A new method to solve the quantum Ablowitz-Ladik system

Physics Letters A 162 (1992) 381-384

PHYSICS LETTERS A

North-Holland

A new method to solve the quantum Ablowitz-Ladik system M. Salerno Department of TheoreticalPhysics, Universityof Salerno, 841O0Salerno, Italy Received 13 August 1991;revisedmanuscript received 18 November 1991;acceptedfor publication 18 November 199I Communicatedby D.D. Holm

We use an alternativeapproach to solvethe quantum problemof the Ablowitz-Ladiksystem.This approachis simplerthan the standard quantum inversescattering method and it is based on the existence,in the symmetryalgebra, of an invariant operator with finite dimensional eigenspaces. This makes the quantum problem equivalent to the diagunalization of finite matrices, a problem which is easilysolvedon a computer.

Nowadays there is a general procedure, known as the quantum inverse scattering method (QISM) [ 14 ], which allows one to solve the quantum problem of a classically integrable system. This method was introduced in connection with some models in quantum statistical mechanics and in quantum field thepry which are defined on the chain and represents the quantum version of the well-known inverse scattering method (ISM) for solving soliton equations. The energy spectrum and the eigenvalues of the higher integrals of motion are expressed, by this method, as finite series of the solutions of the socalled Bethe equations. Due to the complexity of these equations, however, it is in general difficult to get explicit values for the energy levels of the system. Sometimes, it is possible to solve the quantum problem of an integrable system by an alternative method. Indeed, the existence in the symmetry algebra of an integrable system of an operator with finite dimensional eigenspaces allows the reduction of the infinite dimensional eigenvalue problem for the Hamiltonian to the problem of diagonaiizing finite matrices. This method is quite standard in quantum mechanics and it was used to solve the quantum problem of a nonintegrable lattice version of the nonlinear Sehr6dinger equation which is known as the DST equation [ 5-7 ]. The two lattice point DST system is integrable in terms of a Lax pair, so it could be quantized using both QISM and the present standard approach [8 ]. While each method ultimately ElsevierSciencePublishersB.V.

leads to the same set of energy eigenvalues, the standard one was shown to be much easier to use than the QISM. The aim of the present paper is to show that the standard method can be effectively used to solve the quantum problem of the Ablowitz-Ladik system [ 9 ] iAj+Aj+ 1-2Ay+A)_I +½7]Aj[2(Ay+, +Aj_I) = 0 , j = I, 2, ...,f,

(1)

with periodic boundary conditions: Af+l=Ab Ao =Af This system represents an integrable discrete version of the nonlinear Schr'6dinger equation and it was quantized by Gerdjikov, Ivanov and Kulish by using the QISM [ 10]. The quantum problem of this system, however, was solved by these authors only in a formal manner, i.e. although the Bethe equations were written, no explicit solutions for the energy levels were derived either analytically or numerically. In the following we show that the proposed approach is easier, both analytically and numerically, than the QISM and some explicit formulas for the first energy eigenvalues of the system are presented. Furthermore, we show, for the first excited states, that the energy levels derived by the proposed approach coincide with those derived by extending the QISM analysis of Gerdjikov et al. Let us start by introducing the Hamiltonian operator of the quantum Ablowitz-Ladik system

3 81

Volume 162, number 5

H=-

PHYSICS LETTERS A

Ef 0 f (AJ+ l + A j - l ) - - ln(l 2 +/?)

j=l

× l n ( l + ½YA+Aj)),

(2)

with A +, A operators satisfying the following nonstandard commutation relations [ 10 ],

[A+,A+]=[Ai,Aj]=O, [Ai, A + ] = ( I + ½?A+A~)tSo.

(3)

Note that these commutation relations can be seen as a deformation of the usual ones for bosonic creation and annihilation operators; indeed, the latter are obtained from eqs. (3) by shrinking the deformation parameter y to zero. By using eqs. (3) one gets from the Heisenberg equations of motion i,4i = [ a , H I ,

(4)

the quantum version of the Ablowitz-Ladik (QAL) system, if4i + (Ai+ 1 -- 2Ai + A i - 1)

+½yA~-Ai(Ai+l+ A i _ l ) = 0

•

(5)

We introduce the Fock space ~ corresponding to the creation operator A ~- and its usual basis [0) k, I 1 ) k, .... IJ)k, -.- defined as Akl0)k=0,

A~'ln)k=v/-~, In+l)k.

A k i n ) k = X/-~, I n - - l ) k .

(6)

By using the commutation relations (3) and requiring the orthonormality of the above basis, we easily get the following relationships between the coefficients ot and//, a , =/~,_~,

ot,+t = 1 +or,(1 +½y).

(7)

The general solution of this system of difference equations is readily obtained as ot,=//,_~ = 2 ( l + ½ Y ) " - 1, Y

(8)

from which the action of the A ~, Ak operators on the basis states is derived,

382

Note that these relations reduce to the usual ones for bosonic creation and annihilation operators when the deformation parameter y goes to zero. Due to the nonstandard commutation relations (3) the number operator of the QAL system is given by 1

f

N = ln(l+½Y)j=~ I n ( I + ½ ? A + A j ) .

(10)

Indeed, by expanding the logarithm in powers ofA +A and using relations (9) one readily obtains

Nljt ...Jf > = (J, +...+Jf)IJ~ ...Jf ) •

( 11 )

We note also that in the limit of ? going to zero this operator reduces to the usual bosonic number operator. An important property of this quantum system is that the number operator in eq. (10) commutes with the Hamiltonian, [N,H] = 0 .

(12)

This is a consequence of the fact that N is the quantum correspondent of the second conservation law of the classical Ablowitz-Ladik system [ 5,6]. Due to this property one can exploit the decomposition of the Hilbert space of quantum states as the direct sum ~ = ~ ~

... ~ ( 3

....

(13)

where af. denotes the eigenspaces of N with n as eigenvalue whose dimension is

( n + f - 1 )! dim (~e~) = n ! ( f - 1)! "

(14)

The Hamiltonian can then be separately diagonalized in the finite dimensional eigenspaces ~ , thus reducing the infinite dimensional eigenvalue problem for H to a simple algebraic problem. For a finite number of degrees of freedom f the Hilbert space of the quantum states can be identified with the tensor product

ae=~®~® ...®~,

(15)

with the generic element of the corresponding product basis given by

A~-In>k= x/2[(1 +½Y)n+l--1]/Y I n + l ) k ,

A k i n ) k = x/Z[(1 + ½ r ) " - - 1 ] / r I n - - l ) k .

24 February 1992

(9)

[Jl J2 "'Jf) = ]Jl > 1® ]J2 ) 2 ® ' " ® ]Jf>f"

( 16 )

Volume 162,number5

PHYSICSLETTERSA

In this representation the matrix elements of the restriction of the Hamiltonian H to ~e~ are given by

= 2 nJ~t/lJ~2...dejf f2

24 February 1992

suitable from a numerical point of view. In dosing this paper we wish to compare the above results for f = 2, n = 1, 2 with those one can derive by using the QISM. Following ref. [ 10] we have that the spectrum of the QAL system is obtained as

- k~=t~ {X/[ (1 + ½ryk-'+' -- 1 ] t (1 + ½y)Jk-- 1 l

E=2n-

(17)

X aik+ l,jk+ l + l (~ik,jk+ l_ l } ,

where i~+i2+...+i:=j~+jz+...+j:=n. From eq. (17) the eigenvalues and eigenvectors of H are readily obtained by diagonalizing the corresponding (f+ n - 1 )!/[ 0 r- 1 )!n! ] matrix representation. For example let us compute the first excited levels for the two degrees of freedom (f= 2) QAL system. For n = 1 the secular equation corresponding to the 2 × 2 matrix (17) is readily written as

det12_-ff 2-2 =0,

(is)

from which we have the energy levels El = 0, E2 = 4. For n = 2 the secular equation for the 3 × 3 matrix (17) is - 2x/~- ~ 4-E - 2 ~

0 ] -2~1=0, 4-E I (19)

from which we get Et = 4, E2.3 = 4 + 24 ~ . For arbitrary n we have to diagonalize the symmetric tridiagonal (n + 1 ) × (n + 1 ) matrix with diagonal elements

Hi,,=( i - l , n - ( i - l

)lHli-l,n-(i-1)

)

=2n

(20)

and with the upper diagonal ones given by

Hi,i+t = ( i - 1, n - ( i - 1 ) l H l i ,

n-i)

= ( - 4 / y ) x/ [ ( l + ½y)i-1] [ ( l + ½Y)n+t-'- l ] , i=1 ..... n.

(22)

where the Zk'S satisfy the Bethe equations

+ x / [ (1 + ½yyk+'+l -- 1 l [(1 + ½yy*-- 1 l

4-E det - 2 x / ~ ½ y 0

~ (z2+z~-2), k=l

X ~ik-l,jk-l+ l(~ik,jk--I

FI sinh(2k--2j +~/) ,

k = 1, 2, ..., n,

(23)

with r/= ½In(1 + ½y) and Zk=eXp(2k). In ref. [7 ] it was shown that for n = 1 system (23) reduces to the simple equation z~f= 1 which implies, by using eq. (22) for f = 2 that E t = 0 , E2=4, in agreement with the above analysis. For n = 2 system (23) becomes equivalent to the two identical equations (see ref. [71) z ~ +4 - t ( 1 + ½y)z~I - (1 + ½y)z~ + t = 0 , k= 1, 2,

(24)

with the constraint (ztz2)2:=t where t ( = + 1 ) is an eigenvalue of the translation operator. Ifz~ is a root ofeq. (24), then t/z~ is also; thus from eq. (22) the independent values of the energy are E = 4-z],( 1 + l/t) - z ~ 2 ( 1 +t).

(25)

For f = 2 and t = + l , the energy levels are E2,3= 4+2 4 ~ and for t = - I there is a single energy level at Et = 4. We see that these levels coincide with those derived above. As n grows, however, the Bethe equations (23) become very complicated and the QISM appears to be less effective than the proposed approach in the numerical computations of the spectrum. A detailed comparison between these two methods is presently under investigation [ 11 ]. I wish to thank Professor A.C. Scott for interesting comments. Financial support from INFN sezione di Salerno is acknowledged.

(21)

In general forfdegrees of freedom the Hamiltonian matrix will have a band structure which will make the problem of computing the spectrum particularly 383

Volume 162, number 5

PHYSICS LETTERS A

References [ 1 ] L.D. Faddeev and L.A. Takhtajan, Hamiltonian methods in the theory of soliton (Springer, Berlin, 1987 ). [2] L.D. Faddeev and L.A. Takhtajan, Usp. Mat. Nauk 34 (1979) 13. [ 3 ] L.D. Faddeev, in: Les Houches Lectures, eds. J.B. Zuber and R. Stora (North-Holland, Amsterdam, 1984) pp. 719-756. [4] P.P. Kulish and E.K. Sklyanin, Lecture notes in physics, Vol. 151. Integrable quantum field theories, eds. J. Hietarinta and C. Montonen (Springer, Berlin, 1982) pp. 61-119. [ 5 ] A.C. Scott and J.C. Eilbeck, Phys. Lett. A 119 (1986) 60. [ 6 ] S. De Filippo, M. Fusco Girard and M. Salerno, Nonlinearity 2 (1989) 477.

384

24 February 1992

[7] M. Salerno and A.C. Scott, Quantum theories for two discrete nonlinear Schr6dinger equations, to appear in Nonlinearity. [8 ] V. Enorskii, M. Salerno, N.A. Kostov and A.C. Scott, Phys. Scr. 43 (1991) 229. [9] M.J. Ablowitz and J.F. Ladik, J. Math. Phys. 17 (1976) 1011. [ 10 ] V.S. Gerdjikov, M.I. Ivanov and P.P. Knlish, J. Math. Phys. 25 (1974) 25. [11]V. Enol'skii, M. Salerno, A.C. Scott and J.C. Eilbeek, Alternate quantizations for discrete nonlinear wave equations, submitted to Physica D.