Group theory approach to scattering

ANNALS

148, 346-380 (1983)

OF PHYSICS

Group Theory

Approach

to Scattering

Y. ALHASSID A. W. Wright

Nuclear

Structure

Laboratory,

Yale University,

New Haven,

Connecticut

06511

F. G~~RSEY J. W. Gibbs

Laboratory,

Yale University,

New Haven,

Connecticut

06511

AND F. IACHELLO A. W. Wright Nuclear Structure Laboratory, and Kernfysisch Versneller Instituut,

Yale University, Rijksuniversiteit

New Haven, Groningen,

Connecticut Nederland

06511.

Received January 4, 1983

We show that both bound and scattering states of a certain class of potentials are related to the unitary representations of certain groups. In this class, several potentials of practical interest, such as the Morse and Poschl-Teller potentials, are included. The fact that not only bound states but also scattering states are connected with group representations suggests that an algebraic treatment of scattering problems similar to that of bound state problems may be possible.

1. INTRODUCTION Recently, algebraic techniques based on group structures have proven to be useful in the description of bound state problems in a variety of fields. These include, among others, the rotation-vibration spectra of nuclei [ 1] and the rotation-vibration spectra of molecules 121. In all these applications, the groups used were compact, so that their unitary representations could reproduce the observed discrete and finite dimensional spectra. The algebras associated with the group were used to generate the spectra, while the representation matrices of the group could be used to calculate time-dependent excitations of the bound levels [3-51. However, many of these systemshave both discrete and continuous spectra. For example, diatomic molecules may dissociate, above a certain energy, in two parts. Or one may wish to study atom-atom scattering. In this article, we provide the first steps towards the extension of algebraic techniques to the description of scattering processes,by connecting the scattering states of certain potentials to the representations of some groups. At present, only one potential problem has been analyzed with group theory, both in its bound and scattering parts, the Coulomb problem [6]. Here one makes use of the groups SO(4) and SO(3, 1) in order to describe the degeneraciesof the bound 346 0003.4916/83 $7.50 Copyright All rights

tQ 1983 by Academic Press. Inc. of reproduction in any form reserved.

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[7-91 and scattering [ 10, 111 states. These groups are imbedded into the dynamical group SU(4,2) [ 121. It is important to note that the Coulomb problem is peculiar in having an infinite number of bound states, so that its dynamical group is noncompact. This is in contrast with most applications in physics, such as those mentioned above [ 1,2], where the number of bound states is finite and thus describable by a compact dynamical group. The results of our investigation are that, for problems whose bound state solutions are describable by a compact group G, the scattering solutions are simply obtained by an analytic continuation into the complex plane of some of the quantum numbers charaterizing the bound states in G. At the same time, the compact group G is analytically continued into a non-compact group G *. The scattering states are related to the unitary infinite-dimensional representations of G* [ 131. In order to keep the discussion simple, we concentrate our attention in this article to one-dimensional problems. The group structure of the corresponding bound state problem is that of SU(2). Scattering states are obtained by a proper analytic continuation of SU(2) to SU( 1, 1). The group SU(2) can be realized either on a twodimensional harmonic oscillator space or on a sphere. We show that the first realization is connected with the Morse potential [I4 1. This potential plays an important role in molecular physics and it has been the subject of several other group theoretical investigations [4, 151, although always confined to the bound state part of the spectrum. We also show that the second realization is connected with the Poschl-Teller potential [ 161. This potential emerges in a variety of problems in physics, such as the soliton solutions to the Korteweg-de Vries equation [ 171, the Hartree mean field equation of a many-body system with a d-function two-body interaction [ 181, the non-relativistic limit of the sine-Gordon equation [ 191, and in connection with completely integrable many-body one-dimensional systems [ 201. In addition to these two potentials, a larger class of one-dimensional potentials can be related to groups. This larger class of soluble potentials will be discussed in detail in another article [ 2 11. An interesting and somewhat simpler realization of the bound and scattering states is provided by the use of coherent states. We shall briefly review the properties of these states for SU(2) and SU(1, 1). Finally, the group describing bound states, SU(2), and that describing scattering states, SU(1, l), can be imbedded into a larger group that contains both. This group, Sp(4, R), contains still a third subgroup that we shall call the potential group, SU,( I, 1). The importance of Sp(4, R) is that it provides a unification of the various approaches to one-dimensional potential problems.

2. THE BOUND STATES A. Algebraic Approach A convenient way to construct a spectrum-generating algebra for systems with a finite number of bound states is by introducing a set of boson creation and

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annihilation operators. The number of boson operators needed in this construction, n, is related to the dimension of the space, r, by n = r + 1 [ 1,2]. Therefore, for onedimensional problems (r = 1) we shall introduce two boson operators, a and b. These operators obey the usual commutation relations [a, a+] = [b, b+] = 1;

[a, b+] = [b, a+] = [a, b] = 0.

(2-l)

The bilinear products utu, utb, btu, and btb generate the group U(2), the unitary group in two dimensions. It is possible to recast these four generators in a more familiar form by introducing the three generators of SU(2), J, = u+b, J- = b+u,

P-2)

J, = $(a+~ - b+b),

or, since A = (i) transforms as the spinor representation of SU(2), J, = A +$A,

i = x, y, 2.

(2.3)

Here J, = J, f iJy, and ui denote the Pauli matrices. Equations (2.2) or (2.3) are just the Schwinger representation [22] of SU(2). The fourth generator is the linear Casimir invariant of U(2), fi=u+u

+ b+b,

(2.4)

i.e., the total boson number operator. This operator commutes with all the others. The quadratic Casimir invariant of SU(2), C = Jz + Jf, + J:, is related to fi by c = $qfi

+ 2);

(2.5)

therefore, if we denote the eigenvalues of the operator C by (C) = j(j + l),

(2.6)

the irreducible representations 0, of SU(2) are characterized by a total boson number N = 2j.

(2.7)

We shall consider, in this article, systems characterized by a Hamiltonian, H, quadratic in the generators. Up to an additive function of the operator !?, the most general form of this Hamiltonian is H = a,J: + a,Jz + a,Jt.

This H commutes with fi, and thus it can be diagonalized

(2.8)

within the representation

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[N]. In general, the spectrum of H cannot be calculated in closed form. This happens only when H can be written in terms of Casimir invariants of a complete chain of groups [ 1,2]. For U(2) two possible chains exist,

(2.9)

\ U(l)

(II)*

However, since the algebras of O(2) and U(1) are isomorphic, these two chains are essentially identical. This is in contrast to problems in a higher number of dimensions (r = 3, r = 5) [ 1,2] when the various possible chains are no longer identical. An example of a dynamical symmetry corresponding to the group chain U(2) 3 O(2) is provided by the Hamiltonian H'"

=-a

The eigenstates of this Hamiltonian

J* Y Y'

ay

(2.10)

> 0.

can be denoted by IN, my), where

A(N,m,)=~Im,); (2.11)

The corresponding eigenvalues can obviously be written in the closed form Emy = -ay rn: .

(2.12)

The Hamiltonian H”’ thus has a finite number of bound states (2j + 1 = N + 1) and a quadratic spectrum. The eigenstates JN, my) can be easily written using boson operators, acting on a vacuum state 10). One first introduces the eigenstates of J, [22], IN, m,) = CN,,z(a+)N’*+m~(b+)N’*-m~l O), CN,,,,,=

[ (;+mz)!

(2.13)

(+)!]-“‘,

and then rotates them by 7r/2 around the x axis. The final result is (A&b+ IN,

my>

=

y*+.,

(

ia;+)“imi

CN.m,

where c,,,,~ is the same as in (2.13) but with m, replaced by my,

lo),

(2.14)

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B. Potential Approach

The algebraic approach of the preceding Section 2A can be connected with exactly soluble Schriidinger equations with certain potentials. This connection opens the way to an algebraic treatment of a large class of potentials of practical interest. In this section, we discuss the connection to one of these, the Morse potential. A second example, the Poschl-Teller potential, will be discussed later on in this article and other examples in Ref. [ 2 11. A simple connection between the abstract boson algebra and the potential approach can be obtained by realizing the boson creation and annihilation operators as differential operators in a two-dimensional harmonic oscillator space, a = (x’ + ip,,)/fi

= (x’ + &)/fi,

a+ = (x’ - ip,,)/\/z

= (xl --$/Jz,

b = (x” + ip,n)/fi

= p

b+ = (x” - @,,,)/fi

+&)/A

=

the differential operator corresponding to an operator B being defined by @v)(x) = (x PI w>, in the coordinate representation operators N and J,, become Iv:; Jy=-?

i

(2.16)

{x). In terms of the variables x’, x”, the two

x”+.r”‘-~-&2

3

x = (x’, x”),

v(x) = (x I vl>,

a2 )

xrgx’g

1

)

.

(2.17)

It is now convenient to make a change of variables (point transformation)

to

x’ = r cos 4,

O
Og#C27~,

(2.18)

x” = r sin f$. Then

la a fq=$(- --r--7y+r2 r f3r f?r i 13 Jy=-mT

la2 r a4

i

- 1, (2.19)

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The eigenvector equations (2.11) are solved by (2.20)

VI~,~,(~~ 4) = RN,m,,W ezimT where 2m, = integer, since v/ should be periodic satisfies 11 i ---r--f1 a 2 Br The final transformation

a i3r

in d with period 27r, and R,h,,,,,(r)

4m:, + rz RN+.(r) Yz i

= (N + 1) RN,&-).

(2.21)

is r’=(N+

(2.22)

l)em9,

which leads to

[--$+ (qq2 (ep2’- 2e-‘)] RN.,,,@) = -m.$,y,y@).

(2.23)

This is just the Schrodinger equation for a single particle in a one-dimensional Morse potential. Introducing the appropriate units, Eq. (2.23) can be rewritten as [ 141

I

h2 d2 2~ dx2 + D(e-2-,,lld

-

2e-“V,.d)]

ii2

=----rm;R(x); 24

Rtx)

(2.24)

thus the depth of the potential D is related to N by (2.25) The fact that the bound state solutions of the one-dimensional Morse potential are related to representations of the group SU(2) z SO(3) was already pointed out in Ref. 141. The quantity (2.25) is there called “reduced anharmonicity.” In conclusion, by imbedding the original one-dimensional problem into a twodimensional space, we have been able to connect the bound state eigenfunctions of the Schrodinger equation with a Morse potential (2.24) to the representations Dj (j = N/2) of SU(2). The energy eigenvalues are then given by the eigenvalues of the Hamiltonian (2.10). In general, this imbedding into a higher-dimensional space appears to be a general procedure to connect potential problems to abstract group structures.

595/148/2-l

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3. THE SCATTERING

STATES

A. Algebraic Approach

The bound state spectrum of the Hamiltonian (2.10) is given by Eq. (2.12), where my is an integer or half-integer, -N/2 ,< my < N/2. Equation (2.12) suggests that the continuous part of the spectrum can be obtained by analytically continuing my -+ ik,, in such a way that -a,m: + +a,k:, where k,, proportional to the momentum, should be continuous 0 < k, < CO. One possible way to continue the algebra of SU(2) would be by introducing the operators K; = iJ,, K; = iJ,,

(3.1)

K;=Jz.

These three operators generate the algebra of SU(1, 1) [23,24], [K’+,KI_]=-2K;, [K;,K;]

(3.2)

= rtK’,.

However, this continuation suffers from the fact that both KG and K: are not hermitian and thus cannot correspond to physical observables. A SU(1, 1) algebra more appropriate to the present problem can be obtained by using the same set of boson operators as in (2.1)-(2.2), and introducing the three operators [ 2.51 K + =atbt 1 K- = ab,

(3.3)

K, = i(ata t btb + 1).

The reader can easily check that the K’s satisfy SU(1, 1) commutation relations. Also, K: = K- , Kj = K,, and thus K,, K,, K,, are all explicitly hermitian. Equation (3.3) can be rewritten, in analogy with Eq. (2.3), as K.=B+A’B I 2

’

i = x, y, 2,

(3.4)

where B = (t+), A, and A, are the Pauli matrices A, = u, and 1, = uz and A, is the 2 x 2 unit matrix, Aj = I. The boson representations (2.3) and (3.4) can then be unified in the single formula Jo

ki(E)

= A +(E)2A(~),

i = x, y, 2,

(3.5)

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where E = +I and E = -1 are associated with SU(2)

and SU(1, l), respectively,

and

A(E) = ((J%)bI

(J$)bj3

(3.6) The scattering then

Hamiltonian

corresponding

to the bound state Hamiltonian

H(” = a Y K=Y’ with dynamical

a,, > 0,

(2.10) is

(3-7)

symmetry U(1, 1)x O(1, l),

(3.8)

which is the non-compact chain that arises from the analytic continuation of the bound state chain U(2) 3 O(2). In order to complete the algebra of U(1, 1) one needs a fourth operator. While in the compact case, this operator was the sum, #, of the boson number operators, Eq. (2.4) in the present case it is the operator &f=a+a-btb,

(3.9)

i.e., the difference of the number operators. The eigenvalues A4 of the operator A? characterize the irreducible representations of SU(1, 1). The quadratic Casimir invariant of SU( 1, l), C = Kz - Ki - K:, is related to it.? by c-g*+

l)(k-

1);

therefore, if we denote the eigenvalues of the operator

(3.10) C by

(C) = .G + 11,

(3.11)

M=-(2j+

(3.12)

we find that 1).

The unitary representations of the non-compact group SU(1, 1) are infinite dimensional [ 13 1. The generator K, is still compact and has a discrete spectrum in such a representation. However, the generator K, is non-compact and will have a continuous

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spectrum [26]. In our case, we need the simultaneous and K,, denoted by (M, ky),

eigenvalues of the operators ti

~l~,k,)=WW,),

(3.13)

K, IM, ky) = k, IM, &,). The expectation value of the Hamiltonian

Ek,= a&,

(3.7) in these states is then k, = continuous.

The eigenstates ]M, k,,) can be constructed using boson operators way. Consider first the eigenstates of the (compact) generator KL,

(3.14) in the following

Kz IM, k;) = kL IM, kJ,

(3.15)

k; = -j, -j + 1, -j + 2 ,... . We take A4 to be a non-negative integer (so that j < 0) and both the number of Q and b bosons are non-negative integers. This is known as the discrete representation 0.7 of SU(1, 1) [ 13, 271. The (unbounded) discrete states ]M, kL) can be written as

IM, k,) = CM,kl(at)(M-‘)/2+k,(b3-(M+1’/2+k: I()), (3.16) c M,k,=

-

[( ___

Some care is needed to go from (3.16) to the eigenstates of K, (the scattering states), since K, is non-compact. Under an SU(1, 1) rotation eieKXKz

e - i@K,

=

cash OK, + sinh BK, ;

thus the transformation of K, into K, cannot be accomplished (0 = real). However, for 8 = -h/2 [an SL(2, C) rotation]

(3.17) within

RK;R -’ = -iK P with R = exp((rc/2)K,) hermitian. In order to find the transformation (bat) is an SU(1, 1) doublet

e iOK,

properties

of the boson operators

SU(1, 1) (3.18)

we note that

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For 8 = i(7r/2) we find the transformed operators (;)=-&-(; Similarly.

(3.20)

i)(y)+).

since (a, -bt) transforms with the inverse matrix, we find

(6 -b’) = (a,-b+) (Ii ;).

(3.21)

Since commutation relations are conserved by a similarity and g are new boson operators with a new vacuum state @)=R

(0);

transformation

iIa)=sla)=o.

Since the states R IM, k;) are the eigenstates of -iK,, obtained from (3.16) with the substitution kz + - ik,, IM, k,) = c,,,.k8(~+)‘“~l)“~‘ky(~~-(dft

(3.18), a’ (3.22)

the eigenstates of K, can be l)/z-ikv 10).

(3.23)

In terms of the boson operators a and b, Eq. (3.23) can be rewritten as (a:

)(M-ll/2-iki

(jaAbtj

-Wtll/2-ikv

IM, k,) = c.w,k,, x

&d4)(a+b++ob)

,o>.

(3.24)

Comparing (2.14) with (3.24) one can see that the algebraic approach can be simply extended from bound states to scattering states by including complex powers. The scattering states (3.24) can be expanded into eigenstates IM, k,) of K, 1231, IM, kJ =;

4JM

ky) IM, k,),

(3.25)

where the coefficients A,: are simply obtained from (3.24). However, we shall see that, with the use of the coherent state representation, Section 4, this expansion can be accomplished in a simpler way. B. Potential Approach

In the same way in which the representations of SU(2) are connected with the bound state solutions of the one-dimensional Schrddinger equation with a Morse potential, the representations of SU(1, 1) are connected with the scattering solutions in the same potential. In order to show this explicitly, we perform the transformation (2.15) and write the operators A and K, in terms of the variables x’, x”,

(3.26)

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In contrast with (2.18) we now introduce hyperbolic coordinates x’ = T cash 4, O
x” = r sinh 4.

O
(3.27)

In terms of the new variables, the operators A and K, are

i

(3.28)

a

Ky=~~* The solutions of the equations (3.13) can be written in the form (3.29)

where RM,k,(r) satisfies 1 z

---

The final transformation

1a

a 4k2 r - - -2 + r2 RM,k,(r) = MRMsk,(r). r Br ar r2

(3.30)

is r2

= Mepp

(3.3 1)

9

which leads to [- -$

+ (T)’

(ew2’ - 2f?:P)] RM,ky@)=

kf,RM,k.@).

(3.32)

This is again the Schrodinger equation for a simple particle in a one-dimensional Morse potential. However, Eq. (3.32) now describes the scattering solutions (k: > 0) rather than the bound state solutions. Comparing Eq. (3.32) with (2.23) one can see that, in order that both equations arise from the same potential, one must have M=N+

1.

(3.33)

Note also that k, is a continuous variable, since 0, being a hyperbolic angle, is not periodic (with real period), in contrast with 0 in Eq. (2.18), which is periodic. As a result, no periodic boundary condition must be imposed on the wave function, and k, can be any real number. Finally, we note that, as already pointed out in the bound state case, imbedding the one-dimensional scattering problem into a two-dimensional space (introducing the redundant variable 4) has enabled us to link the scattering problem to an algebraic structure [SU(l, l)]. The idea used here is similar to the so-called projection method

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[ 28-3 1 ] in which a complicated equation of motion can be projected from a simpler equation in a higher-dimensional space. Before discussing other properties of the algebraic structures introduced above, it is convenient to recall the explicit solutions of the Schrodinger equation with a Morse potential. The substitution R(r)

= (ry

(3.34)

2 = r2,

e -(1’2)rZF(r2),

transforms Eq. (2.21) into the Laplace equation (see Ref. 132. p. 5.511)

[

z$

+ (c - z) f

- a F(z) = 0, I

(3.35)

with

.=,,-N.

2’

c=2m,+

1.

(3.36)

The bound state solutions of the Schrodinger equation with a Morse potential can be written then in terms of the confluent hypergeometric functions RN,Jr)

= C(rZ)my e -(‘/2)r2F m, -N

2’

2m, + 1; r2 ;m,>O,

(3.37)

where r2 is given by Eq. (2.22). The boundary conditions RN,,,@ + f co) = 0 are satisfied by Eq. (3.37), since, according to Eq. (2.22), p + co corresponds to r -+ 0 and p + -co corresponds to r + co. The constant C is determined by imposing the normalization condition. For m,, < 0, a similar expression holds with m, replaced by -my.

The scattering solutions hypergeometric functions

can also be written

in terms

of the confluent

RMM.ky(r)= A(r2)iky e- (1’2’r2F ik, - $ + +, 2ik, + 1; r2 + B(r2)-iky ep (“2)r2F

-ik, - F + +, -2ik, + 1; r2 ,

(3.38)

where we have used M = N + 1 instead of N in order to conform with Eq. (3.30). The ratio A/B can be obtained by imposing the boundary condition RM,ky@+ -co) = 0, since at p + -co the Morse potential goes to infinity. This gives

B = -A

(3.39)

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One can easily check that as p--t +co, the solution (3.38) behaves as a combination of ingoing and outgoing waves Rw.k,@ + +a)

=AMiky

e-ibp

+ BM-‘ks

eibpe

(3.40)

The reflection amplitude is T(2ik,+

l)T

R = M-2&

(3.41)

r(-2ik,

+ l)r

The transmission amplitude vanishes since the potential goes to infinity as p + -co,

T= 0.

4. COHERENT

(3.42)

STATE APPROACH

A. Bound States In addition to the algebraic and potential approach, another, alternative, approach is possible in terms of coherent states. In this section, we shall review the coherent state approach to SU(2) [33] and SU(1, 1) [34]. For bound states, we need the coherent states of SU(2). These can be introduced as the most general state that can be obtained from the state /j, mz = -j) under a group operation. In this section we shall use the notation 1j, m) for the states of Section 2, with j= N/2. A general rotation will rotate the state 1j, -j) into the direction 0 G (0,$). It is convenient to introduce the parametrization In> = u(n) Ij, -j>, u(l) = exp(ti+ - L*J-) = exp(ia+b - L*b+a),

(4.1)

For any fi on the sphere, Eq. (4.1) defines a coherent state IL) [35]. Using the Baker-Campbell-Hausdorff relation one can rewrite the exponential in (4.1) as exp(ti+ -L*L)=exp(zJ+)exp[(ln(l

+ ]z12)JZ)] exp(-z*L),

(4.2)

where

e z=e-imtgT

(4.3)

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is the projection of Q from the sphere onto the complex z-plane. An alternative definition of a coherent state follows from (4.2) [35], lz)=(l

+Izl’)-jexp(d+)Ij,-j).

(4.4)

The state Iz) is normalized (z I z) = 1,

(4.5)

since u(d) in Eq. (4.1) is unitary. The unnormalized

coherent state Iz) is defined by

lz) = ewW+I lj, -I?.

(4.6)

The main advantage of using Iz) is its analyticity in z. Yet another definition of Iz) that makes apparent its connection with the algebraic approach (see Appendix A) is Iz)=

,t2i)ll,,:*

(za+ t b+)‘j IO).

The coherent state Iz) can therefore be visualized as a boson condensate. The coherent states form an overcomplete set, in the representation j, (2’4z ‘) jsinBdBd$

IA)@ = (2’T

‘) 1 (1 +“;:lI)Z

IZXZI

= (Y t 1) d*z Iz>(zl =I, 71 1 (1 + lZI*)*j+2

(4.8)

where (4.9)

m:= -j is the identity operator in the space of the representation described by an analytic function f(z) defined as

j. Any state If)

f(z) = cz* If)* Using the completeness relation, (4.8), it follows the coherent state representation

that f(z)

can be

(4.10) is just the wave function in

If) = j 4+)f(z) l z>q

(4.11)

with measure

&(z) = (2j + ‘) (1 + /~/*)-~j-* 7c

d*z.

(4.12)

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The scalar product of two such functions is

u-1g>= I W) (fIz*>(z* I g> d2z = (V-t 1) i (1 t Iz/2)2j+* f*(z) i-t

g(z).

(4.13)

A coherent state Iz) can be expanded in eigenstates I j, m,) of J,, 2j

lz)=exp(zJ+)I.L-.)=

n

1 >J;

Ij,-j)

n=0

l/2 (z)"z+j

lj,

m,),

(4.14)

so that

fj,&)=(z*lj,m,> =(j :jmI)I’*(zP+j

(4.15)

are the eigenfunctions of J, in the z-representation. The bound state wave functions of Section 2 are eigenfunctions of J,,. To find their z-representation, we note that, under an SU(2) transformation U, f(z) transforms as [24]

(4.16)

from which we can deduce the differential representation, Jx=jz+2-, Jy=-‘jr+iTdr,

1-z’ ltz*

form of the generators

in the z-

a’ dz d

(4.17)

J,=-jtz$. The equation (4.18)

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361

is then solved by =

hj.,y(Z)

1+

Cj,,,(

1-

iZ)j-“‘(

to be compared with the boson representation B. Scattering

iZ)j+

my,

(4.19)

(2.14).

States

For the scattering states of Section 3 we need the coherent states of SU(1, 1). The coherent states of SU(2) are defined by points on the sphere xf + xi + x: = 1. In the analytic continuation to SU(1, 1) the sphere is transformed into the hyperboloid -xi - xg + xi = 1. On the hyperboloid we can introduce quasi-polar coordinates Q = (0, 4) by x, = sinh 19cos 4, x2 = sinh 0 sin 4, x3 = cash 0,

(4.20) o
03,

O,
where 0 is now unbounded. For any 0 on the hyperboloid, for the representation D,? ,

we define a coherent state,

j A> = u(J) I j, -j>, u(l) = exp(X+ A=

e-i@

-

- A*K_)

= exp(Aa+b+ - A*ab),

(4.21)

8

2’ Here ]j, -j)

is the state with the lowest eigenvalue of K, in the representation (4.22)

and we shall use the notation ]j, k) for the states of Section 3 withj= - (M + 1)/2. The hyperboloid can be projected onto the interior of the unit circle by the transformation 9 z = e-‘@ tgh -, (4.23) IZIG 1. 2 Using the Baker-Campbell-Hausdorff

relation, we find

Iz) = (1 - lzI’)-jexp(zK+)

Ij, ;.).

(4.24)

which is the normalized coherent state of SU(1, 1). The coherent states (4.24) are just the Gelfand-Naimark [36] z-basis, that were shown to be eigenstates of a Z-operator [37]. The unnormalized, but analytic, ]z) is I z> = exp(zK+ > I .L 3).

(4.25)

362

ALHASSID,

GtiRSEY,

The coherent states form an overcomplete (‘j4:

AND

IACHELLO

basis for SU( 1, 1) too (Appendix A),

‘) [sinhBdBdqS(i)(/Zi

= (zi+ 1) 71 i 121<1 (1 -d2z IZI’)’ IZXZI = Pj+ 1) ?t

i ,2,~1 (1

d2z _ Iz12)2j+2 lz)(zl =I,

(4.26)

where I=

S k:=

Any state If) IZIG 1,

(4.27)

Ij,kz)(kz,jl. -j

can be described now by an analytic function f(z)

in the unit circle, (4.28)

f(z) = cz* If>. The expansion in the z-basis is the same as in (4.1 l),

lf)=l;-,<,L , &(z)./-(z) lz),

(4.29)

dp(z) = (2j + 7c ‘) (1 - (z]2)-2jP2 d’z.

(4.30)

but now the measure is

The scalar product of two analytic functions

=W+ 1).r 71

is

d2z

l-l<1 \ (1 - Iz/2)2j+2

f*(z)

g(z)*

(4.3 1)

This is just Bargmann’s representation [27]. Comparing with the compact case, SU(2), one can see that the only difference is the change in sign in the measure and the restriction of the complex plane to the unit circle. Also here j < O! The unnormalized state Iz) can be expanded in eigenstates 1j, kr) of K, , ]z)=exp(zK+)]j,-j)= =r

+m k:y-

k,j

j-

k

+.i

5
(4.32)

GROUP

THEORY

APPROACH

TO

SCATTERING

363

Therefore, the eigenstates of K; are given by

(4.33) k, = -j,

-j

+ l,... .

The scattering states of Section 3 are eigenstatesof the operator K,. To find their zrepresentation, we note that, under an SU(1, 1) transformation u, the function f(z) transforms as [24 ]

(4.34)

from which we can deduce the differential form of the generators

K.y=-jz+Zdr,

l+z’

d

1-z’

d

K,.=ijz+iTZ,

K,=-j+zg.

(4.35)

z

Incidentally, we note that K, + iK, = i(d/dz) is the translation operator, and that the relation K, - iK, = z(K, -j) leads to a definition of a g-operator [37]. whose eigenvalues are z. .g=

K, - iK, K,-j

’

(4.36)

The canonically conjugate operator is fi=

K,, $ iK,.

(4.37)

Using (4.35), we can find the eigenstates of K: and K,. Those of K, are homogeneouspolynomials of degreej + kZ in z, Eq. (4.33). Since these eigenfunctions must be analytic in the unit circle, j + k, must be a non-negative integer. Thus we recover the results that kL is discrete. The eigenfunctions of K, can be obtained by solving the equation ijr+i---------

1-z’ 2

d dz

hj.k,. = kyh.i,k,..

(4.38)

364

ALHASSID,GtiRSEY,AND

IACHELLO

and are given by hj,k,(Z)

=

Cj.k,(

1-

Zr’-iky(

1+

ZY’+

ik’.

Note that hj,k, is analytic for any real k,, so that k, has indeed a continuous spectrum. By comparing Eqs. (4.19) and (4.39) one can see that the scattering states are obtained from the bound states in the z-representation by the substitution z + iz, my + ik,. It is also interesting to compare the coherent state representation (4.39) with the boson representation (3.24). Here the coherent state representaton offers a significant simplification, since the boson creation and annihilation operators in (3.24) are not in normal order. Thus, for example, the computation of the expansion coefficients A,JM, k,,) in (3.25) is much simpler in the coherent state representation than in the boson representation. Finally, one may consider another representation of the coherent states (wrepresentation) by performing the fractional linear transformation (Poincare transformation)

w-i z=------w+i’ This transformation transforms the upper half-plane onto the interior of the unit circle and, incidentally, carries SU( 1, 1) into SL(2, R). The scattering wave function&,,r(z) is transformed into a function x,m,(w) defined for w in the upper half plane,

jj,,(w)

= Fj,k, d’ iky.

In this representation, the scattering wave functions are obtained from the eigenfunctions of J, in the z-representation, by the substitution z + w, m, -+ iky . The metric here is (4.42) The different parametrizations of the coherent states used in this section are related to different parametrizations of the hyperboloid and are discussed in greater detail in Appendix B.

5. OTHER ALGEBRAIC

APPROACHES

In Sections 2 and 3 we have discussed an algebraic approach to SU(2) and SU(1, 1) based on the Schwinger representation with two boson operators, a and b. Still other boson representations are possible. In this section we discuss another of these, a generalized Gelfand-Dyson representation, and find its connection with the Schwinger representation.

GROUP

THEORY

APPROACH

In the Gelfand-Dyson representation, terms of one boson operator, a, as [38]

TO

SCATTERING

365

the generators of SU(2) are expressed in

J:=+(l

-c12)+ja,

J:=-tc+(l

+a’)+&&

(5.1)

Ji = ata -j.

The form (5.1) contains explicitly the constantj that characterizes the representations of SU(2). However, it is possible to make Eq. (5.1) independent of j by introducing a second boson operator, 6, such that the number of b bosons characterizes the representations of SU(2). Substituting $btb forj in Eq. (5.1) gives J;=;d(l

--a2)+;btba,

J;=-+a+(1

+a’)+;b+ba,

(5.2)

J; = ata - + btb.

Equation (5.2) will be called a generalized Gelfand-Dyson representation. The difference between the Schwinger and the Gelfand-Dyson representations is that while in the former the total number of u and b bosons characterizes the representations of SU(2), in the latter it is only the number of b bosons that characterizes the representations. The connection between the two representations is given by a similarity transformation induced by the metric G = b“ tu. (5.3) Since c1and b commute, G is well defined by its operation on a complete set of states In,, n,), where n, and nb are the numbers of a and b bosons

Under G, the creation and annihilation

operators transform according to

Gp’aG=ba, G-katG -_

b-lat

3

(5.5)

G-‘bG=b, G-‘btG

= -@+a) b-’

+ bt.

366

ALHASSID,

The transformation and using

GtiRSEY,

AND

IACHELLO

of a and a’ is easily proven by writing e-4ataae4ata ,-oota,t

pta

b = es, with

= ,o, 1 = e-4a+

(5.6)

The transformation of b follows trivially from its commutativity with non-trivial relation is the last one in (5.5). To prove it, we note that b-“b+b”

= -nb-’

=Ji',

G. The only

+ b+

for any eigenvalue n of at,. Using the transformation G-'JiG

I/?, a] = 0,

properties

(5.7) (5.5) we find that

i = x, y, z,

(5.8)

G-‘liTG = b+b, where J; and Ji are given by Eqs. (5.1) and (2.2) respectively. Equation (5.8) establishes the connection between the Schwinger and Gelfand-Dyson representation. We note that while the Schwinger representation is manifestly unitary (Ji hermitian), the Gelfand-Dyson representation is unitary only with respect to the metric G. Similarly, it is possible to introduce a generalized Gelfand-Dyson representation of SU(1, 1). The explicit form of the generators in the representation D,: of SU(1. 1) is K:+(a’+

1)-j&

K;,=++(o’-

1)-ija,

(5.9)

K;=a+a-j. If we transform

the Schwinger

representation

(3.3) with the metric

I- = (b+)“+“,

(5.10)

we obtain r- ‘KiJ = K!

i = x, y, z,

(5.11)

r- ‘&jr = -atb. Here K, and A? are given by (3.3) and (3.6), respectively, while the transformed define a generalized Gelfand-Dyson representation of SU( 1, 1),

Kl

(5.12)

GROUPTHEORY

APPROACHTO

SCATTERING

361

The representation 0: is now characterized by a fixed number, -2j - 1, of b bosom (j < O!), as is obvious from the second equation in (5.11). Once again the primed representation (5.12) is unitary only if one usesthe non-trivial metric operator K

6. OTHER POTENTIAL

APPROACHES

A. Bound States A realization of the algebra of SU(2) and SU( 1, 1) on a two-dimensional harmonic oscillator space has led to a connection between these algebras and the solutions of the Schrodinger equation with a Morse potential. This is only one in a large number of ways in which abstract algebras can be connected to solutions of Schrodinger equations with certain potentials. In this section, we discussanother possibleway that leads to another potential of considerable practical interest, the Pbschl-Teller potential [ 16] V@)=-

P’

(6.1)

cash’ Such a potential has a finite number of bound states and a continuum of scattering states. The bound states of the Poschl-Teller potential are obtained from the SU(2) algebra by using its familiar realization on the sphere [39 1,

(6.2)

where we have used Z,, I,, I2 instead of Jz, J, , fi in order to distinguish this case from the previous one. In analogy with (2.9) we now look for dynamic symmetries characterized by the group chain U(2) 1 O(2). Rather than the operator J,, it is more convenient here to diagonalize the operator I;. We thus look for the simultaneous eigenfunctions of I2 and Zz, I’xT = j(j + 1) xj”,

(6.3)

to be compared with (2.11). The solutions of (6.3) are of the form x7(19,4) = uy(t9) e’“@,

595/148/2-E

(6.4)

368

ALHASSID,

GtiRSEY,

AND

IACHELLO

where u,?(6) satisfies the familiar equation

[--i&j& (sinO$)+&]u:(8)=j(j+

l)uj”(O).

The solutions of (6.5) are the associated Legendre functions PJ” (cos 19), and the functions xj”(O, 4) are just the spherical harmonics Y,,(B, (s). The substitution cos 8 = tgh p,

--oo
(6.6)

brings Eq. (6.5) to the form

-$-f;s;2b’ [

I qp) = -m2qp).

(6.7)

This is again a dimensionless Schrodinger equation with the Poschl-Teller potential (6.1). We note that, as in the case discussed in Section 2, the strength of the potential is characterized by the invariant of the group j (or the boson number N = 2j in the algebraic approach). Furthermore, the spectrum is given by E, = -am2,

(6.8)

where we have inserted the scale factor a in order to provide the appropriate dimensions. The corresponding Hamiltonian is thus of the type H=

-al:.

(6.9)

We remark that the functions Py(cos 8) satisfy the orthogonality (j+m)!

n Pj”*(cos 8) Pi” ‘(cos 0) sin 8 do = (j - m)! (ja-

i 0

so that the normalized

1

relations (6.10)

eigenfunctions are

These functions satisfy the appropriate boundary conditions qp+

fco)=O,

m f 0.

(6.12)

Since both the Morse and the Poschl-Teller potential appear to be related to representations of SU(2), it is clear that there must be a transformation connecting them. Since this transformation is rather involved and it does not appear to be known, we mention it briefly here. More details are given in Appendix B. We start from the equation (6.13) Ii+ = NW,

GROUP

THEORY

APPROACH

TO

369

SCATTERING

where fi is given by Eq. (2.19). We introduce two new variables t,, t,, through t, = t cos CD, t = fr2, and canonically conjugate two-dimensional Eq. (6.13) can be rewritten as f(l + r2)y=

t, = t sin @,

(6.14)

@ = 24, momenta, z’= (rxr rY). In terms of these,

(Nf

1)1//.

6.15)

We now use the Hylleraas method 1401, developed in the context of the Coulomb problem, that is, we square Eq. (6.15) to obtain lt2(l + r’) - 2is’. ?(l + r2) + 3(1 + r2)]y/ = (N + 1)‘~. Next, we make a canonical transformation,

exhanging coordinates

(6.16)

and momenta (6.17)

This transformation amounts Introducing the wave function

to taking

a two-dimensional

Fourier

transform.

x = (1 + t’*)3’*ly’, where v” is the Fourier transform

-i

s$I'gl+~-

(6.18)

of t,u, we obtain

a2 aV 1 x=

Finally, projecting back the two-dimensional t’ = tg(O/2), @’ = 4, we find

N(N + 2) (,/2

+

plane into

- [&$j (sinegj)+$&]x=n+

1)2X.

the unit sphere

1)x

using

(6.20)

with j = N/2. This is the same as Eq. (6.5). The connection between the solutions of the Schriidinger equation with a Morse and Poschl-Teller potential has an interesting application. Using (6.17) and (6.18) we can write that x(t’, @‘) = (1 + t’2)3’2 j;^ d@ jam t dt ,-ir”cos(@‘-s)

,,/(t, @).

(6.21)

Since I&, @) = R?(t) eim*, X(t’, W) = uj”(t’) eimd’,

(6.22)

370

ALHASSID,

we find, after the CDintegration,

GtiRSEY,

AND

IACHELLO

that

uj”(t’) = (1 + t’ 2)3’2 J--mt dt R;(t)

J,(tt’)(2n(-i)“].

(6.23)

Bessel function of order m. But u,y(t’) is just the where J, is the cylindrical associated Legendre function Py(cos 0) with t’ = tg0/2 and Rjm(t) is given by (3.37). Thus P~~(COS

e)

dtt*e-‘F(m-j,2m+

CC

l,t)J,

(6.24)

or Pj”(cos e) cc

1 cos3

e

cC dt t* e-‘L;?‘,,,(t) J, t tg 4 , c -0 ( )

(6.25)

(T 1

which provides an interesting connection between a Laguerre, L,;!‘,,,. and a Legendre. Pj”, function.

B. Scattering States The scattering states of the Poschl-Teller potential can be connected to an algebraic structure in the same way as those of the Morse potential, by analytically continuing SU(2) to SU(1, 1). One possible way to continue the algebra of SU(2) tc SU( 1, 1) would be to introduce the operators F; = il,, F-l,= Ir ,

(6.26)

F; = iI,. However, in this continuation, both F-k and F; would not be hermitian. In order to keep them hermitian, it is more convenient to realize the algebra of SU(1, 1) on the hyperboloid, x2 - 4” + z2 = 1, rather than on the sphere, x2 + JJ*+ z2 = 1. In terms of the hyperbolic coordinates 0 < 0 < rt and 0 < 4 < co, defined by x = sin 0 cash 4, y = sin 0 sinh 4, z = cos 0, the three generators of SU( 1, 1) can be written as

(6.27) F,=e*O

GROUP

while its Casimir

THEORY

invariant

APPROACH

can be written

F*=-

[&g

TO

371

SCATTERING

as (6.28)

(sine&)-&$].

Here again we have preferred to use F=, F, . F* instead of K:, K, , fi in order to distinguish this case from the previous one. The scattering states of the Poschl-Teller potential are then the simultaneous eigenstates of F* and F,, F*x; = j(j + 1) x,;, F:xj”=

(6.29)

kxjk.

They are of the form

x@,$4) = u’(B)eik@

(6.30)

with k continuous and real (since FZ is non-compact). Using the same substitution before, Eq. (6.6), we find that the function uf@) satisfies the equation d* -2-

4

.iU + 1) uj”@) = k*u.;@). cash’ p I

as

(6.31)

This is the Schriidinger equation for the scattering states in a Poschi-Teller potential. Furthermore, by repeating the same arguments leading to Eq. (6.20), we find that j=-(M+ 1)/2. The scattering solutions UT@) can be obtained from the bound state solutions by an analytic continuation in the quantum number m, i.e., m 4 ik. Using Eq. (6.11) one finds that u;(p) = c;Py(tgh

(6.32)

p).

For j = 0 (no potential), the solution is a plane wave, eikp. In the variable z = tgh p, a plane wave has the form ikp _ e ---

1 + z i(k’2’ ( l-z

1

(6.33)

*

In order to find the transmission and reflection amplitudes, which form the S-matrix for one-dimensional problems, we have to expand asymptotically the group functions for a given j in terms of the functions for j = 0 (plane waves). For this purpose, we write the associated Legendre functions as hypergeometric functions 1321,

Pi”(z)=

r(l

l

-m)

[z)m’2F

c-j, j-t

1, 1 -m;G].

(6.34)

372

ALHASSID,

After analytic continuation

u;(z) = ci”

AND

IACHELLO

we obtain

’ . r(1 - Ik)

In the limit p -+co,z+l

GijRSEY,

(s)ik’2F

(-j,j+

1, 1 -ik;-$).

(6.35)

and u;(z) - I+1 c;

(6.36)

so that on the far right-hand side U;(Z) is a plane wave etikp. To investigate the limit z-t -1, we transform the hypergeometric function in (6.34) using P-+-00, Eq. (5.2.49) of Ref. 1321. This gives

Pyyz)=(5L)“:’ XF

[

-j,j+

=(-m)

r(l-?n+j)z-(--m-j) l,m+

l;---

l+z 2

m> + I-(-j)r(j+

1)

I-mm+,--m-j,++

1;2

1+z

. )I

After analytic continuation,

(6.37)

and taking the limit z + -1, we find ih/2

(-)iW

-i/d2

T(-ik) r( 1 - ik + j) I’(-ik

- j)

T(ik) t-Yk’*

F(-j)

F(j + 1) I ’

i.e., a combination of transmitted wave, e+ikO, and a reflected wave, reflection and transmission coefficients can be simply obtained as

(6.38)

ePik”. The

R = T(ik) T(-ik -j) T(-ik + j + 1) ’ T(ik)T(-j)T(j + 1)

(6.39)

.=T(l--ik+j)T(-ik-j) r( 1 - ik) T(-ik)

(6.40)

’

These expressions can be rewritten in terms of the quantum number M, by noting that j= -(M + 1)/2. In addition, Eqs. (6.39) and (6.40) can be converted to the

GROUPTHEORYAPPROACHTOSCATTERING

373

appropriate units be introducing a scale of length, d, thus changing the potential (6.7) to (6.1) (6.41)

One obtains the same expressions(6.39), (6.40) but with ik replaced by ikd.

7. THE UNIFIED APPROACH

AND Sp(4,R)

In the previous sections the bound states and scattering states of certain potentials were treated as basis for the representations of different groups, connected to each other by analytic continuity. Using the generators of each group, one can have transitions either from one bound state to another, or from a scattering state to another. However, none of the above generators can connect a bound to a scattering state. In order to calculate transitions from bound states to the continuum (for example, molecular dissociation), we need a larger group that includes both the bound and the scattering group as subgroups. The simplest way to construct this group is by using the algebraic boson approach discussed in Sections 2 and 3. The most general quadratic form in the boson operators a and b is formed by the 10 operators ata, atb, bta, btb; atat, aa, btbt, bb; atbt, ab.

(7.1)

One can show that these 10 operators close under the symplectic algebra Sp(4, R). The group Sp(4, R) contains as subgroups both the bound state group SU(2) and its invariant i? [the first four operators in (7.1)], and the scattering state group SU(1, 1) and its invariant &? [given by Eqs. (3.3) and (3.9)]. The group Sp(4, R) thus provides a unified framework within which both bound and scattering states can be discussed. In addition, we note that the group Sp(4, R) contains another interesting SU( 1, 1) group, generated by the three operators L, = i(atat + btbt), L- = f(aa + bb),

(7.2)

Lz = I(ata + btb + 1). To find the significance of this SU( 1, 1) algebra, we compute its Casimir invariant L2 = a(ata + btb + 2atabtb - 1) - a(atatbb + btbtaa) =+a,

(7.3)

where J,, is given by (2.2). Thus the Casimir invariant of the above SU(1, 1) is just the dimensionlessHamiltonian (2. lo), i.e., ff”‘=-r--l

4’

(7.4)

374

ALHASSID,

GtiRSEY,

AND

IACHELLO

On the other hand, the operator L, is related to the number operator N, the invariant of SU(2), by L /+l -. z

(7.5)

2

Thus we see that the energy, determined by H(I), and the strength of the potential, determined by 13, exchange roles when going from the SU(2) group of Section 2 to the SU(1, 1) group of this section. In the representation D,f of the present SU(1, I), we have a basis In, Z) satisfying

H”’ /n, 1)= -(Z + 4)’ 1n, I), (7.6) LzIn,l)=

i

q

1

In, 0.

Within this representation (I fixed), one steps between states corresponding to potentials with different strengths, n, but all having the same energy, -(I + +)‘. Thus, for a given 1, the weakest potential for which a bound state with energy -(I + i)’ appears is characterized by n = -2Z1 bosons. Then, using L + , one can step to deeper and/or wider potentials with n = -2Z+ 1, -2Zf 3,... bosons, all having the same bound state energy. The present SU(1, 1) group can thus be called “the potential group.” The group Sp(4, R) contains as subgroups the bound state group, the scattering state group, and the potential group. Its generators can connect all states (bound and unbound) in the same potential, and all states with the same energy but corresponding to potentials with different strengths. It thus provides a unified approach to one-dimensional problems.

7.

CONCLUSIONS

Many problems in physics are characterized by a spectrum with a discrete, finite dimensional part and a continuous part. Recently, algebraic techniques have been used in a variety of problems [ 1, 21 to describe the discrete part. In this article, we have presented the first steps towards an extension of the algebraic formulation to cover the continuous part of the spectrum. The algebraic approach to the discrete, finite dimensional part of the spectrum consists in writing the Hamiltonian, H (and other operators), in terms of the generators of a compact algebra, G. We find that, in order to describe the scattering part of the spectrum, we must analytically continue the compact algebra, G, to a non-compact algebra, G*. When the Hamiltonian, H, is written in terms of all the generators of G, the eigenvalue problem Hy = Ey/ must be solved numerically in the basis provided by the irreducible representations of G. However, when H can be written in terms only of Casimir invariants of a chain of groups G 113G’ 2 ..., a dynamic symmetry arises and the eigenvalue problem can be

GROUP

THEORY

APPROACH

TO SCATTERING

375

solved in closed form, giving rise to energy formulas. Similarly, we find that, when H can be written, after analytic continuation, in terms only of Casimir invariants of a chain of groups G*IG’* 2 . . . . one can obtain the S-matrix describing the scattering preocess in closed form. In this article, we have extensively discussed one example of this continuation procedure, that in which G = U(2) and G* = U(1, 1). Furthermore, we have shown that, when the algebras G and G* are realized in terms of differential operators, dynamic symmetries of H lead to exactly soluble Schrodinger equations with certain potentials. In the case of U((2) and U( 1, 1), the class of exactly soluble Schrodinger equations includes those with the onedimensional Morse potential [ 141 and the one-dimensional Poschl-Teller potential [ 161. The former is extensively used in molecular physics, while the latter appears in a variety of problems in physics. The more general case in which H contains all the generators of G or G* leads to potentials that are Weierstrass functions [41]. These will be discussed in a forthcoming publication [ 2 11. The next step in the extension of algebraic techniques to include scattering is to study the case G E U(4), G* E U(3, 1). The group U(4) appears to provide a good description of the spectra of diatomic molecules [ 2 ] and it is closely connected with the Morse potential in three dimensions [42]. We expect that dynamic symmetries of H will provide in this case three-dimensional S-matrices that can be written in closed form. At present, only one exactly soluble case is known in three dimensions, that of the Coulomb problem 161. This problem is peculiar in having an infinite number of bound states. Our approach will provide other examples that are more appropriate to discuss systems with a finite number of bound states. Finally, we have found that the algebra describing the bound states, G, and that describing the scattering states, G*, can be imbedded into a larger algebra, G, that contains both. This algebra, Sp(4, R) for one-dimensional problems, provides a unified approach to both parts of the spectrum. The algebras G and G* can be realized in several ways, For practical applications, a convenient realization is in terms of boson creation and annihilation operators [1,2]. The realization in terms of differential operators leads to the usual Schrodinger formulation. In this article, we have also discussed a third realization, in terms of coherent states on a coset space (z-representation). This realization offers several advantages, especially if one wishes to calculate transitions between states 143 I.

APPENDIX

A

Here we add some details to the discussion of Section 4. Since the state 1j, -j) is left invariant by the subgroup of SU(2) generated by J;, the group has the decomposition SUP) where SU(2)/U(l)

= [~~(2)/~(1)10

U(l),

is the coset space (351 described

(A.1)

by the sphere. The coherent

316

ALHASSID,

GijRSEY,

AND

IACHELLO

states of Eq. (4.4) correspond to this coset space. For some purposes, it may be convenient to use a decomposition of the transformations of SU(2) according to u(z,u)=

( -z*l

&i

;)i’gl’

,-Fl.,).

64.2)

Using Eq. (A.2) one can recover Eq. (4.16). Similarly, the coherent states of SU(1, 1) are defined on the coset space SU(1, 1)/U(l), w h ere U(1) is the subgroup generated by the compact generator KZ. This coset space is a hyperboloid. Each element of SU( 1, 1) can be decomposed according to

4z,u)=&

(LI* f )(‘jo/:,-Vu,*).

(‘4.3)

Finally, the completeness relation (4.26) follows from the expansion (4.32) and the orthogonality condition (1 - (z~2)-zj-‘(z*)k~+j(~)k~+jd2z

APPENDIX

= 6,;,.

(A.4)

B

In this appendix, we summarize the different parametrizations of the hyperboloid used in Section 4B to define the coherent states of SU( 1, 1). The hyperboloid -x;-x;+x;=R*

03.1)

is left invariant under 0(2, 1) transformations, just like the sphere is left invariant by O(3) transformations. To the Lie algebra isomorphisms o(2, 1) - su(1, 1) - s/(2, R) there correspond parametrizations of the coherent states appropriate to each case. These are, respectively, the quasi-polar, stereographic, and Poincare coordinate systems that we briefly review here. We give the form of the line element ds* = -dx;

- dx; + dx:

P.2)

appropriate to each case. B.I.

Quasi-Polar

Coordinates

These are defined by x1 = R sinh 8 cos 4, x2 = R sinh 19sin 4, x3 = R cash 8,

03.3)

GROUP

THEORY

APPROACH

TO

SCATTERING

311

with

(‘3.4)

ds2 = R2(dB2 + sinh20 d$2).

Equation (4.20) is obtained from (B.3) with R = 1. B.2. Stereographic

Coordinates

Define z = R epir tgh2.

e

P.5)

Then 2z*

x, +ix,=

3 u3.6)

x3 = R

and P.7)

The upper sheet of the hyperboloid (x3 > 0) is mapped stereographically into the interior of the circle /z) = R. The discrete, unitary representations of SU( 1, 1) are realized by functions f(z) analytic in 1z 1< R. B.3. Poincare’ Coordinates

Let z-R----,

w - iR w + iR

(B.8)

w = x + iy.

This transformation maps the interior of the circle, Iz / = R, into the upper w plane (y > 0) and we have xl + ix, = x3 =

xz$

y2-R2-2iRx 2Y

’

x2+y2+R2 2Y

.

(B.9)

378

ALHASSID,

GtiRSEY.

AND

IACHELLO

The discrete series representations of SL(2, R) are realized by means of analytic functions in the upper w-plane. The metric is the Poincare metric of the upper halfplane, ds2=R2

(dx2ev;dy2)

=4R2

APPENDIX

,z’;,*.

(B. 10)

C

We complete here some details of the transformation connecting PBschl-Teller potential approaches. Squaring Eq. (6.15) we obtain t( 1 +

t( 1 +

72)

the Morse and

= (N + 1)‘I&

r2)V/

(C.1)

Since [t,

721

1 = 2iF. F-----, t

1 t

(C.2)

we have t( 1 +

7’)

t(1

+ 7’)

=

t* +

t’s2

2its’. F+ + 1 (1 +

-

7’).

(C.3)

Using [t,5’. fl =it

(C.4)

we obtain Eq. (6.16). Using coordinates t’ and @‘, the equation for I$, the Fourier transform

+2t’g(I Introducing

+t’2)+3(1

+t’I)

I

v’=(N+

of v, reads

l)?$.

(C.5)

the wave function ,Y, Eq. (6.18). one obtains

i! +$t$

I

(1$f2)‘J2+3(1

(1 + p?)l/? 22 t,2 F +[“)-I/2

I

x= (1(N+ 1): x. + tl?).v2

CC.61

GROUP

After a long calculation, -(l

THEORY

APPROACH

379

TO SCATTERING

this yields

+t’2)2

i

;+p- r

f2

a2 x = N(N t 2)x. acDt2 1

cc.71

This is the same as Eq. (6.19). The sphere can be projected on a plane tangent to it in the following way. A straight line from the south pole through any point (0, $) on the sphere will intersect the plane at a point with coordinates t’ = fg(8/2), @’ = 4. Using 1% 2T 1 a a sin 0 88 sin 0 a8 = 7sin’ 0 t’ F t’ F’

(C.8)

4tt2 sin2 @= (I + tr2)23 Equation

(6.19) is easily transformed

(C.9)

into (6.20).

ACKNOWLEDGMENTS We wish to thank 1. Bars. A. E. L. Dieperink, A. D. Jackson. C. Tze. and 0. van Roosmalen for interesting discussions. This work was supported in part by Department of Energy Contracts DE-AC 02. 76 ER 03074 and DE-AC 02-76 ER 03075. and. in part, by the Stichting voor Fundamenteel Onderzoek der Materie (FOM).

REFERENCES 1. A. ARIMA

AND

2. F. IACHELLO, (1982),

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(1982).

32.

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0. S. VAN

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(N.Y.)

99 (1976),

253; 111 (1978).

78 (1981).

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