Periodic Toda lattice in quantum mechanics

ANNALS

OF PHYSICS

220, 300-334 (1992)

Periodic

Toda

Lattice

in Quantum

Mechanics

A. MATSUYAMA Department oj Physics, Faculty of Liberal Arts, Shizuoka University, 836 Ohya, Shizuoka 422, Japan

Received March 18, 1992; revised June 5, 1992 The quantum mechanical periodic Toda lattice is studied by the direct diagonalization of the Hamiltonian. The eigenstates are classified according to the irreducible representations of the dihedral group D,. It is shown that Gutzwiller’s quantization conditions are satisfied and they have a one-to-one correspondence to the irreducible representation of the D, group. We have also carried out the semiclassical quantization of the periodic Toda lattice by the EBK formulation. The eigenvalues of the semiclassical quantization have a one-to-one correspondence to the integer quantum numbers, and those quantum numbers also have a close relationship to the symmetry of the state. Numerical calculations have been done for N = 3, 4, 5, and 6 particle periodic Toda lattices. The distributions of the eigenvalues are systematic and distinguished by the symmetry of the state. As illustrative examples, amplitudes of the wave functions and density distributions are shown. 0 1992 Academic Press, Inc.

1. INTR~OUCTION

The Toda lattice is a one-dimensional dynamical system of equal masses connected by nonlinear springs of exponential type. It has been studied extensively more than two decades since it has exceptional properties [ 11. For example, it has a solitary wave solution and corresponds to a discrete version of the nonlinear continuous system described by the KdV (Korteweg-de Vries) equation. These unique properties are due to the complete integrability; i.e., the system has as many constants of motion as the degree of freedom. The complete integrability of the Toda lattice was noticed numerically by Ford et al. [2] and was proved by Htnon [3] and Flaschka [4]. There are two kinds of lattices depending on boundary conditions. One is the open Toda lattice and the other is the periodic Toda lattice. In the open lattice, the first and the last particles are not coupled, while they are coupled in the same way as others in the periodic lattice. In classical mechanics, an initial value problem of the infinite open lattice is solved by the inverse scattering method [S], and the solution of the periodic Toda lattice is given by Kac and Moerbeke [6] and Date and Tanaka [7]. In quantum mechanics, the open Toda lattice has a continuous spectrum of unbound states, while the periodic Toda lattice has only bound states and a discrete spectrum. For the N-particle open Toda lattice, the exponential potential vanishes in the asymptotic region where particles are far apart from one another and the quantum state approaches that of N free particles with definite momenta. These N asymptotic momenta specify the eigenstate, and the eigenfunc300 0003-4916192 $9.00 Copyright All tights

0 1992 by Academic Press, Inc. of reproduction in any form reserved.

PERIODIC TODA LATTICE

301

lion can be given by a function with the N momenta as parameters [8]. On the other hand, the potential does not vanish for the periodic Toda lattice. One can imagine that the periodic Toda lattice consists of N particles on a circle where neighbouring particles are coupled by a spring which always tends to expand. Then, all the springs tend to expand to minimize the energy, but ultimately some of the springs must shrink due to the periodicity and the energy is increased. Therefore the system is always bounded both in classical and in quantum mechanics. The quantum version of the periodic Toda lattice was first studied by Gutzwiller [9]. He has developed a formulation which is parallel to the method of Kac and Moerbeke in classical mechanics. His method is to expand the eigenfunction of the N-particle periodic Toda lattice in terms of the product of the (N - 1 )-particle open Toda lattice and a free particle. As examples, he has explicitly constructed the eigenfunctions of the N= 2, 3, and 4 periodic Toda lattices. The quantization conditions which the eigenvalues should satisfy are fairly simple and it can be conjectured that the similar quantization conditions hold generally for the N-particle periodic Toda lattice. Although Gutzwiller formulated a systematic way of constructing the eigenfunction more than a decade ago, numerical works have not come out until recently. We have studied the quantum mechanical three-particle periodic Toda lattice by the direct diagonalization of the Hamiltonian in Ref. [lo], and a similar work was also reported by Isola et al. [Ill]. In Ref. [to], we classified the eigenstates according to the representation of the permutation group S3 under which the Hamiltonian is invariant. We have found that the eigenvalues fulfill Gutzwiller’s quantization conditions and the semiclassical quantization is a good approximation even at the ground state. Furthermore, the energy spectra were shown to have the same universal statistical properties as other integrable systems, i.e., the nearestneighbour spacing distribution P(s) is close to the Poisson distribution P(s) - r ’ [12], and Dyson-Mehta rigidity d,(L) [13] shows d,(L)-L/15. The purpose of this paper is to extend the study of Ref. [IO] in a more systematic way, generally for the N-particle periodic Toda lattice. We will briefly describe the Hamiltonian and the conserved quantities in Section 2 and discuss the symmetric properties of them in Section 3. For the purpose of later discussion, Gutzwiller’s quantization conditions and the semiclassical EBK quantization are reviewed in Sections 4 and 5. Numerical methods are discussed in detail in Section 6, and the results are presented in Section 7 for N = 3, 4, 5, and 6 periodic Toda lattices. Section 8 will be devoted to summary. 2. HAMILTONIAN The Hamiltonian of the periodic dimensionless form as

AND CONSERVED QUANTITIES

N-particle

Toda lattice can be written in a

H=tCp~+Cexp(q,-qq,+,), II ,i

(2.1)

302

A. MATSUYAMA

where we set qN+ 1 = ql. The potential is of exponential type and an nth particle interacts only with (n f 1)th particles. According to Ref. [4], the equations of motion can be written in a Lax form

z=

CB,Ll,

(2.2)

where L and B are the N x N matrices

b, a, a1

aN

b2

0

. . b n-1

L=

a,-1

a,-, b,

a,

a,

b,+,

(2.3)

0

biv-1

aN-l

aN-l

aN

0

biv

-a1

aN

0

a1

0

. .

0 a n-1

B=

-a n-1

0 a,

(2.4)

-a,

0

0

. .

0 aN-l

-aN

and the quantities

-aN-l

0

a, and b, are defined as

a,=~exp(q”-~ntl),

b,=fp,.

(2.5)

Therefore the eigenvalues of the matrix L are constants of motion and thus the coeffkients of the characteristic polynomial of the matrix L are also constants of motion. Let us define the coefficients Ai as follows (I is N x N identity matrix): det(21Z-

2L) = (21)N + A,(21)N-’

+ A,(21)N-2

+ “* +&,-,(21)*+&-,(2il)+‘d,-2.

The quantities Ai are conserved and in involution [ZZ, Aj] = 0, and, furthermore, mutually in involution

(2.6)

with the Hamiltonian [A,, Aj] = 0. ([, ] is the

303

PERIODIC TODA LATTICE

Poisson bracket.) The quantity generally expressed as

A, is an &h-order

polynomial

of the momenta

and

(2.7)

A.=(-l)"Cp,,pj,...pi,(-xj,)(-X,Z)".(-X,,),

where Xj = exp(g, - qi+ r). and the sum is taken over all terms with different indices (il, i,, .... i,, j, , j, + 1, .... j,, j, + 1) and n = k + 21. In the center-of-mass (C.M. ) system (P = 0), A, = -P = 0, A z = 1P’ - H = -H; thus the conserved quantities A , and A z have physical meanings of the total momentum and the energy, respectively. However, the quantities A i (i > 3) do not have physical interpretation. In quantum mechanics, the Hamiltonian and the quantities A, are operators and are given by the canonical quantization, i.e., the momentum pi is replaced by the operator - ir?(8/@,). Since we have used three parameters (the mass of each particle m,, the range a,, and the strength V, of the potential) as units, Dirac constant i7 cannot be scaled out. This scale-noninvariance is one of the features of the quantum Toda lattice. Although h is basically arbitrary, we set h = 1 in the following arguments. The qualitative feature of the results does not depend on the size of h.

3. SYMMETRIC PROPERTIES

Let us define the operators s and t as follows: dql,

422 ...> qN- I > qN) = (42,

t(q, >q2, ..., qN--l,qN)=(-qN,

43,

...? q,v,

-q*K

(3.1)

41)

1, ..‘. -qz,

-41).

(3.2)

The operator s shifts the coordinates, while the operator t reverses the coordinates and changes the sign at the same time. Then one can see that the Hamiltonian (2.1) is invariant under the transformations of the coordinates generated by the operators s’r’ (O
t’=

1

tst-~‘=sV’.

(3.3)

Geometrically, D, represents the group of symmetry of the N-sided regular polygon, where s represents 271/N rotation around the center and t is a reflection with respect to a line of symmetry. The D, group has two kinds of representations depending on whether N is odd or even, and we will describe them separately (see Ref. [14], for example). If N is

304

A. MATSUYAMA

odd (N= 2n + l), the D, group has 2 one-dimensional representations and (N - 1)/2 = n two-dimensional representations. Using the notation of the point group, we will call the one-dimensional representations A, and A,, and the twodimensional representations Ek (1
c, = is*,P- I},

c, = {s,P},

co= (11,

C n+1= The one-dimensional

{t, st, A, s3t,

representations

. ..)

....

c, = (f, sn+l},

s2”t}.

(3.4)

are

Al:s=l,t=l

(3.5)

A,:s=l, and the matrices of the two-dimensional

t= -1

(3.6)

representations are (3.7)

Once the representation matrices are given, the characters are easily calculated and are shown in Table I. If N is even (N= 2n + 2), the D, group has 4 one-dimensional representations and (N- 2)/2 = n two-dimensional representations. The onedimensional representations are A,, A,, B,, and Bz and the two-dimensional representations are Ek (I< k d n) in terms of the point group notation. In this case, TABLE I Character Table of the D, Group” N=odd

co

Cl

Al A2

1 1

1 1

&

2

N=

even

A, A2

B, & Ek

co

c2

1 1

Xkb)

Cl

1 1

1 1

xkts2)

c2

1 1 1 1

-1 -1

1 1 I 1

2

x*(s)

XkW)

a xk(sm) is defined as xk(sm) = cmk+ cmmk= 2 cos(2nmklN).

c n+L

cn

. . .

. . .

1 -1 0

XkW

C “+I

C “+2

1 (-*)+I

-1

(-l)n+’

-1

c

1 1

.”

xkb”+‘)

“+3

0

1 -1 -1 1 0

PERIODIC

TODA

305

LATTICE

the D, group has n + 4 different irreducible representations and the number of the class is n + 4. The classes are denoted by Co, C,, . ... C,l+3 as follows: co= {l),

c, = {S,S2*+‘},

...) c,=

{S”, .Yn+2], c,,+, = (s”+‘},

C ,,+2--- {I, s2t, s4t, ...) S29}, Cn+3=

(St, s”t. s’t, .... s2”+‘t).

(3.8)

The explicit representation matrices are the same as those of N= odd case, in which we should add the one-dimensional B, and B2 representations B,:s=

-1, t= 1

(3.9)

B2:s=

-1, t= -1.

(3.10)

The characters are also shown in Table I. Once the representation matrices are given, one can construct the representation basis by applying the projection operator. The projection operator Pzm, takes out TABLE Reduction

of the Product

kl=k2=k

II

Representation

of the D,

Group

Wn+

1)

k, fkz (k, >k2)

(k,+kzn+l)

N = even

Bz

&:

A, A2 B, & Eki

BZ B, A2 A, E ri+,

Ekl E,, E-ni-1 E nil ***

-k,

***

k,=k?=k

Al+A2+B,+Bz Al+Az+EZk A,+Az+&n+z-2ir

(?k=n+l) (2<2k
k,#kz (k, >k,)

B,+&f&-k? E~.-kz+~k,+k: EL,-,1+Ezn++z-k,-~:

(k,+k,=n+l) (k,+k?n+2)

595.220

2. I I

k: kJ

306

A.MATSUYAMA

the Ith component of the basis in the representation the representation matrix as

S and is defined in terms of

(3.11) where R is an element of the D, group, D!;‘(R) is the matrix element of R in the unitary representation S, d, is the dimension of the representation, and g is the order of D, (g= 2N). The projection operator is very useful, not only for constructing the basis with definite symmetry, but also for studying the symmetric property of the wave function. Let us call A ,/A2 (A, or AZ) symmetry as A-symmetry as a whole since A, and A, symmetries come out in a pair when we discuss the symmetry of the states. In the same way, we will call B,/B, symmetry as B-symmetry, and Ek (k = 1, 2, ...) symmetry as E-symmetry. The Hamiltonian has the A,-symmetry and the conserved quantities have either A, or A, symmetry; i.e., Aj (i=even) has the A,-symmetry, while Ai (i = odd) has the AZ-symmetry. Since we will calculate the eigenvalues of the operators Aj in Section 7, it is necessary to reduce the product representation to irreducible representations. In Table II, we tabulate the reduction of the product representations. Since the reduction of Ek, x Ek2 is complicated, they are shown separately in Table II.

4. GUTZWILLER'S

QUANTIZATION

CONDITIONS

About a decade ago, Gutzwiller developed a systematic way of constructing simultaneous eigenfunctions of the operators Z and 4 [9]. (Script letters will be used for the quantum mechanical operators hereafter.) His method has an advantage that the wave functions are given in explicit forms. The basic idea is to expand the eigenfunction of the N-particle periodic Toda lattice in terms of the products of the eigenfunctions of the (N- 1)-particle open Toda lattice and a free particle. He obtained quantization conditions by claiming that the wave functions should be bounded and also independent of a particle chosen for a free one. Those are necessary conditions that the eigenvalues of the operators Z? and 4 should satisfy. In Ref. [9], he has explicitly constructed the eigenfunctions for N= 2, 3, and 4 periodic Toda lattices. Since the formulations are almost identical, they are easily extended generally for the N-particle Toda lattice. However, a proof for the N-particle case has not been given so far because of difficult algebraic manipulations and we could not overcome this unsatisfactory situation, either. Therefore Gutzwiller’s quantization conditions for N> 5 still remain conjectures, but we will show that the eigenvalues indeed satisfy those conditions for N= 5 and 6 in Section 7. Gutzwiller’s formulations are rather involved and we will briefly summarize his algorithm for later use. Suppose we have N- 1 real numbers (E, A,, A,, .... AN)

PERIODIC

TODA

307

LATTICE

and we want to examine whether they are the eigenvalues of the operators X and .g in the CM. system. First, we should solve the following Hill-type equation:

fl

1

A(K)=

0

D(K-1)

1 D(K) 0

+1 L D(K) 1

1 1

0 = 0.

+1

D(K+ 1)

D(K+ 1)

1

0

1

D(K + 2)

.

.

(4.1) where D(K)=KN+E~N~Z-iA3~N~3+~4tifV.--j+j~?K.~

-“+

__.

+i,vA,;

(4.2)

is a complex number (K = ip + k), and the double sign is - ( + ) for N = odd (even). Equation (4.1) is independent of k, i.e., if K = ip is a solution, K = ip + k is also a solution, and it has generally N different purely imaginary solutions K; (Cjtii=O) in -l/2 6 Re(rc) d l/2. In the special case of N= odd and A,=A,= ~~~=O,A(lc)=OhasN-lsolutions,andti=Oshouldbeaddedtothem since K = 0 automatically satisfies the necessary condition of the wave function. In practical calculation, it is useful to rewrite Hill’s determinant as K

A(ip)=r,,ri*,~,fr,*,r,,,+,/~D(ip)D(ip+l)~ and Y, is defined by the recursion relation r K-

I

= rK *

I

D(K) D(K + 1) ” ”

with the boundary condition rip+k -+ 1 (k -+ co ). The double sign is + ( - ) for N= odd (even). Let K = ip be one of the solutions of Eq. (4.1). Then, second, we should calculate r!,, by solving Eq. (4.4) and obtain r=

H(l+ip-E)

(4.5 1

r ‘6’

where n (1 + ip -E) is the product of N gamma functions: i.e., n (1 +ip-.s)=n

Ql +ip-E,)

(4.6)

308

A. MATSUYAMA

and .si (1 d i < N) is the zero of the polynomial D(E). Note that D(E) = 0 has N different purely imaginary solutions for classically allowed values (E, A,, A,, .... AN). Finally, “Gutzwiller’s phase angle do” is given by

dG= arg(r)

(4.7)

#o’s are the same for N different solutions tci and Gutzwiller’s tions are expressed as

quantization

condi-

q&,.= 0,

+ n/N, &-271/N, .... + (N - 1) 7c/2N (mod rc)

for

N=odd

(4.8)

&=O,

+x/N, +2n/N,..., k(N-

for

N= even

(4.9)

2)71/2N,71/2(mod rc)

The angle do characterizes the symmetric property of the eigenfunction and it has a one-to-one correspondence to the irreducible representations of the D, group. This correspondence can be proved explicitly by making use of the Gutzwiller’s formulation for N= 3,4 cases. By applying the projection operator P$,,, (Eq. (3.11)) to the eigenfunction Y, one can obtain the following results: For N=3

,$A'= Re(y), 4”’ = Im( Y), qE*’ = @** = 0, p+4w-Jp= y, p=() p-2 = y

p+q)p4

for

&=O

for

&=7r/3

for

do = --7113

(4.10)

If A, = 0, one of the solutions ICYis zero, and $o = 0, &tA2= Im( Y) = 0. For N=4 &4l=@2=(),

4”l=Im(y),

@‘=Be(Y),

qjE1=qjE**=O

for

f&=0

4”’ = Re( y), 4”’ = Im( Iy), 4”’ = dB2 = 0, 4”’ = qjE,* = 0

for

I& =7cJ2

(j”’ = 4‘4’ = 0, 4”’ = 4”’ = 0, p

= 0,

p

for

do= 7114

4”’ = 4”’

= y,

p*=o

for

f& = -n/4.

= 0, 4”’

=

4”’

= 0, p1

=

y

(4.11)

If A, = 0, two of the four solutions K~ are complex conjugates of the other two, i.e., K;, rcg= ~2, for example. In this case, one can show that the wave function has either A, or B, symmetry by using the explicit form of the wave function. In general cases, one can conjecture the following relations: For N=odd (N=2n+l)

K1 =

A-symmetry

k-=0, &. = fh/N

(k = 1, 2, .... n),

E,-symmetry

(4.12)

309

PERIODIC TODA LATTICE

For N=even

(N=2n+2)

dG=O,

B-symmetry

4c = 742,

A-symmetry

c$~= +kn/N (k = 1, 2, .... n),

E,,+ , ,-symmetry.

(4.13 )

By an analogy of N = 3,4 cases, the state with A, = A, = .. = 0 will have the A ,-symmetry for N = odd and either A, or B, symmetry for N = even. One of the important results which can be immediately seen from Gutzwiller’s formulation is that the eigenstates generally degenerate in two-fold. If we denote the purely imaginary solutions of d(lc)=O as (K,, x2, .... K,,,), then (K:, K:, .. . . ~2) are solutions of A*(K)=O. The function A*(K) is given by Ll(~)+D*(ti) in Eq. (4.1) and

Thus the eigenvalue (E, A,, A,, A,, .... A,v) changes to (E, -A,, A4, -A,, ( - 1)N AN) and the wave function changes to its complex conjugate, i.e., y(q,, q2, .... qN: Kf, K;, .... K;) = y(q,, q?. .... q,v: h-1>h-7,.... xv)*.

_..,

(4.15)

Since the energy eigenvalue E is the same, the wave functions !P and !ZJ*degenerate in twofold unless Y is a real function. The eigenvalues of the conserved quantities L4 are (A3, A4, A,, .... A,,,) and (-A,, A,, -A,, .... ( -l)‘“AN) for degenerate states. When the eigenvalues of the 4 (i = odd) operator is zero, i.e., A, = A, = . = 0, the solutions rci come out in pairs. If we arrange xi such that Im(k-,)
5. SEMICLASSICAL

QUANTIZATION

The Toda lattice is integrable and the Hamiltonian can be rewritten in terms of the action-angle variables (Zi, Sj) by the canonical transformations. The difficulty is that the Hamiltonian is not separable; i.e., there is no way to separate coordinates into explicit integrating variables. However, one can still define the canonical conjugate variables (v<, pi) (i= 1, 2, .... N - 1) such that the action I, is expressed as an integral over one period of the motion. The variable p, is called an auxiliary spectrum and is defined by the eigenvalues of the reduced matrix L* which represents the (N - 1)-particle open lattice. The matrix L* is the (N - 1) x (N - 1) matrix given by removing the first row and the first column from the matrix L (Eq. (2.3)).

310

A. MATSUYAMA

The variable pi moves only a bounded region of /d(p)1 B 2. The function d(p) is given by the characteristic polynomial of the matrix L d(p) = det(2p1-

2L) + 2

(5.1)

which is the same as (2.6). d(p) should not be confused with the Hill-type nant A(K) (Eq. (4.1)). In the C.M. system, d(,u) is given by d(p)=(2py-E(2py+

determi-

A3(2p)NP3

+ .'. +A,_,(2~)2+AN-1(2~)+AN,

(5.2)

where E is the.energy and the Ais are the constants of classical motion. One should note that L!(P) and D(K) (Eq. (4.2)) have the relation D(2k) = PA(c);

(5.3)

d(p) is the Nth order polynomial of p and it has N- 1 intervals satisfying [d(p)] > 2. The variables pj are confined in these N- 1 intervals and are labeled as ... -c/4&,. As examples, we show the graphs of d(p) for N= 3 and Pl
li(d(Pj)f

{A(~i)2-4}“2)l~

(5.4)

N=4

I

\ A

A \

/

ti

FIG.

1.

Function

d(p)

(Eq. (5.2))

I 44

4 0 -2

for N=

P \ A-

3 and N = 4 in the case of A, = 0.

PERIODIC

TODA

311

LATTICE

where the double sign is + for i = odd (even) and - for i = even (odd) when N is odd (even). Then, the action I; can be expressed as the integral

It=

vr(Pi)

(5.5)

dP,.

+

The region of the integral is the closed interval of p satisfying ld(p)l >, 2. Let us denote the roots of d(p) = 2 by p,! (p’, < PL;< .
(A(p)+

{A’(p)-4)“‘)

dp

cc(WG) = -4 jpig /I;,-, {A2(p)-4ydp Zzj=2

(5.6)

i~2’i’210g)~(A(~)-(A2(~)-4jL,2)

dp

(5.7)

For N=even {A2(p)-4;~‘i2)

dp

(5.8) z2i = 2

fl’2,+1 .r.

210g ;(A@)+

{A’(p)-4)“‘)

d,u

A’?,

=-- 4

p( dA/dp) {A2(p)-4]‘:’

dp’

(5.9)

In this way, the energy E and the conserved quantities A, are given implicitly in terms of the actions Ii. The semiclassical quantization can be carried out through EBK (Einstein Brillouin-Keller) formulation. If a classical Hamiltonian is expressed in terms of actions Ii, the semiclassical quantum energy is given by the substitutions Ii + (ni + $) h. (In the case of the Toda lattice, Maslov index is 2). We will use the letters E’ and a: for the energy and the conserved quantities of the semiclassical quantization. Then the values (E’, a:) have a one-to-one correspondence to the quantum numbers (n,, n2, .... nN- i) through the EBK quantization. For the later

312

A.MATSUYAMA

convenience, we will change the indices of the quantum N=odd (N=2m+ 1) (n,

and for N=even

3 n2?

eq.3 n&l)+

(n,,

n,-1,

. ..> n,,

n-1,

numbers as follows. For

..*, n-,+1,

n-d

(5.10)

(N=2m+2)

(n,, n2, .... nN--l)-’

(n,, nmel, .... n,, no, n-l, .... n-,+,,

n-,).

(5.11)

For the degenerate states, the eigenvalues are (E’, +a;, ai, + a;, ...). If one of the quantum numbers of the degenerate states is (n,, n,- ,, .... n-,+ it n-,), then the quantum numbers of the other state is the reverse of the sequence, i.e. n- m+l, -., n,-,, n,). For the non-degenerate state, the eigenvalues are ;;ymp,a3 = 0, ai, a; = 0, ...) and the quantum numbers have the relation n--m = n,. We will find a close relationship between the symmetric property of the wave function and the quantum numbers in Section 7.

6. NUMERICAL

METHODS

6.1. Orthogonal Bases

Since the periodic Toda lattice has only bound states, we have diagonalized the Hamiltonian in terms of orthogonal bases. In practical calculation, we used onedimensional harmonic oscillator (h.o.) basis for each particle and thus N product of h.o. basis for the diagonalization of the Hamiltonian. In this case, one should be careful about the C.M. motion since the h.o. bases explicitly break the translational invariance. Once the eigenfunction is obtained, it is easy to calculate the eigenvalues of the conserved quantities 4. The basis for N-particle is hqn2...?&L where q,(x) is one-dimensional we will use the abbreviation hn2

927

...T qiv) = cpn,(q1) %,(42) .-.(Pn,(qlvh

(6.1)

h.o. wave function with extension parameter w, and .. .n,)=4

n,n2...nN(ql,

923

...YCJN).

(6.2)

The quanta n,, n2, .... nN should not be confused with the semiclassical quantum numbers (n,, n2, .... nN-, ) of Section 5. The total quanta is No = n, + n, + . . . + n,.,. Since the bases with different total quanta are orthogonal, we will fix the total quanta No hereafter. The Hamiltonian of the periodic Toda lattice has the symmetry of the dihedral group D,; therefore the eigenfunctions can be classified according to the irreducible representation of the D, group. Let us explain how the bases of a definite symmetry are constructed. First, we will fix the partition of No, i.e., fix the quanta and n, an,> ... >n,. The basis Cn,, n2, .... n,] such that n, +n2+ ... +n,=N,

PERIODIC

TODA

313

LATTICE

(n, n2 . . nN) consists of many symmetries and we will project the definite symmetry state by employing the projection operator Pc,n, of Eq. (3.11). The basis with definite symmetry S is given by qF’=

Pc,,(n,n2

.-.n,);

(6.3)

@,’ transforms as the Ith component of the vector space of the irreducible representation S. By changing the partition [n,, n2, .... n,] throughout all the possibilities and the column index m of P&,), one can obtain all the independent orthonormal bases cp:’ (i= 1, 2, ...). The function VT’ is a linear combination of (n, nl . n ,%). cpF’=

C Cy’(i;

{nkJ)(n,n,.-.n.).

(6.4)

I )Jk: Using the Jacobi coordinates, one can separate the coordinates into the C.M. coordinate (yO, to) and the intrinsic coordinates (v];, 5;) (i = 1, 2, .... N- 1 ), and the basis (n,n?.. . nN) can be expanded as (n,n2...n,)=

1 (n,, n,, .... nN;mo, ml, .... mv- l > :mki X4 mam,...mN-,(50~ 4,2 ...* 4,

,I?

(6.5)

where

and (n,, n2, .... nN; ma, ml, .. . . function cp?’ is expressed as

mN-

1

) is the transformation

qs’= 1 c c?‘(i;{nk})(nl?

n2,

in!%) X4

. . . . izN;

m0,

ml,

coefficient. Thus the

. . . . mhipl

>

:mkl momi--.m,vm,(tO?

51,

(6.7)

...Y tN-1).

The trouble is that the basis cp:’ has an excitation of the C.M. coordinate t,,. Since the system is translationally invariant, the excitation of the C.M. motion has nothing to do with the intrinsic motion. Therefore one must remove the excitation of the CM. motion from the bases. Removing the C.M. motion from the bases cp>,’ can be carried out by diagonalizing the Hamiltonian of the C.M. motion &&, in terms of cp:‘: %wtt;.i= 42: i is the eigenfunction states. The eigenfunction

(ma + f) wit$;

(6.8)

with quantum number nz, and i distinguishes degenerate dz,,‘, i is a linear combination of the bases cp;\‘.‘: q52,‘,i=C

C:‘(m,,

i;,j) 9:‘.

314

A. MATSUYAMA

Since the state with m, # 0 has the C.M. excitation, we will keep only m, = 0 states as orthogonal bases, which will be denoted by by’. The basis #?’ consists of the basis (nln2...nN) by (6.4) q+T’=C

Cc’(m,=O,

i;j)

j

1 Cs”(j; (nk})(nlnZ..-nN) InkI

= c cyi; {n,})(n,n,~..nN), I”kl

(6.10)

cyi; {Q})=c cs”(m,

(6.11)

where = 0, i; j) CF’(j; {nk}).

So far, the total quanta No is fixed, and if No-dependence is shown explicitly,

g(;=

1 @‘(No, i; {nk})(nln2.. .rzN).

(6.12)

ifi&1 Thus the orthogonal bases are given by dzi,i (i = 1, 2, .... imaX) and i,,, is a function of S, I, and No; i.e., i,,, = i,,, (No, S, I). The normalization is

dq, ...&d’,$q~ 7...> q,v)* h?;,:(q,, .+.p qN) sdql (6.13) 6.2. Matrix Elements If one uses the Jacobi coordinates, one can easily see that the Hamiltonian (2.1) has the C.M. momentum ‘lo explicitly. In usual calculation, coordinates are measured in the C.M. system (i.e., the CM. momentum q. = 0) and only relative coordinates are relevant: therefore one need not care about the C.M. motion. However, in the many-body system, the expressions in terms of the relative coordinates are very involved and not suitable for numerical calculation. On the other hand, if one uses the single particle coordinate (pi, qi) measured from a fixed origin, the calculation becomes simpler, but one should be very careful about the treatment of the C.M. coordinate. Since the CM. motion of the Hamiltonian is not fixed, we must remove the kinetic energy of the C.M. motion from (2.1) and make the Hamiltonian translationally invariant. The intrinsic Hamiltonian Xint is xint

=

#

-

flri

=tCP~+Cexp(q,-qq,+,)-~II~. n n

(6.14)

PERIODIC

The basis d;i,;

TODA

has no C.M. excitation d;;,;h


and can be written

...?qN)=(Po(~ob/c&G~

q2,

The Hamiltonian 2’“’ representation S is

has no C.M. coordinate

IxintI

315

LATTICE

(6.15)

423’T 5.c1).

and thus the matrix element in the

GG:,,> = ~cpo(~o)l cpo(irowP~i:., = Mi~.i

as

/~“‘“‘I

I.X1”‘l ‘p;,;.,) (6.16)

‘p?(.,).

Since 4;:. i is expressed as a linear combination of the basis (nln2 ...n,v), all we need is the matrix element of the momentum p,, = -i(d/dq,) and the potential exp(q,, - q,,+ 1). The matrix element of the potential term is ((rn,rn~~

..,nN) le41pY?I (n,n,...n,v))

= (m, /ey’I n,)(m,

le my21n,)

x ~,,z,,,, ” (5,,1,,1y.

(6.17)

The explicit forms of the matrix elements between the h.o. bases are (,w,I,)I-i~~~n~q,)=

-i{(~)“6.,..,,~,-(~)‘i2g,,~,,,+,~

(CpAq) letql cpJq)> = ($)“2exp kmax xpil k=O

(6.18)

(A) n, + II ?h ( JG 1

1

k!(nz-k)!

(n-k)!’

(6.19)

where k,,, = Minim, n>. Using (6.18), (6.191, one can easily calculate the matrix element of the Hamiltonian and obtain the eigenvalues and the eigenfunctions with the definite symmetry S. In A or B symmetry, the representation is one-dimensional and the states are either non-degenerate or doubly degenerate. It has been shown in Section 4 that the non-degenerate states belong to either A I or B, representation. Since the product representation is A, x A, = A, and B, x B, = A,, the eigenvalue of the G& (c$ with i = odd) operator is zero. The eigenvalue of the z&,,, (4 with i = even) operator is given by the matrix element of the eigenstate 4”’ or 4”’ A C”LT”=

(4”’ L&,“I 4”‘)

A eYen= (I”’ The degenerate states the degenerate states. Hamiltonian %, [Y?, Table II, one can see

wevenl P)

for

A ,-symmetry

(6.20)

for

B,-symmetry.

(6.21 )

of the A-symmetry are obtained as follows: Let 4”’ be one of Since the conserved quantities -4 are in involution with the 4.1 = 0, the function 4.4 ‘I is also an eigenfunction. From that “Cg,ddd.41(&‘Y,,4A’) has the A2 (A,)-symmetry since the

316

A. MATSUYAMA

&Odd (J&,,) operator has the A, (A,)-symmetry. Thus the other eigenstate is LZ$,,~~~’ which has the AZ-symmetry. Therefore one of the eigenstates belongs to the A,-symmetry and the other to the AZ-symmetry. In the actual calculation, we diagonalize the Hamiltonian separately in each symmetry. Thus most of the eigenvalues coincide in A, and A, symmetries except for those which do not denegerate. Since the product representations (A, x A, = AI, A, x A, = A,) do not include the AZ-symmetry, the eigenfunction with definite AI/A, symmetry is not the eigenfunction of the JZ&, operator which has the AZ-symmetry. The simultaneous eigenfunction of J? and LX&, can be given by q+” = (l/&$4”’

f $A’).

(6.22)

It is also an eigenfunction of the Sg,,,, operator. The eigenvalues of the conserved quantities 4. are given by the matrix element of the eigenstate as A odd=

fIrn

(4”’

b&ddl

A eYe”= $1 GA’ I4”,“I

(6.23)

iA’>

4”‘) + (4”’ lJ;l’,“,“l dAZ)l.

(6.24)

If &id is the eigenvalue of one of the degenerate states, then -Aodd is the eigenvalue of the other state. The eigenvalue A,,,, is the same for both the degenerate states. The same argument also holds for B-symmetry. In E-symmetry, the representation is two-dimensional and the component 1 takes two values (,= 1, 2). One can see that the matrix element between the different components vanishes by applying the s operator of the D, group (#z;.,lj pP”tI

qq$

= (S#fgj

IssP”tl sqy$$)

Ek.1 = (i k 4M,,i lx’“‘I ip $No,j) = c-2” (qs$,li ppl qjf$?) k

Ek.2

= 0,

(6.25)

since cdZk # 1, and the matrix of &“I in the E-symmetry reduces to the two blocks which are complex conjugate. Since cPk= (ck)*, +$$ = (+2$)* apart from a constant factor. Therefore the complex conjugate representation E* is equivalent to the E-representation. For simplicity, we denote the wave function with the first component (I= 1) of the basis as 4” and that with the second component (I= 2) as dE* = (+“)*. Thus the eigenstates always doubly degenerate. The eigenvalues of the 4 operator are A odd = (4” b%ddl (6.26) 4”) = ($E* Idddl dE* > A eYen = (4” l4”mll 4”) = Y

(6.27)

where we used the fact that the 4 operator is the ith order polynomial of the momenta and integrations are done by parts. Thus the signs of the eigenvalues of dodd are opposite for the degenerate states, while they are the same for &,,.

PERIODIC

TODA

317

LATTICE

In Table III we summarize the eigenfunction of the states and the eigenvalue of the 4 operator. Since the d operator is a polynomial of p, and ekyi, the matrix element can be calculated by using (6.18), (6.19). The conserved quantities 4. are the functions of (p,, qi) and also rewritten in terms of Jacobi coordinates (vi, r,). The coordinate q, always appears in the form of the relative coordinate, i.e., ey~P”l+i, and thus the C.M. coordinate to does not appear. However, the C.M. momentum q. appears and the matrix element has a contribution of the C.M. motion. Since the quantities col:of the intrinsic motion are free from the C.M. motion, we must evaluate the matrix element in the C.M. system, i.e., y10= 0. The quantity &; is the ith order polynomial of the momenta and thus the ith order polynomial of y10when -4. is expressed in Jacobi coordinates. Let us expand c4 as the power series of vu as

The intrinsic operator &‘y is given by setting q0 = 0, i.e., &r’ = diO. Since the C.M. motion of the basis is not excited, the matrix element of the C.M. coordinate is

and it vanishes for k = odd. Thus k = odd terms of Z& do not contribute to cg and can be neglected. We have calculated dk for N= 3, 4, 5, and 6, and it turns out that -E& can be expressed in terms of A?“’ and G?:,’ with j< i: TABLE Eigenfunction Sym. (1)

W.f.

Non-degenerate

stale

A B (2)

Degenerate

state

and Eigenvalue .4 Odd

III for Each Symmetry A CICII

318

A. MATSUYAMA

(1)

N=3 &g3= &q’

(6.30)

(6.31)

(6.32)

It seems that similar relations will also hold for the N-particle case. The matrix elements of qz, qi, and ~8 in the state qO (to) are o/2, $02, and yw3, respectively. Using the eigenvalues E and Ai of the operators Xint and 4, the eigenvalues a, of the intrinsic operators JS’~?~can be recursively expressed as (1)

N=3 a,=A,

(2)

(6.34)

N=4 a,=A,

(6.35)

a4=A,-&o’+$oE (3)

N=5 a3=A3 a,=A,-&co2+&wE a*=A,-hoa,

(6.36)

PERIODIC

(4)

TODA

319

LATTICE

N=6

(6.37)

a,=A,-&w*+iwE a,=A,-iwa, ab = A, - km3

+ &w’E

- &oa,.

If the eigenfunction is given by a linear combination of the basis (n, n2 . n,,,), the matrix element of 4 (Table III) can be easily calculated, and one can obtain the eigenvalues of the intrinsic operators &F’ recursively. 6.3. Gutzwiller’s

Phase Angle and Semiclassical

Quantization

The calculation of the Gutzwiller’s phase angle q5o can be carried out by using the eigenvalues which are obtained by the direct diagonalization in terms of the bases q5;:;,‘., The calculation consists of two parts. One is to obtain the solutions ti, satisfying d(tii) = 0, and the other is to calculate the angle q5o. The calculation of the Hill-type determinant d(ti) is done by using the formula (4.3). Since rip + k + I as k-i 1x1, we set ri,,+k= 1 for k = 100 and used the recursion to obtain the values r r,,+k for k = 1, 0, - 1. In Fig. 2, we show the typical behaviour of A(@) for N = 3 case. In the case of A, = 0, K = 0 is the solution which gives us a well-defined wave function and A(K) = 0 has only two solutions ti?, K) (ti? + ~~ = 0). In general, A, # 0 and A(K) = 0 has three solutions ui, rc2, and ICY(ti, + K: + JC~= 0). These solutions are very close to the poles si of A(K) (i.e., D(E,) = 0). and one can numerically easily

FIG. 2. Hill-type determinant and A, # 0 cases, respectively.

A(@)

for N=

3. The upper

and lower

graphs

correspond

to the .4 1 = 0

320

A.MATSUYAMA

solve the Hill-type equation A(K) = 0 by the bisection method. Once the solutions ~~ are obtained, one can calculate the value Y of (4.5) and obtain the Gutzwiller’s phase angle & as do = arg(r). Calculation of the semiclassical quantization is not easy for a many-body system. We must search the N- 1 semiclassical eigenvalues (E’, a;, .... ah) which satisfy the semiclassical quantization conditions simultaneously. We have exact eigenvalues and it seems that the semi(~5 a3, .... a,,,) obtained by the direct diagonalization classical eigenvalues (E’, a;, .... ah) are close to the exact values. Therefore, we used the simplex method with the exact values as initial values and obtained the semiclassical eigenvalues. In practice, we must calculate the integral of (5.6)-(5.9). Since IA(p)1 > 2, v(p) > 0 and the integral is well-defined. In the actual calculation, we use the second formula given by the partial integration. In this case, the integrant diverges both at the lower and upper bounds as l/,,& + cc (E + 0). These singularities of l/& type can be integrated as [ (&/A) = 2 ,,k and are harmless. In the numerical calculation, we have changed the variable p = ,LL,!f t2, explicitly removed the divergence of the integrant, and used the Gauss-Legendre integration formula.

7. RFXJLTS AND DISCUSSION We have studied the periodic Toda lattice of N = 3, 4, 5, and 6. Once the eigenvalues (E, ai) are obtained, one can easily examine whether those eigenvalues satisfy Gutzwiller’s quantization conditions. The correspondence between the Gutzwiller’s phase angle do and the symmetry of the state has been shown in Section 4, and we will numerically demonstrate that the relation holds indeed. Concerning the semiclassical quantization, we have noted that the quantum ) and the symmetry of the state has a close numbers (~1,,n,-,, .... n-,+,,n-, relationship as follows. For N = odd (N = 2m + 1 ), we define the integer N, as N,=(rZ,-n-,)+2(n,~,-n~,+,) . . . +m(n,

+WL-~-~-~+~+

-n-,)

(mod N).

(7.1)

Then N,=O N,= For N=even

+k

for

A-symmetry

for

E,-symmetry.

(7.2)

(N=2m+2),

N,=(n,-n-,)+2(n,-,-n-,+,)+3(n,-,-n-m+,)+ +m(n,-~n,)+(m+l)n,(modN).

... (7.3)

PERIODIC

TODA

321

LATTICE

Then N,=O

for

A-symmetry

N, = N/2 N,= +k

for

B-symmetry

for

E,-symmetry.

(7.4)

Although the relations (7.1 k(7.4) are deduced from the numerical calculation, it seems natural and we can conjecture that it will hold, in general. In the following subsections, we will discuss the numerical results of the N = 3, 4, 5, and 6 periodic Toda lattices in detail. 7.1 N=3 We have already reported the results of the N = 3 periodic Toda lattice in Ref. [lo]. where we used the Jacobi coordinates and removed the CM. motion from the beginning. In the present calculation, we used the basis 4s,‘,i for the diagonalization. In Table IV we list the number of the basis for each symmetry A,. A,, and E. In the practical calculation, we used the basis up to N, = 30; thus the total numbers of the basis are 91, 75, and 165 for A,, Al, and E symmetries, respectively. The extension parameter o is set w = 2.0. If the exponential potential is approximated by the second-order polynomial, i.e., ey = 1 + q + $q’, the Hamiltonian is reduced to the sum of two independent harmonic oscillators with normal mode w = $. Since the exponential potential grows more rapidly than the quadratic potential, the value o larger than $ is preferable. We could reproduce the previous results exactly. In Table V, we show the lowest 10 eigenvalues (E, u3) TABLE Number

of the Basis for N = 3

NO

A,

AZ

E

No

A,

AL

E

0

1 0 1 1 1 1 2 1 2 2 2 2 3 2 3 3

0 0 0 1 0 1 1 1 1 2 1 2 2 2 2 3

0 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

3 3 4 3 4 4 4 4 5 4 5 5 5 5 6

2 3 3 3 3 4 3 4 4 4 4 5 4 5 5

6 6 6 I 7 7 x x x 9 9 9 10 I0 10

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

595,‘220’?-

IV

I2

322

A. MATSUYAMA TABLE Eigenvalues

and Quantum

V Numbers

for N = 3

(1) A-symmetry

Level 1 2 3 4 5 6 7 8 9 10

(2) E-symmetry

(HI, n-1) 4.8374 (4.7748) 8.6386 (8.5854) 10.9089 (10.8558) 12.7787 (12.7310) 15.1741 (15.1272) 17.2328 (17.1887) 18.0295 (17.9800) 19.7540 (19.7107) 21.9793 (21.9378) 22.6891 (22.6452)

0.0 (0.0) 0.0 (0.0) 9.1890 (9.2294) 0.0 (0.0) 11.4083 (11.4350)

(O,O) (1, 1) (3,O)

(Z2) (4,l) (3,3)

(& 24.3683 (24.4328) 13.6433 (13.6635)

(60) (5,2) (474)

(:I:, 29.0128 (29.0562)

(7,l)

@I> n-1) 6.7265 (6.6686) 8.7553 (8.7002) 10.6942 (10.6438) 12.8764 (12.8280) 13.1772 (13.1256) 14.9907 (14.9448) 15.5527 (15.5023) 17.3185 (17.2741) 17.5794 (17.5336) 19.5909 (19.5482)

2.3934 (2.4110) -5.4596 (-5.4897) 3.1107 (3.1219) -6.9160 (-6.9356) 13.5811 (13.6302) 3.8373 (3.8454) -18.6391 (-18.6962) -8.3868 (-8.4014) 16.5866 (16.6194) 4.5751 (4.5814)

(170) a21

(2,1) (1,3) (4,O) (3,2) (0.5) (224) (5>1) (433)

semiclassical eigenvalues (E’, a;), and the quantum numbers (n,, n-r). The accuracy of the calculation is examined by changing the number of the basis and the extension parameter w, and the eigenvalues are shown within the reliable digits. One should note that N, = n, - n _ 1 = 0 (mod 3) for the A-symmetry and N, = 1 for the E-symmetry. The state with N, = -1 corresponds to the eigenfunction bE’. In Fig. 3, we show the contour plot of the potential in the Jacobi coordinates G 12) (5, = (1/&k - q2), t2 = (l/,/h, + q2 - 2qd. It has the D3 group symmetry and the minimum is at t1 = t2 = 0; i.e., the bottom of the potential is at the center of mass. In Fig. 4, we show the plot of the eigenvalues (E, a3). The distribution is systematic and one can assign them integer quantum numbers (n,, nPl) of the semiclassical quantization. The region 2(E/3)3i2 < la,] + 2 is where the classical motion is forbidden; i.e., at least one of the intervals of [d(p)1 2 2 does not exist. In Figs. 5 and 6, we show the contour plot of the squared wave function )YYJ’ for the lowest two levels. When we use the polar coordinates (r, 0) such that c1 = Ycos t?, t2 = r sin 8 as Ref. [lo], the states with the A,-symmetry have the angular momentum 1 = 0, -t 3, &6, .... and those with the AZ (E) symmetry have 1=+3- > +6 - 7 -.’ (1 = +l, f2, +4, ...). Therefore only the wave function !P with the A-symmetry has 1 = 0 and non-zero value at rl = c2 = 0, while those with the E-symmetry vanish at the origin. As illustrative examples, density distributions p(q)

PERIODIC

TODA

323

LATTICE

l-

0

0

-1 !Q

-1 Ftc;. 3. Contour plot contours arc the same.

\ OO

of the potential

I

FIG. 4. Distribution of the eigenvalues is the region where the classical motion quantum number

0

1

of the three-particle

I 20

I

52

periodic

I LO

Toda

I

lattice.

Spacings

of the

I

60

a3

in (E, a,)-plane for N = 3. Shaded area (2(E/3)31’ < 1~~1 + 2) is forbidden. The dashed lines show the series of the same

B ::oQ

324

A.MATSUYAMA

0

000

0

-1 -

0 ,

,Q,, -1

FIG. 5. Contour plot of the squared wave function 1YyI* for the ground (B) of the A-symmetry. The contour spacing is A 1VI2 = 0.1.

0

state

1

%

(A) and the first excited

state

hA l-

O-

-1 -

IB

k,_

Giii@ Q l-

00° Q0

O-

-1 -

FIG. 6. Contour plot of the squared wave function 1VI2 for the ground state (A) and the first excited state (B) of the E-symmetry. The central region is a hollow, while there are three hills outside. The contour spacing is d I Y(’ = 0.05.

PERIODIC

TODA

LATTICE

325

for the lowest four levels are calculated and shown in Fig. 7. One sees that the density of the ground state of the A-symmetry is close to the Gaussian shape and the distribution spreads when the system is excited. For example, the densities of the second and fourth excitations have the typical pattern of oscillation. For the E-symmetry, the density of q = 0 is slightly lower and the oscillation is less eminent than that of the A-symmetry. We have found that Gutzwiller’s quantization conditions are satisfied with an accuracy of 10n4 % for the levels in Table V. In fact, A and E symmetries correspond to do = 0, 7c/3. In Fig. 2 of the Hill-type determinant A(K), the typical values are: The poles si= ie, (D(ci) = 0) are (C,, E2, E3) = (0, -2.1994, 2.1994) and the solutions of A(ti)=O (rci= iC,) are (rCZ, K3)= (-2.1876,2.1876) for the ground state of the A-symmetry (E = 4.8374, a3 = 0) in the upper graph, while (Er , F, , Cj ) = ( -2.7559, 0.3629, 2.3930), (I?~, I?*, LX) = ( -2.7524,0.3694, 2.3830) for the ground state of the E-symmetry (E = 6.7265, a3 = 2.3934) in the lower graph. One can see that the semiclassical eigenvalues are already good approximations of the exact values, even at the ground state. For the low excitation states (total quanta N, d 3) the excitation energy for the quantum number n, , is approximately 0, ‘u 2.0, and the excitation energy is given by AE1n,w,+n_,05,=(n,+n~,)tu,.

(7.5)

FIG. 7. Density distribution p(q) for N = 3. The uppermost graphs show the densities of the ground states, while the lower graphs correspond to the excited states with the energy increased. Note that i’(y) has the symmetry p( -9) = p(q).

326

A. MATSUYAMA

7.2. N= 4

In Table VI we list the number of the basis for each symmetry A,, A,, B,, B,, and E, and, in practice, we used the basis up to N, = 20 for A and B symmetries, N, = 16 for the E-symmetry. The total numbers of the basis are 256, 190, 250, 195, and 240 for A,, A,, B, , B,, and E symmetries, respectively. The extension parameter w is set w = 2.0. In Table VII we show the lowest 10 eigenvalues (E, ~3, G), semiclassical eigenvalues (E’, a;, ai) and the quantum numbers (n, , no, n _ i). The eigenvalues (E, u3, u4) satisfy the Gutzwiller’s quantization conditions with an accuracy of 1O-3 % for the states listed in Table VII. The state with the A (B, E) symmetry has 4o = 7c/2 (0, -7c/4) and N, = (n, -n-r) + 2n, = 0 (2, 1). In Fig. 8 we show the distribution of the eigenvalues in the (a3, a,)-plane. The distribution is systematic, although much more complicated compared to the N = 3 case. The shaded area is the region where the classical motion is forbidden. For N = 4, the function d (FL) of Eq. (5.2) is A(p)=16p4-4Ep2+2A3p+Aq.

(7.6)

The solutions of dA/dp = 0 are pL1= ,/@i

cos(0/3),

p2 = ,,/$

cos((8 + 2x)/3},

pj=&iLOS{(e+47c)/3},

(7.7) TABLE

Number

VI

of the Basis for N = 4

NO

A,

-42

B,

&

E

0 1 2 3 4 5 6 I 8 9 10 11 12 13 14 15 16 17 18 19 20

1 0 2 1 4 2 6 4 9 6 12 9 16 12 20 16 25 20 30 25 36

0 0 0 1 1 2 2 4 4 6 6 9 9 12 12 16 16 20 20 25 25

0 1 1 2 2 4 4 6 6 9 9 12 12 16 16 20 20 25 25 30 30

0 0 1 0 2 1 4 2 6 4 9 6 12 9 16 12 20 16 25 20 30

0 1 1 3 3 6 6 10 10 15 15 21 21 28 28 36 36 45 45 55 55

Led

6

5

4

3

2

I

03

6.329

(6.375)

11.915

0.0

(0.0) 0.0

(0.0)

15.162

(15.680)

16.429

(16.352)

7.802

(7.830)

15.177)

(0.0)

14.048)

15.253

0.0

( - 14.916)

13.176)

14.125

-14.842

13.262

0.0

(0.0)

12.932

(12.851)

(10.858)

10.944

(9.549)

9.636

(6.465)

(11.832)

(4)

0.0 (0.0) 0.0 (0.0) 0.0 (0.0)

E

6.562

W

04

4.727

(a,)

(6.472)

6.834

(53.086)

53.031

(15.263)

15.476

(26.764)

26.873

(1.220)

1.493

(5.911)

6.183

(13.116)

13.241

(22.313)

22.311

(5.312)

5.492

(4.655)

( 1) A-symmetry

0)

(3,(x3)

(0,4,0)

3. I. I)

1.2.1)

(0.0.4)

(2.0.2)

(2. I.

(0.2.0)

(1.0.1)

(0.0.0)

(n,.n,.n

1)

(16.469)

16.543

(16.424)

16.501

15.423)

15.503

15.097)

15.172

14.143)

14.222

13.219)

13.303

12.931)

13.013

11.737)

Il.818

(9.642)

9.731

(8.603)

8.694

(E’)

E

Eigenvalues

(0.0)

0.0

(-8.882)

-8.857

(-15.509)

(0.0) ~ 15.439

0.0

(-6.579)

-6.532

(0.0)

0.0

(7.523)

(0.0) 7.492

0.0

(6.145)

(0.0) 6.098

0.0

c::,

(42.836)

42.929

(5.205)

5.568

( 11.464)

11.670

(16.539)

16.752

(25.399)

25.493

(35.903)

35.874

(4.727)

5.000

(14.346)

14.419

(4.200)

4.316

(12.001)

12.030

(4)

4

Numbers

VII

( 2 ) E-symmetry

and Quantum

TABLE

(1.3.1)

c2,0,4t

(0. 1.4)

(2. 1.2)

(0.3, 2)

(0. 3.0)

(3.0. I)

(1,1.1)

(2.0.0)

(0.1.0)

(n,.n,.c,)

for N = 4

E

(14.839)

14.918

(14.748)

14.828

(14.597)

14.676

(13.587)

13.668

(13.412)

13.490

(12.448)

12.530

(11.369)

Il.456

11.195)

11.219

10.166)

10.252

(8.004)

8.096

W

( -2.919)

-2.947

(-12.272)

- 12.229

(4.108)

4.095

(10.608)

10.549

( -3.567)

-3.550

(2.906)

2.815

(-10.209)

- 10.147

(3.428)

3.411

(-2.825)

-2.796

(2.734)

2.705

(4)

03

04

4.848

(4)

(39.034)

39.066

(3.450)

3.770

(5.888)

6.207

(12.654)

12.820

(15.144)

15.318

124.232)

24.285

(3.050)

3.274

(5.330)

5.557

(12.892)

12.974

(4.721)

(3) E-symmetry

1.2)

(0.3.1)

(1.0.4)

(3.0.2)

(3. 1.0)

(I.

(1.2.0)

(0,0.3)

(2.0.1)

(O.I,l)

(1.0.0)

(n,.n*3K,)

3

3

::

i P

ij

ki 2 8

328

A. MATSUYAMA B

A

k 0 30-

A

l A-sym. %=n12 mB -9ym. gG=o A E- sym. %=a A E-sym. ~=-xnc

A

. A

4

A

.

.

A

A 0 20I 1

I

l

A

II A

I

. A n

. A 8

lo-

A

. PA 4 . .

A

‘A

I

0

10

.

. , 2?4

a3 A

FIG. 8. A. Distribution of the eigenvalues in (aj, a,)-plane for N = 4. The lowest 30 eigenstates are plotted for each symmetry (Es24 for A and B symmetries and Es.21 for the E-symmetry). B. Magnified graph of Fig. 8A. The eigenstates with E 5 20 are plotted.

where

coso= -$gg”* Since pz
-2,

(&G&E’).

one can obtain

4P3)

2

27

the conditions

d(1”1)6 -2.

(7.8)

for the allowed

(7.9)

There are three intervals [A(p)/ 2 2 and each one corresponds to quantum numbers (n,, n,, n-i). In contrast to the N= 3 case, we have two different kinds of the excitations, i.e., n + 1 and no. In the low excitations (N, < 3), the excitation energies of the n+, and ni quanta are 15, ?: 1.60, W, N 2.22, and the excitation energy is approximately given by AEz

(n, +n-,)G,

+n,,cS,.

(7.10)

PERIODIC

Since W,>W,, mode.”

TODA

329

LATTICE

the “central mode” has larger excitation energy than the “peripheral

7.3. N= 5

In Table VIII we list the number of the basis for each symmetry and, in practice, we used the basis up to N, = 14 for the A-symmetry and N, = 12 for the E-symmetry. The total numbers of the basis are 324, 288, 364, and 364 for A,, A ?, E,, and E, symmetries, respectively. The extension parameter o is set w = 2.0. In Table IX we show the lowest five eigenvalues (E, a3, u4, as). semiclassical eigenvalues (E’, a;, ai, a;), and the quantum numbers (n,, n,, K,, n .1). The eigenvalues (E, u3, u4, as) satisfy the Gutzwiller’s quantization conditions with an accuracy of 0.1 % for the listed states. The state with the A (E,, EJ symmetry has do=0 (7c/5,27r/5) and N,=(n,-n_,)+2(n,-n~,)=O (1,2). It is difficult to visually display the distribution of the quantum states (6 u3, u4, us). In the same way of the N = 4 case, the low excitations are approximately given by the excitations of the quanta ~2+ , and n +*. The excitation energy of the lower states (N, 6 3 ) is tiZ z 1.32, Oi rr 2.08. Thus the excitation energy is AE=(n,+n-,)w,+(n,+n

The central mode Wr has larger excitation energy than the peripheral which is the same situation as the N = 4 case. TABLE Number

0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 I5

1 0 2 2 5 6 10 12 19 22 32 36 49 56 12 82

(7.1 I )

,)w,.

mode Oz,

VIII

of the Basis for N = 5

0 0 0 2 2 6 6 12 14 22 26 36 42 56 64 82

0 1 2 4 7 11 17 24 33 44 57 73 91 112 136 163

0 1 2 4 7 11 17 24 33 44 57 73 91 112 136 163

330

A. MATSUYAMA

TABLE Eigenvalues

and Quantum

IX Numbers

for N = 5

( 1) A-symmetry 1 2 3 4 5

0.0 (0.0) 0.0 (0.0) 0.0 (0.0)

(0,o,o,0) (LO,031) (0,1,LO) co, LO)

4.31 (4.35) 0.0 (0.0)

12.50 (12.25) 15.06 (14.61) 32.87 (32.63) 24.91 (24.47) 17.41 (16.78)

9.54 (9.41) 11.56 (11.44) 12.17 (12.05) 12.39 (12.27) 12.95 (12.83)

2.55 (2.58) -1.12 (-1.13) 3.16 (3.18) -3.19 ( - 3.22) - 7.46 (-7.53)

13.48 (13.13) 23.69 (23.35) 15.91 (15.37) 33.79 (33.52) 24.61 (24.17)

- 1.57 (-1.53) -4.83 ( - 5.01) - 1.64 (-1.59) 18.48 (18.77) 13.08 (13.16)

(LO,

10.28 (10.16) 10.89 (10.76) 11.58 (11.46) 12.31 (12.18) 12.86 (12.74)

1.46 (1.47) 5.70 (5.75) -4.14 (-4.19) -9.41 (-9.48) 1.57 (1.58)

21.72 (21.47) 13.70 (13.26) 23.60 (23.25) 13.14 (12.60) 26.21 (25.75)

-6.91 ( - 7.06) -3.13 (-3.03) 10.00 (10.11) 4.61 (4.46) - 7.95 ( - 8.10)

(0, l,O* 0)

8.27 (8.14) 10.81 (10.69) 12.32 (12.20) 12.92 (12.80) 13.52 (13.41)

(i::, 0.0 (0.0) (ii)

2.60 (2.83) 0.0

(0.0)

c&o, 022)

(2) El-symmetry 1 2 3 4 5

(3)

O,O)

(0, LO,

GO,

1)

03 1)

(0, 0,230) (0, 0,

WI

E,-symmetry 1 2 3 4 5

GAO,QO) (030, 1, 1) (0, 0, 0, 3) (1, LO,

1)

PERIODIC

TODA TABLE

Number

2 3 4 5 6 I 8 9 IO 11 12

1 0 3 3 9 10 24 26 50 59 95 112 169

X

of the Basis for N = 6

A2 0

331

LATTICE

B,

0 0 0 3 3 10 14 26 35 59 74 112 141

B2 0

I 5 5 14 17 33 40 69 81 126 150

0 0

2 5 8 17 23 40 54 81 105 150

El

EZ 0

0

1 2 6 11 22 33 57 80 121 164 231 298

5 I3 20 36 54 x5 I16 171 224 307

7.4. N=6 In Table X we list the number of the basis, and, in practice, we used the basis up to No = 11 for A and B symmetries, N, = 9 for the E-symmetry. The total numbers of the basis are 392, 336, 392, 336, 333, and 333 for A,, A,, B,, Bz, E,, and E, symmetries, respectively. The extension parameter w is set w = 1.6. In Table XI we show the lowest five eigenvaues (E, a,, clq, a,, ub ), semiclassical eigenvalues (E’, 4, ai, a;, a:), and the quantum numbers (n,, ?I,, n,, n-i, nPz). The eigenvalues (E, a3, a4, a5, ~7~) satisfy the Gutzwiller’s quantization conditions with an accuracy of 1 % for the listed states. The state with the A (B, E,, E2) symmetry has &=7(/2 (0, -27~16, -7116) and N,=(n,-n_,)+2(n,-n_,)+3n,=O (3, 1.2). The low excitations are approximated by the following formula, as before, dE=(n,+n~2)02+(n,+n_,)w,+n,to,,

(7.12)

where O2 z 1.10, 0, N 1.86, c&, N 2.14. The central mode has larger excitation energy than the peripheral one, i.e., 0,>0, >O,. This is the same as N=4, 5 cases, and one can generally conclude that the central mode has larger excitation energy.

8.

SUMMARY

The Hamiltonian of the N-particle periodic Toda lattice has the symmetry of the D, group, and the eigenstate can be classified according to the irreducible representation of the D, group. The D, group has 2 one-dimensional representations

332

A. MATSUYAMA TABLE Eigenvalues

and Quantum

XI Numbers

for N = 6

(1) A-symmetry 1 2 3 4 5 (2)

9.96 (9.80) 12.12 (11.97) 13.63 (13.48) 14.02 (13.87) 14.25 (14.09)

0.0 (0.0) (2) 0.0 (0.0) 3.04 (3.06) 0.0 (0.0)

23.19 (22.65) 27.71 (26.90) 44.62 (43.95) 38.22 (37.34) 55.79 (55.20)

0.0 (0.0) 0.0 (0.0) 0.0 lO.0 1 2.75 (3.13) 0.0 (0.0)

- 7.29 (-7.14) - 7.90 (-7.66) - 10.31 (-9.62) - 9.25 (-8.74) -54.18 (- 54.65)

B-symmetry 1

2 3 4 5

12.06 (11.90) 12.89 (12.74) 13.38 (13.22) 14.25 (14.09) 14.79 (14.64)

0.0 (0.0) 4.59 (4.65) -8.42 (-8.49) (iii) 2.30 (2.31)

37.57 (37.02) 35.78 (35.04) 26.90 (25.98) 44.34 (43.49) 49.11 (48.27)

0.0 (0.0) - 17.74 (- 17.87) 11.59 (11.28) 0.0 (0.0) - 25.34 (-25.79)

-24.54 ( - 24.69) -6.70 (-6.21) -6.82 ( - 6.60) -27.19 (-27.22) - 6.84 (-6.02)

11.04 (10.88) 12.88 (12.72) 13.28 (13.12) 13.92 (13.77) 14.32 (14.15)

2.31 (2.35) -0.19 (-0.20) 2.83 (2.85) -2.19 ( - 2.22) - 5.24 (-5.29)

25.17 (24.49) 36.00 (35.24) 29.61 (28.60) 49.68 (49.04) 43.08 (42.17)

-3.73 - 3.64) - 7.42 - 7.70) -4.02 - 3.90) 13.84 (14.07) 11.02 (10.84)

- 7.45 (-7.25) -8.59 (-8.11) - 8.05 (-7.75) - 28.46 (-28.33) -26.17 ( - 26.24)

11.79 (11.63) 12.19 (12.03) 13.15 (13.00) 13.71 (13.55) 14.01 (13.83)

2.13 (2.16) 5.13 (5.19) -2.35 (-2.39) -4.62 (-4.67) 2.25 (2.30)

33.04 (32.42) 26.42 (25.61) 40.67 (39.97) 45.14 (44.43) 39.44 (38.43)

-11.70 (-11.90) - 7.63 (-7.44) 5.29 (5.23) 30.05 (30.42) - 13.37 (- 13.71)

-7.53 (-7.10) - 7.28 ( - 7.07) - 25.65 ( -25.74) -4.63 (-3.87) -8.23 (-7.60)

(3) E,-symmetry 1 2 3 4 5 (4)

E,-symmetry

1 2 3 4 5

333

PERIODIC TODA LATTICE

(A,, A,) and (N - 1)/2 two-dimensional representations (Ek, k = 1, 2, ...) for N= odd, and 4 one-dimensional representations (A,, A,, B, , B,) and (N - 2)/2 two-dimensional representations (Ekr k = 1, 2, ...) for N=even. We have diagonalized the Hamiltonian in terms of the orthogonal basesof N products of the one-dimensional harmonic oscillator for each symmetry. The eigenstates generally degenerate in two-fold, i.e., 4, = ( l/,/?)(dAI + i~$“~) and 4” = (l/$)(4”’ + &SE’) for A and B symmetries, respectively, and 4E, 4”’ for the E-symmetry. The non-degenerate states belong to the A ,-symmetry for N = odd and either A, or B, symmetry for N = even. The conserved quantities L4. have the A, (A,) symmetry for i = even (odd), and their eigenvalues have been calculated for each eigenstate. Gutzwiller has developed a systematic way of constructing the eigenfunction and has obtained the quantization conditions in terms of the “Gutzwiller’s phase angle 4 G.” The angle 4o specifies the symmetric property of the eigenstate and it has a one-to-one correspondence to the irreducible representation of the D,V group. We have also carried out the semiclassicalquantization of the periodic Toda lattice by the EBK (Einstien-Brillouin-Keller) formulation. The eigenvalues of the semiclassical quantization have a one-to-one correspondence to the integer quantum numbers. We have noted that the quantum number also has a close relationship to the symmetry of the state. We have reported the numerical calculation of the periodic Toda lattice for N = 3, 4, 5, and 6. The distributions of the eigenvalues are systematic and distinguished by the symmetry or the quantum number. We have confirmed that the eigenvalues satisfy the Gutzwiller’s quantization conditions. The minimum of the potential is at the center of mass and the wave function tends to be localized around it. As illustrative example, we have shown the density distribution of the system for N = 3. The density of the ground state is close to the Gaussian shape, while the distribution spreadswhen the system is excited. When the particle number is increased, the density distributions become similar and have no noticeable structure.

ACKNOWLEDGMENT We thank

the theoretical

nuclear

physics

group

of the University

of Tokyo

for useful discussions

REFERENCES 1. M. TODA, J. Phys. Sot. Japan 22 (1967). 431. M. Toda, “Theory of Nonlinear Lattices.” Berlin, 1981. 2. J. FORD, S. D. STODDARD, AND J. S. TURNER. Prog. Theor. Phys. 50 ( 1973 ). 1547. 3. M. H~NON, Phvs. Rev. B 9 (1974), 1921. 4. H. FLASCHKA, Phys. Rev. B 9 (1974), 1924. 5. H. FLASCHKA, Prog. Theor. Phys. 51 (1974), 703. 6. M. KAC AND P. VAN MOERBEKE, Proc. Nat. Acad. Sci. U.S.A. 72 ( 1975), 1627, 2879.

Springer.

334

A. MATSUYAMA

7. E. DATE AND S. TANAKA, Prog. Theor. Phys. 55 (1976). 457; Prog. Theor. Phys. Suppl. 59 (1976), 107. 8. M. BRUSCHI, D. LEVI, M. A. OLSHANETSKY, A. M. PERELOMOV, AND 0. RAGNISCO, Phys. Left. A 88 (1982), 7; M. A. OLSHANETSKYAND A. M. PERELOMOV,Phys. Rep. 94 (1983), 313. 9. M. C. GUTZWILLER, Ann. Phys. (N.Y.) 124 (1980), 347; 133 (1981), 304. 10. A. MATSUYAMA, Phys. Left A 161 (1991), 124. 11. S. ISOLA, H. KANTZ, AND R. LIVI, J. Phys. A 24 (1991), 3061. 12. M. V. BERRYAND M. TABOR, Proc. R. Sot. London A 356 (1977), 375. 13. M. L. MEHTA, “Random Matrix,” 2nd ed., Academic Press, New York, 1991. 14. C. W. CURTIS AND I. REINER, “Representation Theory of Finite Groups and Associative Algebras,”

Interscience, New York, 1962.