€SDonds to the alsebras v h_ heights 5 ¿2

and 3 3 Originally tt these systems w were e defined for the algebras 0 of height 1 only [63],

taking into account the BC,, type), the systems of types I, and Vo with general root systems (including the noncrystallographic case), the systems of type III, with affine root systems and the systems of types

VI and VI’ with the root systems of Kac-Moody algebras

The potentials of types I-VI’ corresponding to restricted root systems are presented in table 3.1

Additional information about the systems of type VI’ connected with Kac-Moody algebras of heights 2 and 3 is contained in table 3.2, The tables are constructed by means of data collected in appendix A

The noncrystallographic root systems I,,,, H; and H, are not represented in the tables About the systems I,(m) see (2.17) and (2.18) Let us now specify the type of potentials, introducing the following notations

(1) The functions v(q) are of types I-VI (see (2.3), (2.8))

(2) For the systems IE,-VE,,* in the expression

(S) For the systems IF,-VF,,

(7) There are additional constraints on the coordinates for the systems A,-1, Es, E, and Gp, (see appendix A)

(1) In table 3.2 the additional term corresponding to the highest weight 0 is underlined

(2) The system VI’ BAz2, is equivalent to the system VI’ BC,,

4 The proof of complete integrability of the systems

To investigate the integrability of the systems considered in section 2, we study the algebra of the operators J, introduced in [11] (which are analogous to the classical integrals of motion) not only in the classical but also in the quantum case; note that we use a superimposed caret to distinguish the quantum operators from the corresponding classical quantities We prove that the commutators (Je Ji] vanish

Moreover we give the expression of the integrals L, i and L, which are the quantum version of the classical integrals I, = tr(L*), where L is the classical Lax matrix* For the systems of type IA,,_,; these integrals coincide with those obtained previously in the papers [85, 16] Next we give the explicit expression of [, for the systems of type BC,, It is worthwhile to note that I; = 0 for these systems, The complete integrability of the systems of type V follows from the complete integrability of the systems of type I As for the Toda lattice systems VI A,,-;, VI' A,-1, VI BC,, VI’ BC,, the integrals have a similar form as for the previous systems

1 The complete integrability of the systems I-V A,,_, was proved in [41] The proof uses the results

of [11, 93] and [44]

Note first of all that, as it was pointed out in [11], the classical integrals J, can be defined as the coefficients of the characteristic polynomial det[L — AI], where L is the Lax matrix (see [83]) As it was pointed out in [11], they correspond in the quantum case to well-defined operators J, In fact any J,

is written as a sum of terms, containing only commuting operators However, the commutator of two operators [J;, Ji] is not a well-defined operator and hence the fact that these commutators vanish does

not follow from the vanishing of the Poisson bracke h equires_a separate proof, In pz

one must prove that the commutators (Jo, J], or equivalently (A, Ju], vanish

NOtTe Hrs D A ry H ry O-pDrove n oF OoOmmutato vn

easy to show that the following re relation holds:

* It is obvious that J, is a symmetrical polynomial of degree k in the moments py

+ x(é) is connected with the potential by the relation x7(¢) = v(€)+ const (see [83]).

the contribution of the quantity x{2 is fully exhibited by writing

Jn = 1H Jn] = 2(B; ~ Bì)xiax 12 † 5B, 2[ (pi — P›)x 12+ X42(Pi- 8) tree, (4.3)

We see that the first term of this sum is a well-defined operator, while the second term is not

Therefore, after terms of these types are reduced to normal orderings, there arise additional terms and the vanishing of the expression {H, J,} implies the vanishing of the commutator [H, J,] only if all such additional terms cancel

Let us now show that this does indeed happen To this end it is convenient to use another expression

of the operator J, for the systems of types LIV A,_,, which had been obtained in [93], namely

This expression implies that the operator J, contains only terms quadratic in the quantities x; = 8X(% — 4¡) and in particular, quadratic in x:2 Therefore the expression (4.1) for the quantity J,, may be rewritten in the form

where the quantities A:a, Bì, Bz and C12 again do not depend on py, p2 and X12 Now after commuting J,, written in the form (4.5), with H, there arise only well-defined terms, whose exact cancellation is implied by the vanishing of the Poisson bracket {H, J,} We have thus proved that the operators J, are integrals of motion

Therefore the commutators ors LÍ, J] contain only well-defined terms and vanish simultaneously with the

2 In addition to the integrals J, that are obtained from the coefficients of the characteristic

polynomial of the matrix £ there are other conserved polynomials in p; that are also interesting In

particular there are the integrals f, that correspond to the classical expressions J, = tr(L*) We provide below the explicit expressions of 45, f,, 5 We use the notation xi = x(q — q:) and (A’)= tr A’ for a matrix A = {A,,},

B= > pit 5g? Dd xi bi + Bip) + Se*(xks diag(pr, pa))+ 5g” S7 (xu ip?~ (xi pì, k=1 k#l k#l

* The formula (4.4) was obtained previously for the potentials of type I in [44].

3 For the systems of type BC,, the odd integrals Io,+, vanish This fact is the consequence of the vanishing of the classical quantities I2,, (see [83] p 390) We write down the expression of J,:

=2 > Pit 8g7 Dd) [X7(Ge — 1) + X7(Qu + Q)) pi + 887 > x°(q)pi

In particular, one can see that the systems I-IV BC, are completely integrable

For the other classical root systems (B,, C, and D,,) the integrals L, are obtained from (4.7) if the constants g, and (or) g2 vanish

4, For the systems of type V, the integrals I, and J, for the systems of type I allow one to construct, by the substitution p, > p, +iwgq, the rising and lowering operators Bt , Dt „ and B,., Dy

Ôt = Ï(Ð +ioqe g), — Ôš= X(p+ieq,4),

(4.8) B¿ = ÍL( — i4, q) Ô, = ⁄(p— lø4, 9)

With the help of these operators one can obtain the whole spectrum for the problems considered here and all the corresponding eigenfunctions (see section 6) Let us note that the operators B;, B, and B3,

Bi were constructed in [85] and the operators B, and Bi in [16] Now we have also obtained the complete set of operators D, and Di for the systems of type V A,,_, and the operators B, and Bi for the other systems of classical type

The formulae (4.6), (4.7) in the limit g2 +0 give the quantum integrals for the corresponding billiards

Below we shall show that the systems VI for all types of roots (the generalized nonperiodic Toda

lattices) are completely integrable

systems of types I-III for special values of the coupling constants gz and the complete integrability of the systems of type VI Moreover, as it will be shown in the following sections, this connection allows to obtain some formulae for the eigenfunctions and S-matrices for the systems I-III and VI

In the last part of this section some special properties of the integrals are reported

1 Let us consider the restricted root system R in the n-dimensional space = {q1, gn}

Theorem 5.1 [42] Let H be a Hamiltonian,

(|œ[ is the length of the root œ)

In the formula (5.2) the sign ‘‘+’’ corresponds to the systems of type II, the sign the systems of type III, and the term p” must be omitted for the systems of type I

The proof of the Theorem is given in appendix D

oe? corresponds to

*Tt is worth noting that the constants 4g are equal for the roots of the same length.

Remark 1 The Theorem is valid for any restricted root system R, including the nonreduced system BC,, At the same time the constants 4, are arbitrary numbers, which in general differ from half of the multiplicities of the roots (appendix B, table 1) If 24, do coincide with the multiplicities of the roots, then B is the radial part of the Laplace-Beltrami operator (cf (5.4) and (C.5))

Remark 2 The Theorem is valid also for the noncrystallographic root systems (1,(n), Hs, H,), but only for potentials of type I

Let A, (k = 1, n) be the radial parts of the Laplace operator on the symmetric space X with the restricted root system R Note that 4:=B (see the statement in appendix C) The operators 4, commute pairwise and are functionally independent Then the operators I, = &(q) 4, €~'(q) are in- tegrals of the system with the Hamiltonian H (5.2) Thus the systems of types I, II and III are completely integrable if the coupling constants g2 have the form (5.6) and 2, are the multiplicities of the roots

a ER The possible values m, = 241 for symmetric spaces are presented in table 1 in appendix B The complete integrability of the system I, I,(n) will be proved in the section 7

The complete integrability of systems of type V is obtained by means of the trick used for the systems of types V A,,_; and V BC, (section 4)

It is natural to expect that the systems of types I-III are completely integrable for all values of g2

2 The systems of type VI are connected with the symmetric spaces of negative curvature X” = G/K, where G are normal real forms It implies that the root multiplicities are equal to 1

Let x = (h(x), z(x)) be horospheric coordinates of a point x € X” (B.22) and f be the character of the nilpotent subgroup ZC G:

aER+ aellT

Consider the space of functions (C.19)

FT se OTe QA 6 ry Ù 2 bo the pesctristian_- 0 0 1e _Laprace-be ¬ lap Re cl ¬ 1712 aAnPratr Operato RQ Do O C space

W,, and H the Hamiltonian of the system of type VI:

H'= —4exp(—q,) B exp(q,)+ p”,

where p= > a

It follows from the statement in appendix C page 398 that the systems of type VI are completely

ntegrabdie, inougn we do not nave the e Dlicit formso he integrals beside hose-that-were-presentedi section 4

: s worthwhile ton 1at the integrals J, < k, whic ave calculate y

I-IV A,, and I-IV BC,, have special properties We generalize them to the abstract case as follows

(a) The integrals are polynomials with respect to 6, which are invariant under the transformations of the Weyl group (W-invariant)

Ï (sô, Sq) = LAB, q)

(b) The integrals are asymptotically homogeneous’:

l(AT p.,Àq)~A “k(p.4), — q>0

Using these properties we establish certain facts concerning the integrals I,

Proposition 1 The integrals are completely determined by their terms of highest degree

Proposition 2 The integrals commute pairwise

Proposition 3, The algebra of the integrals (with respect to the usual multiplication) is isomorphic to the algebra of W-invariant polynomials on the Cartan algebra 6

The proofs of the Propositions are given in appendix D

These Propositions allow one to prove independently the complete integrability of the systems

I Aa-i-IV Aa-¡ On the other hand the explicit expressions of the Laplace operators were calculated by means of these Propositions

6 Wave functions The purpose of this section is to investigate the solutions of the Schrédinger equation

expressions for ¥,(q) Other properties of ys (g) will be discussed later

4, For the systems I-III the following conditions have to be fulfilled SE

(6.3)

vn(sq)= (gq), sEW, (te =0)

We remind the reader that for one-dimensional n-body systems (the systems of types I-III A„_;) the

| lự(4) dạ: - - - dạ„<=,

đj>4j+I

(4ì - đa) = Ứ(Q;, đ,„) for any permutation of indices if =0

The relation between the Laplace-Beltrami operators and the Hamiltonians of the systems I-III (theorem 5.1) allows one to obtain some explicit expressions for the wave functions by means of the results

of appendix C It is natural to seek solutions to (6.1) in the form

Here B and p are defined v via theorem 5.1 (see also (C.7) and (C.4)) If the coupling constants g2 6 6)

C eroup vaiue aD appendix B Cll @al(q) are Zonal spne al TU 1O CC remal

after theorem 5 1)

Another general formula, (C.17), takes the following form for the systems of type A,-1

n AlfAs—=»-0)/2a—m LiGr An 22a~m ee AiQ2-Adi2a~m

in the corresponding algebra of numbers

The formulae for the wave functions for the systems of type I A,_, and HI A„_¡ may be derived from (6.9) and (6.10) putting a = 0 and substituting a > ia respectively In particular for the systems of type I A,,-1, (6.9) leads to the following expression

kg

R

aft

3 It is not difficult to write down the wave functions in the simplest case gi = 0 As it follows from

conditions (6.3) we get for all root systems, including the noncrystallographic ones

_ `

where |W| is the order of Coxeter group W;

sEw

It is worth noticing that ,(q) (6.13) is the eigenfunction for the billiards problem Ip (A € *) and IIIy (A € P(R), the weight lattice (C.3)) In the crystallographic case the last expression is equal, up to the factor &(q) (6.4), to the zonal spherical function ¢,(q) on SS with a complex group of motion For the

SS SL(n, C)/SU(n), the zonal spherical function

II _ II _i 1 -(n—1)!2 đẹt s ex i(sr Ó„a(44) 7 Ủ/a(44)/£ (aq) Ij<x (A; — À„)/a Tex sinh a(q — qx) (6 14)

may be obtained also by means of direct integration of (6.9), under the condition m = 1 [19]

4 It is possible to find an explicit expression for the eigenfunctions for the systems of types I BC,,—III BC,, (2.9) for special values of the coupling constants: g*=0, gi and g3 arbitrary To this purpose we use the results of [62], where zonal spherical functions on the SS of type A_ III=

SU(r, t)/S(U(r) ® U(t)) were calculated They are presented in the form of a determinant,

b2(q) = /(4) (A) det{F(g)}, (G = 1, 1, THD) (6.15)

where the matrix elements F,,(q;) are expressed through special functions

FInd? peers PUG 4) Pt n—}) TL (À2— À2 \/1121 / 1)J I

| Tjek (AG TAWA Vi T l1) -

¿=1 j<k 4

It may be easily verified that these formulae are valid not only for the group-valued constants y= 2(r—s); 7, sEZ, p2=2, but also for arbitrary 4, w2, provided = 1

5 For the systems of type VI theorem 5.2 enables one to seek the solutions to (6.1) in the form

us (q) the following representation holds:

7 Systems of type I

Systems of type I are described by the Hamiltonian

If a, 2a ER, then a ER,, but 20 € R,

Sa = 28„(a ~~ U|aƑ ’ ta = fra + H2a- (7.3)

Here R is a subsystem of positive roots We are interested in the properties of the solution of the Schrédinger equation:

2 The energy spectrum is continuous and covers the semi-axis 0 = FE, <™

3 It has been proved that the systems of types I A,-, and I BC, are completely integrable for aribitrary values of the coupling constants g2 If g2 take the group values then the systems of type I are completely integrable for any root system The function x(é) in the expressions for the integrals (4.6), (4.7) has the form x(€) = £”

The functions ¢,(q) are analytic in g and are W-invariant Therefore they can be expanded as a series in s,,(q), the invariants of the Coxeter group (see appendix A):

In the group case this corresponds to the formula (C.15) for the zonal spherical function @, (q) Note

that the factorization (C.11),

c(A)= |[ c(A„)= const- [] Àz“=, (7.10)

a&Ry œ€cR-

derived in its general form first in [20], holds true in the group case

We conjecture that factorization occurs even for non-group values of the constants 4 (for systems of the type A,,_,, see [30])

7 We shall now consider a different class of solutions to the Schrédinger equation (7.4); that is, in the same way as in [66,67], where systems of the type A,_, have been studied, we shall seek the solution in the form:

where @ x is given by (1.9) Equations (7.12) and GP) were derived in [67] for systems of type A,,-1

8 3 Eq (7 2 has polynomial solutions which ai are called | generalized harmonic polynomials Unlike € the

Since for all Coxeter systems there exists an invariant of order 2 (see appendix A, table Al), the generating function

respectively The asymptotic behaviour of h,(/) for 1/>n is determined by the strongest singularity of the function H,,(t) at the point t= 1 Hence

["- 1

Comparing (7.22) and (7.19) we obtain for the case A„_¡

The explicit expressions for the simplest generalized harmonic polynomials for the case A„_¡ [8Š] are

P3 = $3

Ps = (n + 5+ n(n — 1)w)ss— 5(2(1 — 1/n) + (n — 2)w.)$a5z

The explicit expression of P; for arbitrary / ¡s not known

9 For systems with two degrees of freedom (systems of rank 2) a procedure similar to (7.11) can be used to separate the variables and thus to solve the problem The cases A2, Bo, G2, and I(n) (n = 5, 7, 8) are of this type This problem was solved in [65] for the system Az, and in [10, 59] for the system Gz

We shall write the solution in general form For odd n the Coxeter group of the systems L,(n) as well as

Az has one orbit, and for even n and in the cases of the systems B, and Gy», two orbits (example 1 in appendix A) For all these systems n is the number of positive roots By going over to polar ‘i Fad that the following identiti isfied (see (2.17) and (2.18)) (1) odd n

Using these identities, and writing the wave function ¢,(q) in the form Y.(q) = Ru(r) F;(@) (r = |qÌ)

we obtain the equations:

Let us obtain the solutions of the last equations By means of the same substitutions used in the case of

a system of type Az [65] and G [10, 59], and taking into account the equality:

A,(sin”! @ - cos“? @) = 3(¡ + mạ)” (sin”! @ - cos”? ø) where

we arrive at the solution (n = 2», even)

Fi(¢) = sin”! Vp ° cos? VD ° PH NAH2-12(cog 2up) , (7.36)

10 As before condide a system of rank 2 We shall expand the wave function Wr (4), which has the

mDtO behaviou 9), 1m CfñmsSO Ệ un ior i £ TIO CADah ion in-pa

the form of Jacobi transform:

eA*sete = 28 T(w) S (1+ w)ï! (Ar)ˆ* Treu(Ar) C#(cos @)

where Œƒ'(cos @) is the Gegenbauer polynomial

After making the substitution ¿ = ởa ứo, for the function ở we obtain an expansion similar to (7.39) In the group case this gives the expansion of zonal spherical functions in partial waves

considering in particular the solutions to the eigenvalue problem (6 IH 3) We restrict ourselves to the

ny Q AOrnnh arr 2 ^@œ<o O a are O r1 2 On a ¬ VO =

— I

xg) = bo (9) bx(4) (8.3)

wo(q)= [] sinh“ a„ (8.4)

œCR-+

The properties of /a(4) related to the decomposition (8.3) were discussed in section 6

5 It can be proved that, if the system is completely integrable, the wave functions have the following asymptotic behaviour

of type I

The formula (8.5) corresponds to (C.15) for the group-valued constants gz We recall that in this case the factorization of c(A) holds, see (C.12) In the case of the systems of type II A,,_1 the formula (C.12) takes the form

‘Ft US, 2 `Ca eT 6 AT DIO 2 3 đ II r2 wir $ ) ay a F d 2 6 WO-Da r +7 r2 O TŒT O « Php U 2 or đ Ar DIOD art]

of the factorized S-matrix in the relativistic problem were investigated in [60]

Apparently the factorization and the asymptotic formulae are valid for all values of the constants g

In the group-theoretical case the asymptotic relation (8.5) was obtained in [24] and the factorization was

* This relation is a self-evident generalization of the expansion of zonal spherical functions considered in [24].

er(g)= > Nae ieL (89)

The coefficients Ï; are determined by a recurrent procedure

{dD—2i0,A)})D(A)=4 Dd pa YS Pi-ama(A){(1 + p — 2ma, ø)— Ì(À, ø)}

œ€=R mz=œ1

9 Systems of type III The systems of type III describe the motion of a particle in the Weyl alcove A, (3.4) under the potential

The systems of type III have the following properties

1 The Hamiltonian H is selfadjoint if ¢2 + 23,/4 = —|al’/8

2 Because of compactness of the configuration space A, (3.4) the energy spectrum is discrete The energy is characterized by the vector A = (Ai, A,) which belongs to the weight lattice P(R) (C.3)

3 The systems of type III are completely integrable, just as the systems of types I and II The function x(é) in the formulae (4.6), (4.7) can have either one of the following two expressions: