Table Of ContentUnified derivation of exact solutions for a class of
quasi-exactly solvable models
Davids Agboola and Yao-Zhong Zhang
∗ †
School of Mathematics and Physics, The University of Queensland,
Brisbane, QLD 4072, Australia
2
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2
n
a
J Abstract: We present a unified treatment of exact solutions for a class of four quantum
3 mechanical models, namely the anharmonic singular potential, the generalized quantum
1
isotonic oscillator, the soft-core Coulomb potential, and the non-polynomially modified
] oscillator. We show that all four cases are reducible to the same basic ordinary differential
h
equation, which is quasi-exactly solvable. A systematic and closed form solution to the
p
- basic equation is obtained via the Bethe ansatz method. Using the result, general exact
h
expressions for the energies and the allowed potential parameters are given explicitly for
t
a
each of the four cases in terms of the roots of a set of algebraic equations. A hidden sl(2)
m
algebraic structure is also discovered in these models.
[
2
v PACS numbers: 03.65.-w, 03.65.Fd, 03.65.Ge, 02.30.Ik
0
Keywords: Quasi-exactly solvable systems, Bethe ansatz
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0
1
.
1
1
1
1 1 Introduction
:
v
i Over the years, much efforts have been devoted to the determination of closed form so-
X
lutions to (Schro¨dinger) differential equations of the form HΨ = EΨ, where H is the
r
a Hamiltonian and E the eigenvalue. In certain cases, the concerned equation can be re-
ducedviasuitablesubstitutions andtransformations toa wellknown differential equation,
from which solutions to the original problem can be easily obtained. For instance, closed
form solutions to many exactly solvable models in quantum mechanics have been studied
in connection with some well known classical differential equations. However, due to lim-
ited applications of exactly solvablesystems inquantummechanics, recentattentions have
been on systems with partially, algebraically solvable spectra. Such systems are said to
bequasi-exactly solvable. Thus a quantum mechanical system is called quasi-exactly solv-
able if only a finite number of eigenvalues and corresponding eigenvectors can be obtained
exactly through algebraic means [1, 2, 3].
∗[email protected]
†[email protected]
1
An essential feature of a quasi-exactly solvable system is that the coefficients of the
power series solutions to the underlying differential equations satisfy three- or more step
recursion relations, in contrast to the two-step recursions for the exactly solvable cases.
The complexity of the three- or more step recursion relations makes it very hard (if not
impossible)togetexactpowerseriessolutionsofsuchsystems. However,onecanterminate
the series at certain power of the variable by imposing certain constraints on the system
parameters. By so doing, exact (polynomial) solutions to the system can be obtained, but
only for certain energies and for special values of the parameters of the problem.
Solutions to quasi-exactly solvable systems have mostly been discussed in terms of
the recursion relations of the power series coefficients. However, this approach does not
generally allow one to give explicit, closed form algebraic expressions for the allowed
potential parameters of the systems. In this paper, we give a systematic and unified
algebraic treatment to a class of four quantum mechanical systems, namely,
1 e d
1. Singular anharmonic potential [4, 5] : V(r) = ω2r2+ + ,
2 r4 r6
1 r2 a2
2. Generalized quantum isotonic oscillator [6]: V(r) = ω2r2+g − ,
2 (r2+a2)2
G F
3. Soft core Coulomb potential [7] : V(r)= ,
r − r+β
1 λr2
4. Non polynomially modified oscillator [8] [15] : V(r)= ω2r2+ .
− 2 1+δr2
Model 1 was investigated in [4] using the Laurent series ansatz and continued fractions,
and its ground state was found in [5]. Solutions to models 2, 3 and 4 were studied in
[16]-[20] on a case by case basis by means of the recursion relations. We will show that all
four models are reducible to the same basic differential equation, which is quasi-exactly
solvable. We solve these models exactly by using the functional Bethe ansatz method in
[21]. Our method allows us to obtain the explicit, closed form expressions of the energies
and the allowed potential parameters for all cases once and for all in terms of the roots of
the algebraic (Bethe ansatz) equations. Particularly interesting are our exact results for
model 1, in that to our knowledge this model was not previously recognized to be quasi-
exactly solvable. We also find a underlying sl(2) algebraic structure in all these models,
which is responsible for the quasi-exact solvability.
Theworkis organized as follows. Insection 2, wedefinethefourmodelsandtransform
the correspondingdifferential equations into the same basic form. In section 3, we present
exact polynomial solutions to the basic equation underlying the four cases by using the
Bethe ansatz method. We then give the general, closed form expressions of both the
energies and the allowed parameters for each of the four cases. Hidden sl(2) algebraic
structures underlying the four models are presented in section 4. In section 5 we provide
some concluding remarks.
2 The four models and the underlying differential equations
In this section, we give a description of the four models to be considered in this paper.
For each case, we reduce the underlying differential equation to the basic equation which
is quasi-exactly solvable.
2
2.1 Singular anharmonic potential
We consider a quantum mechanical model with the singular anharmonic potential [4, 5],
1 e d
V(r)= ω2r2+ + , (2.1)
2 r4 r6
whereω,e,d > 0 are constant parameters. Thecorrespondingradial Schro¨dinger equation
is given by
d2Y(r) ℓ(ℓ+1) 2e 2d
+ω2r2+ + Y(r)+2EY(r)= 0, (2.2)
dr2 − r2 r4 r6
(cid:20) (cid:21)
where ℓ = 1,0,1, , and E is the energy eigenvalue.
− ···
Making the substitution,
ω √2d 1
Y(r) = r3/2+e/√2d exp r2 y(r), (2.3)
−2 − 2 r2
" #
we obtain
2 3 e √2d e
y (r)+ ωr2+ + + y (r)+2 E ω 2+ y(r)
′′ ′
r − 2 √2d r2 ! − √2d
(cid:20) (cid:18) (cid:19)(cid:21)
2 2
1 1 e
= 2ω√2d+ ℓ+ +1 y(r) (2.4)
r2 " 2 − √2d #
(cid:18) (cid:19) (cid:18) (cid:19)
Then the change of variable t = r2 transforms the above differential equation into the
form,
e t e
t2y (t)+ ωt2+ 2+ t+√2d y (t)+ E ω 2+ y(t)
′′ ′
− √2d 2 − √2d
(cid:20) (cid:18) (cid:19) (cid:21) (cid:20) (cid:18) (cid:19)(cid:21)
2 2
1 1 e
= 2ω√2d+ ℓ+ +1 y(t). (2.5)
4 " 2 − √2d #
(cid:18) (cid:19) (cid:18) (cid:19)
2.2 Generalized quantum isotonic oscillator
The generalized quantum isotonic oscillator has recently attracted a number interests.
This system is interesting because it is endowed with properties closely related to those of
the harmonic oscillator. The radial Schro¨dinger equation reads
d2Ψ(r) ℓ(ℓ+1) r2 a2
+ω2r2+2g − Ψ(r)+2EΨ(r) =0, (2.6)
dr2 − r2 (r2+a2)2
(cid:20) (cid:21)
where ω, g > 0 and a are constant parameters of the systems; ℓ = 1,0,1, , and E is
− ···
the energy eigenvalue. Making the variable change z = ωr2, (2.6) becomes
d2Ψ(z) 1dΨ(z) z ℓ(ℓ+1) g z ωa2 E
z + + + − Ψ(z) = Ψ(z) (2.7)
dz2 2 dz2 −"4 4z 2((cid:0)z+ωa2)(cid:1)2# −2ω
This equation can be transformed, by the substitution,
Ψ(z) = (z+ωa2)b+1 zℓ+21e−z/2 ψ(z), (2.8)
3
into the form
7 2(b+1)ωa2 g/2+(b+1)(ℓ+ωa2+3/2)
zψ (z)+ +2b+ℓ z ψ (z)+ ψ(z)
′′ ′
2 − z+ωa2 − z+ωa2
(cid:20) (cid:21) (cid:20) (cid:21)
E ℓ 7
= +b+ + ψ(z), (2.9)
−2ω 2 4
(cid:20) (cid:21)
where
1 1
g = b(b+1) b = b = 4g+1. (2.10)
⇒ − −2 − 2
We find it convenient to work with the new variable t = z+ωpa2. Then Eq.(2.9) becomes
5
t(t ωa2)ψ (t)+ t2+ 4g+1+ℓ+ωa2 t+ωa2 4g+1 1 ψ (t)
′′ ′
− − 2 − −
(cid:20) (cid:18) (cid:19) (cid:21)
p (cid:16)p (cid:17)
t E 5 g 1 3
+ + 4g+1 ℓ ψ(t) = + 4g+1 1 ℓ+ωa2+ ψ(t).
2 ω − − 2 −2 2 − 2
(cid:18) (cid:19) (cid:20) (cid:18) (cid:19)(cid:21)
p (cid:16)p (cid:17) (2.11)
2.3 Soft-core Coulomb potential
We now consider the soft-core Coulomb potential [7]
G Z
V(r)= , (2.12)
r − r+β
where G = Z > 0 and β > 0 are constant parameters. Such potential is of interest in
6
atomic and molecular physics. The G = 0 case simulates the field of a smeared charge
and is usefulin describingmesonic atoms. Solutions to the G = 0 case has been studied in
[19] using the method of solving the recursive relations. The parameter β can be related
to the strength of a laser field, within the range of β = 20 - 40 covering the experimental
laser field strength [22].
The corresponding reduced Schro¨dinger equation is
d2Φ(r) ℓ(ℓ+1) 2G 2Z
+ Φ(r)+2EΦ(r) = 0, (2.13)
dr2 − r2 r − r+β
(cid:20) (cid:21)
where E < 0 is the energy eigenvalue and ℓ = 1,0,1, . Making the substitution
− ···
Φ(r)= (r+β)rℓ+1e c(r+β)φ(r), (2.14)
−
we have
ℓ+1 1 (ℓ+1)(c 1/β)+G Z c (ℓ+1)/β
φ′′(r)+2 + c φ′(r)+2 − + − − φ(r)
r r+β − − r r+β
(cid:20) (cid:21) (cid:20) (cid:21)
= (2E +c2)φ(r). (2.15)
−
Setting
1
E = c2, (2.16)
−2
we obtain
t(t+β)φ (t)+2 ct2+( cβ +ℓ+2)t+(ℓ+2)β φ(t)+2t[Z G c(ℓ+2)]φ(t)
′′ ′
− − − −
(cid:2) = 2[(ℓ+1)βc+βG ℓ (cid:3)1]φ(t), (2.17)
− −
where we have introduced the variable t r for later convenience.
≡
4
2.4 Non-polynomially modified oscillator
The non-polynomially modified oscillator potential has been found to be useful in several
aspectsofphysics. Inlaser physics,itarisesoutof theFokker-Planck equation forasingle-
mode laser [8]. In field theory it provides a simple zero-dimensional model possessing a
non-polynomial Lagrangian [9]. The δ > 0 case is useful in toroidal plasma [14].
The corresponding radial Schro¨dinger equation reads
d2Ξ(r) ℓ(ℓ+1) r2
+ω2r2+2λ Ξ(r)+2EΞ(r) = 0 (2.18)
dr2 − r2 1+δr2
(cid:20) (cid:21)
where ω,δ > 0 and λ and constant parameters; ℓ = 1,0,1, , and E is the energy.
− ···
With the variable change z = ωr2, Eq.(2.18) becomes
d2Ξ(z) 1dΞ(z) 1 ℓ(ℓ+1) 2λ z E
z + z+ + Ξ(z)+ Ξ(z) = 0 (2.19)
dz2 2 dz − 4 z ω ω+δz 2ω
(cid:20) (cid:21)
Making the transformation,
ω ℓ+1 1
Ξ(z) = z+ z 2 exp z ξ(z), (2.20)
δ −2
(cid:20) (cid:21)
(cid:16) (cid:17)
Eq.(2.19) becomes
2z 3 λ/2δ2 +ω/δ+ℓ+ 3
zξ (z)+ z+ +ℓ+ ξ (x)+ 2ξ(z)
′′ ′
− z+ω/δ 2 z+ω/δ
(cid:18) (cid:19)
1 E λ 7
= + +ℓ+ ξ(z). (2.21)
2 −ω ωδ 2
(cid:20) (cid:21)
It is more convenient to work with the new variable t = z+ω/δ. Then Eq.(2.21) becomes
ω ω 7 2ω t E λ 7
t t ξ (t)+ t2+ +ℓ+ t ξ (t)+ ℓ ξ(t)
′′ ′
− δ − δ 2 − δ 2 ω − ωδ − − 2
(cid:20) (cid:18) (cid:19) (cid:21) (cid:20) (cid:21)
(cid:16) (cid:17)
λ ω 3
= + +ℓ+ ξ(t). (2.22)
− 4δ2 δ 2
(cid:20) (cid:21)
3 The basic equation and solutions to the four models
ItwasdemonstratedinthelastsectionthattheSchro¨dingerequationsforthefourcasescan
be transformed to Eqs.(2.5), (2.11), (2.17) and (2.22), respectively, after the appropriate
substitutions and variable changes. These four equations have the same basic form,
t(t α)S (t)+ b t2+b t+b S (t)+c tS(t) = c S(t), (3.1)
′′ 2 1 0 ′ 1 0
−
where α,b ,b ,b ,c ,c are cons(cid:2)tants. This equ(cid:3)ation has regular singular points t = 0,α
2 1 0 1 0
and confluently irregular singular point t = . It is related, via a simple transformation,
∞
to the so-called generalized spheroidal wave equation (GSWE) (see e.g. [23, 24]),
t(t t )X (t)+(B +B t)X (t)+ Ω2t(t t ) 2kΩ(t t )+B X(t) = 0, (3.2)
0 ′′ 1 2 ′ 0 0 3
− − − −
where t ,B ,B ,B ,k,Ω are (possibly c(cid:2)omplex) constants. Indeed, ma(cid:3)king the substitu-
0 1 2 3
tion X(t) = eiΩtS(t) in the generalized spheroidal wave equation yields
t(1 t )S (t)+ 2iΩt2+(B 2iΩt )t+B S (t)+Ω(iB 2k)tS(t)
0 ′′ 2 0 1 ′ 2
− − −
(cid:2) (cid:3)
5
= (iΩB +2kΩt +B ) S(t). (3.3)
1 0 3
−
This is nothing but the basic equation (3.1) with the identifications,
i i 2c
1
t = α, Ω = b , k = b +αb ,
0 2 1 2
−2 2 − b
(cid:18) 2 (cid:19)
b
B = b , B = b +αb , B = αc c 2 b +αb +α2b . (3.4)
1 0 2 1 2 3 1 0 0 1 2
− − 2
Note that for the four models considered in this paper, the c(cid:0)onstants α,b ,b(cid:1),b ,c ,c in
2 1 0 1 0
the basic equation are real numbers.
The basic equation (or more generally the GSWE) is quasi-exactly solvable for certain
values of its parameters, and exact solutions are given by degree n polynomials in t with n
beingnon-negative integers. In fact, the basic equation is a special case of the general 2nd
order differential equations solved in [21] by means of the Bethe ansatz method. Applying
the results in [21], we have
Proposition 3.1 Let n be any non-negative integer. The basic differential equation
Eq.(3.1) has degree n polynomial solutions
n
S(t)= (t t ), S(t) 1 for n =0 (3.5)
i
− ≡
i=1
Y
with distinct roots t ,t ,...,t only if the constant coefficients in (3.1) satisfy the con-
1 2 n
straints
c = nb , (3.6)
1 2
−
n
c = n(n 1)+b t +nb , (3.7)
0 2 i 1
−
i=1
X
where the roots t ,t ,...,t are determined by the Bethe ansatz equations
1 2 n
n 2 b t2+b t +b
= 2 i 1 i 0, i =1,2,...,n. (3.8)
t t − t (t α)
i j i i
j=i − −
X6
In the following, we apply the above results to derive general, explicit and closed form
expressions for the energies and the allowed potential parameters of the four models.
3.1 Singular anharmonic potential
Inthiscase,S(t) = y(t),α= 0,b = ω,b = 2+ e ,b = √2d,c = 1 E ω 2+ e
2 − 1 √2d 0 1 2 − √2d
and c = 1 2ω√2d+ ℓ+ 1 2 e +1 2 . Then by Eqs. (3.6) anhd (3.7),(cid:16)we obtai(cid:17)ni
0 4 2 − √2d
(cid:20) (cid:21)
the closed form express(cid:0)ions o(cid:1)f the(cid:16)energies(cid:17)and wave functions
e
E =ω 2n+2+
n
√2d
(cid:18) (cid:19)
n ω √2d 1
Y (r) = r3/2+e/√2d (r2 t ) exp r2 (3.9)
n − i −2 − 2 r2
" # " #
i=1
Y
6
and the constraint for the potential parameters
n 2 2
1 e e
2ω √2d+2 t + ℓ+ = 4n n+1+ + +1 , (3.10)
i
! 2 √2d √2d
i=1 (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
X
where the roots t are determined by the Bethe ansatz equations,
i
{ }
n ωt2 2+ e t √2d
2 i − √2d i−
= , i = 1,2, ,n. (3.11)
t t (cid:16) t2 (cid:17) ···
j=i i− j i
X6
It is easy to see that y = 1 is a solution of (2.5) provided the potential parameters
satisfy aconstraint. Thissolution correspondsto the n = 0 case in the general expressions
above. Indeed, from Eqs. (3.9) and (3.10), we obtain
e
E = ω 2+
0
√2d
(cid:18) (cid:19)
ω √2d 1
Y (r)= r3/2+e/√2d exp r2 . (3.12)
0 −2 − 2 r2
" #
and the constraint
2 2
1 e
2ω√2d+ ℓ+ = +1 . (3.13)
2 √2d
(cid:18) (cid:19) (cid:18) (cid:19)
This wave function has no nodes and so the state described by it is the ground state of the
system. Eqs. (3.12)and(3.13)reproducethegroundstatesolution andthecorresponding
constraint for the parameters found in [5].
For n = 1, Eqs. (3.9), (3.10) and (3.11) give rise to the energy
e
E = ω 4+ (3.14)
1
√2d
(cid:18) (cid:19)
and the constraint for the potential parameters
2 2
1 2e e
2ω √2d+2t + ℓ+ = 8+ + +1 , (3.15)
1
2 √2d √2d
(cid:18) (cid:19) (cid:18) (cid:19)
(cid:16) (cid:17)
where the root t is determined by the Bethe ansatz equation,
1
e
ωt2 2+ t √2d = 0
1− √2d 1−
(cid:18) (cid:19)
2
1 e e
t = 2+ 2+ +4ω√2d . (3.16)
1
⇒ 2ω √2d ±s √2d
(cid:18) (cid:19)
It follows that the potential parameters obey
2
1 e2 4e 1 2 e 2
2ω√2d+5+ ℓ+ = 2+ +4ω√2d. (3.17)
4 "2d − √2d − 2 # √2d
(cid:18) (cid:19) (cid:18) (cid:19)
The corresponding wave function is
ω √2d 1
Y (r) = r3/2+e/√2d r2 t exp r2 (3.18)
1 − 1 −2 − 2 r2
" #
(cid:0) (cid:1)
with t given by (3.16) and ω,e,d constrained by (3.17). This wave function gives rise to
1
the first excited state of the system.
7
3.2 Generalized quantum isotonic oscillator
Inthiscase,S(t) = ψ(t),α= ωa2,b = 1,b = 5 √4g+1+ℓ+ωa2,b = ωa2(√4g+1
2 − 1 2− 0 −
1), c = 1 E +√4g+1 ℓ 5 and c = g/2+ 1(√4g+1 1)(ℓ+ωa2+3/2). Here
1 2 ω − − 2 0 − 2 −
b is given in (2.10). Then by Eqs. (3.6) and (3.7), we obtain the general, closed form
(cid:0) (cid:1)
expressions of the energies and wave functions
5
E = ω 2n+ℓ+ 4g+1 ,
n
2 −
(cid:18) (cid:19)
p
n
t
Ψ (r) (r2+a2)b+1 rℓ+1e ωr2/2 r2+a2 i , (3.19)
n −
∼ − ω
" #
i=1(cid:18) (cid:19)
Y
and the constraint for the potential parameters,
n
3
n n+ 4g+1+ℓ+ωa2 t
i
2 − −
(cid:18) (cid:19) i=1
p X
g 1
= + ( 4g+1 1)(ℓ+ωa2+3/2) (3.20)
−2 2 −
where the roots t ,t ,...t obey thepBethe ansatz equations
1 2 n
n 2 t2 ℓ+ 5 √4g+1+ωa2 t +ωa2(1 √4g+1)
= i − 2 − i − ,
t t t (t ωa2)
j=i i− j (cid:0) i i− (cid:1)
X6
i= 1,2, n. (3.21)
···
It is easily seen that ψ = 1 is a solution of (2.11) provided that a constraint is satisfied
by the parameters. indeed, this solution corresponds to the n = 0 case in the general
expressions above. From Eqs.(3.19) and (3.20), we have
5
E = ω ℓ+ 4g+1 ,
0
2 −
(cid:18) (cid:19)
p
Ψ (r) (r2+a2)b+1 rℓ+1e ωr2/2, (3.22)
0 −
∼
and
3 3
g+ℓ+ +ωa2 = ℓ+ +ωa2 4g+1
2 2
(cid:18) (cid:19)
p
g = 2 ℓ+1+ωa2 2ℓ+3+2ωa2 . (3.23)
⇒
Hence the energy and wave function(cid:0) of the syste(cid:1)m(cid:0) can be writte(cid:1)n as
5
E = ω +3ℓ+4ωa2 (3.24)
0
− 2
(cid:18) (cid:19)
and
Ψ0(r) rℓ+1 r2+a2 −2(ℓ+1+ωa2)e−12ωr2. (3.25)
∼
The wave functions Ψ0(r) does not h(cid:0)ave nod(cid:1)es and so the state described by this is the
ground states.
For n = 1, Eqs.(3.19), (3.20) and (3.21) give
9
E = ω ℓ+ 4g+1 , (3.26)
1
2 −
(cid:18) (cid:19)
p
8
13 7
g+ +3 ℓ+ωa2 = 2t + ℓ+ωa2+ 4g+1 (3.27)
1
2 2
(cid:18) (cid:19)
(cid:0) (cid:1) p
and
1
t = 2s +s +ωa2 (2s +s )2+ωa2(2s 4s +ωa2) , (3.28)
1 1 2 1 2 2 1
2 ± −
where s = 1 1 √(cid:16)4g+1 and s =pℓ+3/2. Substituting Eq.(3.28) into(cid:17)Eq.(3.27), we
1 2 − 2
have the following constraint for the allowed parameter g,
(cid:0) (cid:1)
5
g+2(ℓ+ωa2+2) +ℓ+ωa2 4g+1
− 2
(cid:18) (cid:19)
p
2
5
= +ℓ 4g+1 +ωa2 2ℓ+ωa2+1+2 4g+1 ,
±s 2 −
(cid:18) (cid:19)
p (cid:16) p (3(cid:17).29)
which can be solved to give
2
1 1 1 B
g = + 2A3 + +C (3.30)
−4 36 (cid:18) A13 (cid:19)
where
A = 15ℓ+93ωa2 30ℓ2 60ℓωa2 30(ωa2)2
− − − −
24ℓ2ωa2 24ℓ(ωa2)2 8ℓ3 8(ωa2)3+53
− − − −
+3 1380ℓ 108ωa2 1443ℓ2 3246ℓωa2 507(ωa2)2
− − − − −
2556ℓ2ωa2 3276ℓ(ωa2)2 624ℓ3ωa2 1224(ℓωa2)2
(cid:2)
− − − −
1
1008ℓ(ωa2)2 612ℓ3 1332(ωa2)3 108ℓ4 300(ωa2)4 450 2 ,
− − − − − −
B = 38+8ℓ2+16ℓωa2 +20ℓ+8(ωa2)2+20ωa2,
(cid:3)
C = 8ωa2+8ℓ+19.
Hence we obtain the energy and wave function
9 1 1 B
E1 = ω(cid:20)ℓ+ 2 − 3(cid:18)2A3 + A13 +C(cid:19)(cid:21),
Ψ1(r) rℓ+1 r2+a2 b+1e−12ωr2 r2+a2 t1 (3.31)
∼ − ω
(cid:18) (cid:19)
(cid:0) (cid:1)
where
2
1 5 5
t = ℓ+ωa2+ x ℓ+ x +ωa2(2ℓ+1+ωa2+2x) (3.32)
1
2 2 − ±s 2 −
(cid:18) (cid:19)
with
1 1 B
x = 2A3 + +C . (3.33)
3 (cid:18) A13 (cid:19)
9
3.3 Soft-core Coulomb potential
In this case, S(t) = φ(t), α = β, b = 2c, b = 2( βc + ℓ + 2), b = (ℓ + 1)β,
2 1 0
− − −
c = 2[Z G (ℓ+2)c] and c =2[β(ℓ+1)c+βG ℓ 1]. Then by Eqs.(3.6) and (3.7),
1 0
− − − −
we have for the energies, wavfunction and the constraint for the parameter Z,
2
Z G 1 Z G
c = − E = − ,
n
n+ℓ+2 ⇒ −2 n+ℓ+2
(cid:18) (cid:19)
n
Φ (r)= (r+β)rℓ+1e c(r+β) (r t ) , (3.34)
n − i
−
" #
i=1
Y
n
n+3
n +ℓ βc c t = β(ℓ+1)c+βG ℓ 1, (3.35)
i
2 − − − −
(cid:18) (cid:19) i=1
X
where t satisfy the Bethe ansatz equations
i
n 1 ct2 ( βc+ℓ+2)t (ℓ+1)β
= i − − i − , i= 1,2, ,n. (3.36)
t t t (t +β) ···
i j i i
j=i −
X6
Note that c = 0 as Z = G.
6 6
It is easily seen that φ = 1 is a solution to (2.17) provided c and Z satisfy certain
constraints. This solution corresponds to the n = 0 case of our general expressions (3.34)-
(3.36), from which we obtain the solutions
2
1 Z G
E = − ,
0
−2 ℓ+2
(cid:18) (cid:19)
ℓ+1 βG
Φ (r) = (r+β)rℓ+1exp − (r+β) . (3.37)
0
− (ℓ+1)β
(cid:20) (cid:21)
and the allowed values for the parameter Z,
ℓ+2 G
Z = . (3.38)
β − ℓ+1
This wave function does not have nodes and so the state described by it is the ground
state of the system.
For n = 1, (3.34)-(3.36) become
2
Z G 1 Z G
c = − E = − , (3.39)
1
ℓ+3 ⇒ −2 ℓ+3
(cid:18) (cid:19)
ct +β(ℓ+2)c = 2ℓ+3 βG, (3.40)
1
−
where the root t obeys
1
ct2 ( βc+ℓ+2)t (ℓ+1)β = 0
1− − 1 −
1
t = βc+ℓ+2 β2c2+2ℓβc+(ℓ+2)2 . (3.41)
1
⇒ 2c − ±
It follows that the parameter(cid:16)Z satisfies the cponstraint (cid:17)
(ℓ+3)2(βZ ℓ 1)(βZ 2ℓ 3)+(2+ν)β(Z G)( 3βZ+4ℓ+6)+2β2(Z G)2 =0. (3.42)
− − − − − − −
10
