Acta Polytechnica


doi:10.14311/AP.2018.58.0118
Acta Polytechnica 58(2):118–127, 2018 © Czech Technical University in Prague, 2018

available online at http://ojs.cvut.cz/ojs/index.php/ap

QUASI-EXACTLY SOLVABLE SCHRÖDINGER EQUATIONS,
SYMMETRIC POLYNOMIALS AND FUNCTIONAL BETHE

ANSATZ METHOD

Christiane Quesne

Physique Nucléaire Théorique et Physique Mathématique, Université Libre de Bruxelles, Campus de la Plaine
CP229, Boulevard du Triomphe, B-1050 Brussels, Belgium
correspondence: cquesne@ulb.ac.be

Abstract. For applications to quasi-exactly solvable Schrödinger equations in quantum mechanics, we
consider the general conditions that have to be satisfied by the coefficients of a second-order differential
equation with at most k + 1 singular points in order that this equation has particular solutions that are
nth-degree polynomials. In a first approach, we show that such conditions involve k − 2 integration
constants, which satisfy a system of linear equations whose coefficients can be written in terms of
elementary symmetric polynomials in the polynomial solution roots whenver such roots are all real and
distinct. In a second approach, we consider the functional Bethe ansatz method in its most general form
under the same assumption. Comparing the two approaches, we prove that the above-mentioned k − 2
integration constants can be expressed as linear combinations of monomial symmetric polynomials in the
roots, associated with partitions into no more than two parts. We illustrate these results by considering
a quasi-exactly solvable extension of the Mathews-Lakshmanan nonlinear oscillator corresponding
to k = 4.

Keywords: Schrödinger equation; quasi-exactly solvable potentials; symmetric polynomials.

1. Introduction
In quantum mechanics, solving the Schrödinger equation is a fundamental problem for understanding physical
systems. Exact solutions may be very useful for developing a constructive perturbation theory or for suggesting
trial functions in variational calculus for more complicated cases. However, very few potentials can actually be
exactly solved (see, e.g., one of their lists in [1]).1 These potentials are connected with second-order differential
equations of hypergeometric type and their wavefunctions can be constructed by using the theory of corresponding
orthogonal polynomials [3].

A second category of exact solutions belongs to the so-called quasi-exactly solvable (QES) Schrödinger
equations. These occupy an intermediate place between exactly solvable (ES) and non-solvable ones in the sense
that only a finite number of eigenstates can be found explicitly by algebraic means, while the remaining ones
remain unknown. The simplest QES problems, discovered in the 1980s, are characterized by a hidden sl(2,R)
algebraic structure [4–8] and are connected with polynomial solutions of the Heun equation [9]. Generalizations
of this equation are related through their polynomial solutions to more complicated QES problems. Several
procedures are employed in this context, such as the use of high-order recursion relations (see, e.g., [10]) or the
functional Bethe ansatz (FBA) method [11–13], which has proven very effective [14–17].

The purpose of the present paper is to reconsider the general conditions under which a second-order differential
equation X(z)y′′(z) + Y (z)y′(z) + Z(z)y(z) = 0 with polynomial coefficients X(z), Y (z), and Z(z) of respective
degrees k, k − 1, and k − 2, has nth-degree polynomial solutions yn(z). Choosing appropriately the polynomial
Z(z) for such a purpose is known as the classical Heine-Stieltjes problem [18, 19]. Such a differential equation
with at most k + 1 singular points covers the cases of the hypergeometric equation (for k = 2), the Heun equation
(for k = 3), and generalized Heun equations (for k ≥ 4), and plays therefore a crucial part in ES and QES
quantum problems.

Here we plan to emphasize the key role played by symmetric polynomials in the polynomial solution roots
whenever such roots are all real and distinct. This will be done by comparing two different approaches: a first
one expressing the polynomial Z(z) in terms of k− 2 integration constants, and a second one based on the FBA
method.

In Section 2, the problem of second-order differential equations with polynomial solutions is discussed and
k− 2 integration constants are introduced. In Section 3, the FBA method is derived in its most general form. A
comparison between the two approaches is carried out in Section 4. The results so obtained are illustrated in

1Here we do not plan to discuss the recent development of the exceptional orthogonal polynomials and the associated polynomially
solvable analytic potentials (see, e.g., [2] and references quoted therein).

118

http://dx.doi.org/10.14311/AP.2018.58.0118
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vol. 58 no. 2/2018 Quasi-Exactly Solvable Schrödinger Equations

Section 5 by considering a QES extension of the Mathews-Lakshmanan nonlinear oscillator. Finally, Section 6
contains the conclusion.

2. Second-order differential equations with polynomial solutions and
integration constants

On starting from a Schrödinger equation (T + V )ψ(x) = Eψ(x), where the real variable x varies in a given
domain, the potential energy V is a function of x, and the kinetic energy T depends on d/dx (and possibly on x
whenever the space is curved or the mass depends on the position [20]), and on making an appropriate gauge
transformation and a change of variable x = x(z), one may arrive at a second-order differential equation with at
most k + 1 singular points,

X(z)y′′(z) + Y (z)y′(z) + Z(z)y(z) = 0, (2.1)

where

X(z) =
k∑
l=0

alz
l, Y (z) =

k−1∑
l=0

blz
l, Z(z) =

k−2∑
l=0

clz
l, (2.2)

and al, bl, cl are some (real) constants. Polynomial solutions yn(z) of this equation will yield exact solutions
ψn(x) of the starting Schrödinger equation provided the latter are normalizable.

On deriving equation (2.1) k − 2 times, we obtain

Xy(k) +
[(
k − 2

1

)
X′ + Y

]
y(k−1) +

[(
k − 2

2

)
X′′ +

(
k − 2

1

)
Y ′ + Z

]
y(k−2) + · · ·

+
[
X(k−2) +

(
k − 2

1

)
Y (k−3) +

(
k − 2

2

)
Z(k−4)

]
y′′ +

[
Y (k−2) +

(
k − 2

1

)
Z(k−3)

]
y′ + Z(k−2)y = 0, (2.3)

which is a kth-order homogeneous differential equation with polynomial coefficients of degree not exceeding
the corresponding order of differentiation. Since all its derivatives will have the same property, it can be
differentiated n times by using the new representation y(n)(z) = vn(z). In such a notation, equation (2.3) can
be written as

Xv
(k)
0 +

[(
k − 2

1

)
X′ + Y

]
v

(k−1)
0 +

[(
k − 2

2

)
X′′ +

(
k − 2

1

)
Y ′ + Z

]
v

(k−2)
0 + · · ·

+
[
X(k−2) +

(
k − 2

1

)
Y (k−3) +

(
k − 2

2

)
Z(k−4)

]
v′′0 +

[
Y (k−2) +

(
k − 2

1

)
Z(k−3)

]
v′0 + Z

(k−2)v0 = 0. (2.4)

Its nth derivative can be easily shown to be given by

k∑
l=0

(
n + k − 2
k − l

)
X(k−l)v(l)n +

k−1∑
l=0

(
n + k − 2
k − l− 1

)
Y (k−l−1)v(l)n +

k−2∑
l=0

(
n + k − 2
k − l− 2

)
Z(k−l−2)v(l)n = 0. (2.5)

When the coefficient of vn in (2.5) is equal to zero, i.e.,(
n + k − 2

k

)
X(k) +

(
n + k − 2
k − 1

)
Y (k−1) +

(
n + k − 2
k − 2

)
Z(k−2) = 0, (2.6)

there only remains derivatives of vn in the equation. It is then obvious that the latter has a particular solution
for which vn = y(n) is a constant or, in other words, there exists a particular solution y(z) = yn(z) of equation
(2.1) that is a polynomial of degree n.

On integrating equation (2.6) k − 2 times, we find that this occurs whenever Z(z) = Zn(z) is given by

Zn(z) = −
n(n− 1)
k(k − 1)

X′′(z) −
n

k − 1
Y ′(z) +

k−3∑
l=0

Ck−l−2,n
zl

l!
, (2.7)

where C1,n,C2,n, . . . ,Ck−2,n are k − 2 integration constants. This is the first main result of this paper.
It is worth observing that in the hypergeometric case, we have k = 2 so that the polynomial Z(z) reduces to

a constant λ. Then λ = λn, where, in accordance with equation (2.7),

λn = −12n(n− 1)X
′′(z) −nY ′(z), (2.8)

119



Christiane Quesne Acta Polytechnica

with no integration constant, which is a well-known result [3]. In the Heun-type equation case, we have k = 3
and the linear polynomial Z(z) = Zn(z) is given by

Zn(z) = −16n(n− 1)X
′′(z) − 12nY

′(z) + Cn (2.9)

in terms of a single integration constant Cn, which is a result previously derived by Karayer, Demirhan, and
Büyükkılıç [21].

On inserting now equation (2.2) in (2.7) and equating the coefficients of equal powers of z on both sides, we
obtain the set of relations

ck−2 = −n(n− 1)ak −nbk−1, (2.10)

cl = −
n(n− 1)
k(k − 1)

(l + 2)(l + 1)al+2 −
n

k − 1
(l + 1)bl+1 +

Ck−l−2,n
l!

, l = 0, 1, . . . ,k − 3. (2.11)

The nth-degree polynomial solutions yn(z) of equation (2.1) can be written as

yn(z) =
n∏
i=1

(z −zi), (2.12)

where from now on we assume that the roots z1,z2, . . . ,zn are real and distinct. We now plan to show that
the integration constants C1,n,C2,n, . . . ,Ck−2,n satisfy a system of linear equations whose coefficients can be
expressed in terms of elementary symmetric polynomials in z1,z2, . . . ,zn [22],

el ≡ el(z1,z2, . . . ,zn) =
∑

1≤i1<i2<···<il≤n
zi1zi2 . . .zil, l = 1, 2, . . . ,n, e0 ≡ 1. (2.13)

We can indeed rewrite yn(z) in (2.12) as

yn(z) =
n∑

m=0
(−1)n−men−mzm, (2.14)

so that

y′n(z) =
n−1∑
m=0

(−1)n−m−1(m + 1)en−m−1zm, y′′n(z) =
n−2∑
m=0

(−1)n−m−2(m + 2)(m + 1)en−m−2zm. (2.15)

On inserting these expressions in equation (2.1) and taking equation (2.2) into account, we get

k∑
l=0

alz
l
n−2∑
m=0

(−1)n−m−2(m + 2)(m + 1)en−m−2zm +
k−1∑
l=0

blz
l
n−1∑
m=0

(−1)n−m−1(m + 1)en−m−1zm

+
k−2∑
l=0

clz
l

n∑
m=0

(−1)n−men−mzm = 0. (2.16)

Here l + m runs from 0 to k + n − 2. Let us therefore set l + m = k + n − r, where r = 2, 3, . . . ,k + n.
Equation (2.16) can then be rewritten as

k∑
r=2

{
r−2∑
p=0

(−1)p [ak−r+2+p(n−p)(n−p− 1) + bk−r+1+p(n−p) + ck−r+p] ep

}
zk+n−r

+ (lower-degree terms with k + 1 ≤ r ≤ k + n) = 0. (2.17)

On setting to zero the coefficients of zk+n−r, r = 2, 3, . . . ,k, we obtain the relations
r−2∑
p=0

(−1)p [ak−r+2+p(n−p)(n−p− 1) + bk−r+1+p(n−p) + ck−r+p] ep = 0, r = 2, 3, . . . ,k. (2.18)

For r = 2, we simply get n(n− 1)ak + nbk−1 + ck−2 = 0, which is automatically satisfied due to equation (2.10).
We are therefore left with the k − 2 relations

r−3∑
p=0

(−1)p
[
ak−r+2+p(n−p)(n−p− 1) + bk−r+1+p(n−p) + ck−r+p

]
ep

+ (−1)r−2[ak(n−r + 2)(n−r + 1) + bk−1(n−r + 2) + ck−2]er−2 = 0, r = 3, 4, . . . ,k. (2.19)

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vol. 58 no. 2/2018 Quasi-Exactly Solvable Schrödinger Equations

After substituting the right-hand sides of equations (2.10) and (2.11) for ck−2 and ck−r+p in these relations,
we obtain a system of k − 2 linear equations for the k − 2 integration constants C1,n,C2,n, . . . ,Ck−2,n,

r−3∑
p=0

(−1)p
Cr−2−p,n

(k −r + p)!
ep = −

r−3∑
p=0

(−1)p
{

1
k(k − 1)

[
(r −p− 2)(2k −r + p + 1)n2 − [2pk2 + 2k(r − 2p− 2)

− (r −p− 2)(r −p− 1)]n + k(k − 1)p(p + 1)
]
ak−r+2+p +

1
k − 1

[(r −p− 2)n− (k − 1)p]bk−r+1+p
}
ep

− (−1)r−1(r − 2)[(2n−r + 1)ak + bk−1]er−2, r = 3, 4, . . . ,k, (2.20)

whose coefficients are expressed in terms of elementary symmetric polynomials (2.13) in the roots of the
polynomial solutions of equation (2.1). This is the second main result of this paper.

The determinant of this system having zeros above the diagonal is easily determined to be given by[∏k
r=3(k −r)!

]−1
6= 0. It is therefore obvious that the constants C1,n,C2,n, . . . ,Ck−2,n can be calculated

successively from the equations corresponding to r = 3, 4, . . . ,k.
It turns out that instead of elementary symmetric polynomials el(z1,z2, . . . ,zn), defined in equation (2.13),

it is more appropriate to express the solution in terms of monomial symmetric polynomials in z1,z2, . . . ,zn,

m(λ1,λ2,...,λn)(z1,z2, . . . ,zn) =
∑
π∈Sλ

zλ1
π(1)z

λ2
π(2) . . .z

λn
π(n), (2.21)

where (λ1,λ2, . . . ,λn) denotes a partition and Sλ is the set of permutations giving distinct terms in the sum
[22]. The derivation of the first three constants C1,n, C2,n, and C3,n is outlined in Appendix. In particular, it is
shown there that m(13,0̇) (a dot over zero meaning that it is repeated as often as necessary), corresponding to a
partition into more than two parts and which in principle might appear in C3,n, actually does not occur because
it has a vanishing coefficient. This is a general property that we have observed for the first six constants that we
have computed explicitly and which all agree with the general formula

Cq,n
(k − 2 −q)!

= −
q−1∑
t=0

[2(n− 1)ak−t + bk−t−1]m(q−t,0̇) −
[q/2]∑
s=1

q−2s∑
t=0

2ak−t m(q−t−s,s,0̇)

−
n(n− 1)
k(k − 1)

q(2k −q − 1)ak−q −
n

k − 1
qbk−q−1, q = 1, 2, . . . ,k − 2, (2.22)

where [q/2] denotes the largest integer contained in q/2.
At this stage, equation (2.22) is a conjecture, which might be proved from (2.20) by determining Cq,n for any

q ∈{1, 2, . . . ,k − 2}. This would, however, be a rather complicated derivation. In Section 4, we will proceed to
show that a much easier proof of (2.22) can be found by comparing the results of the present approach with
those of the FBA one.

3. Functional bethe ansatz method
In its most general form, the FBA method also starts from equation (2.1), with X(z), Y (z), Z(z) given in (2.2),
and considers polynomial solutions of type (2.12) with real and distinct roots z1,z2, . . . ,zn [11–13]. Equation
(2.1) is then rewritten as

− c0 =
( k∑
l=0

alz
l

) n∑
i=1

1
z −zi

n∑
j=1
j 6=i

2
zi −zj

+
(k−1∑
l=0

blz
l

) n∑
i=1

1
z −zi

+
k−2∑
l=1

clz
l. (3.1)

The left-hand side of this equation is a constant, while the right-hand one is a meromorphic function with simple
poles at z = zi and a singularity at z = ∞. Since the residues at the simple poles are given by

Res(−c0)z=zi =
( k∑
l=0

alz
l
i

) n∑
j=1
j 6=i

2
zi −zj

+
k−1∑
l=0

blz
l
i, (3.2)

equation (3.1) yields

− c0 =
k∑
l=1

al

n∑
i=1

l−1∑
m=0

zmi z
l−m−1

n∑
j=1
j 6=i

2
zi −zj

+
k−1∑
l=1

bl

n∑
i=1

l−1∑
m=0

zmi z
l−m−1 +

k−2∑
l=1

clz
l +

n∑
i=1

Res(−c0)z=zi
z −zi

. (3.3)

121



Christiane Quesne Acta Polytechnica

On defining

Sm =
n∑
i=1

n∑
j=1
j 6=i

zmi
zi −zj

, Tm =
n∑
i=1

zmi , (3.4)

and observing that S0 = 0, this relation becomes

− c0 = 2
k∑
l=2

al

l−1∑
m=1

Smz
l−m−1 +

k−1∑
l=1

bl

l−1∑
m=0

Tmz
l−m−1 +

k−2∑
l=1

clz
l +

n∑
i=1

Res(−c0)z=zi
z −zi

, (3.5)

or, with q ≡ l−m− 1 in the first two terms and q ≡ l in the third one,

− c0 = 2
k−2∑
q=0

zq
k−1−q∑
m=1

aq+m+1Sm +
k−2∑
q=0

zq
k−2−q∑
m=0

bq+m+1Tm +
k−2∑
q=1

zqcq +
n∑
i=1

Res(−c0)z=zi
z −zi

. (3.6)

The right-hand side of this equation will be a constant if and only if the coefficients of zq, q = 1, 2, . . . ,k− 2,
and all the residues at the simple poles vanish. This yields cq, q = 1, 2, . . . ,k − 2, in terms of the coefficients of
X(z), Y (z), and the roots of yn(z),

cq = −2
k−1−q∑
m=1

aq+m+1Sm −
k−2−q∑
m=0

bq+m+1Tm, q = 1, 2, . . . ,k − 2, (3.7)

as well as the n algebraic equations determining the roots, i.e., the Bethe ansatz equations,
n∑
j=1
j 6=i

2
zi −zj

+
∑k−1
l=0 blz

l
i∑k

l=0 alz
l
i

= 0, i = 1, 2, . . . ,n. (3.8)

The remaining constant leads to the value of c0,

c0 = −2
k−1∑
m=1

am+1Sm −
k−2∑
m=0

bm+1Tm. (3.9)

It remains to find the explicit expressions of Sm and Tm. For the smallest allowed m values, it is obvious that

S1 = 12n(n− 1), T0 = n. (3.10)

For higher m values, Sm can be written as a linear combination of monomial symmetric polynomials in
z1,z2, . . . ,zn. From

Sm =
1
2

n∑
i=1

n∑
j=1
j 6=i

zmi −z
m
j

zi −zj
=

1
2

n∑
i=1

n∑
j=1
j 6=i

m−1∑
p=0

z
m−1−p
i z

p
j , (3.11)

we get for odd m ≥ 3,

Sm =
1
2

n∑
i=1

n∑
j=1
j 6=i

[
zm−1i + z

m−1
j +

(m−3)/2∑
p=1

(
z
m−1−p
i z

p
j + z

p
i z
m−1−p
j

)
+ z(m−1)/2i z

(m−1)/2
j

]

= (n− 1)
n∑
i=1

zm−1i +
n∑

i,j=1
i 6=j

(m−3)/2∑
p=1

z
m−1−p
i z

p
j +

n∑
i,j=1
i<j

z
(m−1)/2
i z

(m−1)/2
j = (n− 1)m(m−1,0̇) +

(m−1)/2∑
p=1

m(m−1−p,p,0̇),

(3.12)

and for even m ≥ 2,

Sm =
1
2

n∑
i=1

n∑
j=1
j 6=i

[
zm−1i + z

m−1
j +

(m−2)/2∑
p=1

(
z
m−1−p
i z

p
j + z

p
i z
m−1−p
j

)]

= (n− 1)
n∑
i=1

zm−1i +
n∑

i,j=1
i6=j

(m−2)/2∑
p=1

z
m−1−p
i z

p
j = (n− 1)m(m−1,0̇) +

(m−2)/2∑
p=1

m(m−1−p,p,0̇). (3.13)

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vol. 58 no. 2/2018 Quasi-Exactly Solvable Schrödinger Equations

Hence,

Sm = (n− 1)m(m−1,0̇) +
[(m−1)/2]∑

p=1
m(m−1−p,p,0̇), m ≥ 2. (3.14)

Furthermore, it is obvious that
Tm = m(m,0̇), m ≥ 1. (3.15)

On replacing Sm and Tm by their explicit values in (3.7) and (3.9), we get

ck−2 = −n(n− 1)ak −nbk−1, (3.16)

cl = −n(n− 1)al+2 −nbl+1 − 2
k−1−l∑
m=2

al+m+1

[
(n− 1)m(m−1,0̇) +

[(m−1)/2]∑
p=1

m(m−1−p,p,0̇)

]

−
k−2−l∑
m=1

bl+m+1m(m,0̇), l = 0, 1, . . . ,k − 3, (3.17)

in terms of monomial symmetric polynomials in the polynomial roots. This is the third main result of this
paper.

4. Comparison between the two approaches
Direct comparison between equations (2.10), (2.11) and equations (3.16), (3.17) shows that ck−2 is given by
the same expression in both approaches, while for l = 0, 1, . . . ,k − 3, cl is written in terms of al+2 and bl+1,
as well as the integration constant Ck−l−2,n in the first one or a linear combination of monomial symmetric
polynomials in z1,z2, . . . ,zn in the second one.

Equating the two expressions for cl, l = 0, 1, . . . ,k − 3, yields

Ck−2−l,n
l!

= −2
k−1−l∑
m=2

al+m+1

[
(n− 1)m(m−1,0̇) +

[(m−1)/2]∑
p=1

m(m−1−p,p,0̇)

]
−
k−2−l∑
m=1

bl+m+1m(m,0̇)

−
n(n− 1)
k(k − 1)

(k − l− 2)(k + l + 1)al+2 −
n

k − 1
(k − l− 2)bl+1, l = 0, 1, . . . ,k − 3. (4.1)

On setting q = k − 2 − l in equation (4.1), the latter becomes

Cq,n
(k − 2 −q)!

= −2
q+1∑
m=2

ak+m−1−q

[
(n− 1)m(m−1,0̇) +

[(m−1)/2]∑
p=1

m(m−1−p,p,0̇)

]
−

q∑
m=1

bk+m−1−qm(m,0̇)

−
n(n− 1)
k(k − 1)

q(2k −q − 1)ak−q −
n

k − 1
qbk−q−1, l = 0, 1, . . . ,k − 3, (4.2)

where we see that the last two terms on the right-hand side coincide with the corresponding ones in equation
(2.22). The other terms can also be easily converted into those of equation (2.22) by changing the summation
indices. With t = q + 1 −m and t = q −m, we can indeed rewrite

q+1∑
m=2

ak+m−1−qm(m−1,0̇) =
q−1∑
t=0

ak−tm(q−t,0̇) (4.3)

and
q∑

m=1
bk+m−1−qm(m,0̇) =

q−1∑
t=0

bk−t−1m(q−t,0̇), (4.4)

respectively. Furthermore, t = q + 1 −m and s = p lead to

q+1∑
m=2

ak+m−1−q

[(m−1)/2]∑
p=1

m(m−1−p,p,0̇) =
q−1∑
t=0

ak−t

[(q−t)/2]∑
s=1

m(q−t−s,s,0̇) =
[q/2]∑
s=1

q−2s∑
t=0

ak−tm(q−t−s,s,0̇). (4.5)

Collecting all the terms shows that equation (4.2) coincides with equation (2.22), which is therefore proved.
This is the fourth main result of the present paper.

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Christiane Quesne Acta Polytechnica

5. Example: QES extension of the Mathews-Lakshmanan nonlinear
oscillator

The purpose of the present section is to illustrate the results of previous ones by considering a QES extension of
the Mathews-Lakshmanan nonlinear oscillator in the simplest case not amenable to an sl(2,R) description, and
which corresponds to k = 4.

The quantum version of the classical Mathews-Lakshmanan nonlinear oscillator [23] is described by the
Hamiltonian [24, 25]

H = −(1 + λx2)
d2

dx2
−λx

d

dx
+
β(β + λ)x2

1 + λx2
, (5.1)

where β plays the role of the frequency ω in the standard oscillator and the nonlinearity parameter λ 6= 0
enters both the potential energy term and the kinetic energy one, giving rise there to the position-dependent
mass m(x) = (1 + λx2)−1. According to whether λ > 0 or λ < 0, the range of the coordinate x is (−∞,∞) or
(−1/

√
|λ|, 1/

√
|λ|). Such a Hamiltonian is formally self-adjoint with respect to the measure dµ = (1+λx2)−1/2dx.

The corresponding Schrödinger equation is ES [24, 25] and its bound-state wavefunctions can be expressed in
terms of Gegenbauer polynomials [26].

Noting that the potential energy term in (5.1) can also be written as

V0(x) = λA−
λA

1 + λx2
, where A =

β

λ

(
β

λ
+ 1
)
, (5.2)

let us extend it to

Vm(x) = λA−
λA

1 + λx2
+ λ

2m∑
k=1

Bk(1 + λx2)k, (5.3)

where m may take the values m = 1, 2, 3, . . . , A, B1, B2, . . . , B2m are 2m + 1 parameters, and the range of x
is the same as before. The starting Schrödinger equation therefore reads(

−(1 + λx2)
d2

dx2
−λx

d

dx
+ λA−

λA

1 + λx2
+ λ

2m∑
k=1

Bk(1 + λx2)k −E
)
ψ(x) = 0. (5.4)

To reduce it to an equation of type (2.1), let us make the change of variable

z =
1

1 + λx2
(5.5)

and the gauge transformation

ψ(x) = xpza exp
(
−

m∑
j=1

bj
zj

)
y(z), (5.6)

where p = 0, 1 is related to the parity (−1)p = +1,−1, and a, b1, b2, . . . , bm are m + 1 parameters connected
with the previous ones. It turns out that k in (2.2) is related to m in (5.3) by the relation k = m + 2.

For m = 2, for instance, equation (5.4) yields{
−4z3(1 −z)

d2

dz2
+ 2[(4a + 3)z3 − 2(2a− 2b1 + 1 −p)z2 − 4(b1 − 2b2)z − 8b2]

d

dz
+ [2a(2a + 1) −A]z2

+ [−4a2 + 8ab1 + 4ap− 2b1 −p− �]z + B1 − 4b1(2a− b1 − 1 −p) + 4b2(4a− 3)
}
y(z) = 0, (5.7)

after setting E = λ(� + A) and

B2 = 4[b21 + 2b2(2a− 2b1 − 2 −p)], B3 = 16b2(b1 − b2), B4 = 16b
2
2. (5.8)

With the identifications
a4 → 4, a3 →−4, a2,a1,a0 → 0, b3 → 2(4a + 3),

b2 →−4(2a− 2b1 + 1 −p), b1 →−8(b1 − 2b2), b0 →−16b2, c2 → 2a(2a + 1) −A,
c1 →−4a2 + 8ab1 + 4ap− 2b1 −p− �, c0 → B1 − 4b1(2a− b1 − 1 −p) + 4b2(4a− 3) (5.9)

in equation (2.2), from equations (2.10) and (2.11) we obtain

2a(2a + 1) −A = −4n(n− 1) − 2n(4a + 3),
−4a2 + 8ab1 + 4ap− 2b1 −p− � = 2n(n− 1) + 83n(2a− 2b1 + 1 −p) + C1,n,

B1 − 4b1(2a− b1 − 1 −p) + 4b2(4a− 3) = 83n(b1 − 2b2) + C2,n, (5.10)

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vol. 58 no. 2/2018 Quasi-Exactly Solvable Schrödinger Equations

where, from (2.22), the integration constants C1,n and C2,n are given by

C1,n = −2[4(n− 1) + 4a + 3]m(1,0̇) + 2n(n− 1) +
4
3n(2a− 2b1 + 1 −p),

C2,n = −2[4(n− 1) + 4a + 3]m(2,0̇) + 4[2(n− 1) + 2a− 2b1 + 1 −p]m(1,0̇) − 8m(12,0̇) +
16
3 n(b1 − 2b2). (5.11)

On the other hand, the direct application of the FBA relations (3.16) and (3.17) yields the equivalent results

2a(2a + 1) −A = −4n(n− 1) − 2n(4a + 3),
−4a2 + 8ab1 + 4ap− 2b1 −p− � = 4n(n− 1) + 4n(2a− 2b1 + 1 −p) − 8(n− 1)m(1,0̇) − 2(4a + 3)m(1,0̇),

B1 − 4b1(2a− b1 − 1 −p) + 4b2(4a− 3) = 8n(b1 − 2b2) + 8(n− 1)m(1,0̇)
− 8[(n− 1)m(2,0̇) + m(12,0̇)] + 4(2a− 2b1 + 1 −p)m(1,0̇) − 2(4a + 3)m(2,0̇). (5.12)

In both cases, on setting n = 0 for instance, we get that the Schrödinger equation (5.4) with m = 2,

A = 2a(2a + 1), B1 = 4b1(2a− b1 − 1 −p) − 4b2(4a− 3), (5.13)

and B2, B3, B4 given in (5.8), has an eigenvalue

E0,p = λ(8ab1 + 4ap + 2a− 2b1 −p) (5.14)

with the corresponding eigenfunction

ψ0,p(x) ∝ xp(1 + λx2)−ae−λ(b1+2b2)x
2−λ2b2x4. (5.15)

The latter is normalizable with respect to the measure dµ provided b2 > 0 if λ > 0 or a < 14 if λ < 0 and it
corresponds to a ground state for p = 0 and to a first excited state for p = 1.

Furthermore, for n = 1, on taking into account that m(1,0̇) = z1, m(2,0̇) = z21, and m(12,0̇) = 0 in terms of the
single root z1 of y1(z), we obtain that equation (5.4) with m = 2,

A = (2a + 2)(2a + 3), B1 = −2(4a + 3)z21 + 4(2a−2b1 + 1−p)z1 + 4b1(2a−b1 + 1−p)−4b2(4a + 1), (5.16)

and B2, B3, B4 given in (5.8), has an eigenvalue

E1,p = λ[8ab1 + 4ap + 2a + 6b1 + 3p + 2 + 2(4a + 3)z1] (5.17)

with the corresponding eigenfunction

ψ1,p(x) ∝ xp(1 + λx2)−a−1[1 −z1(1 + λx2)]e−λ(b1+2b2)x
2−λ2b2x4. (5.18)

The normalizability condition is now b2 > 0 if λ > 0 or a < −34 if λ < 0. Here z1 is a real solution of the cubic
equation

(4a + 3)z31 − 2(2a− 2b1 + 1 −p)z
2
1 − 4(b1 − 2b2)z1 − 8b2 = 0, (5.19)

hence, for instance,

z1 =
2(2a− 2b1 + 1 −p)

3(4a + 3)
+
(
−
v

2
+
√(v

2

)2
+
(u

3

)3 )1/3
+
(
−
v

2
−
√(v

2

)2
+
(u

3

)3 )1/3
,

u =
4

4a + 3

(
−b1 + 2b2 −

(2a− 2b1 + 1 −p)2

3(4a + 3)

)
,

v = −
8

4a + 3

(
b2 +

2
27

(2a− 2b1 + 1 −p)3

(4a + 3)2
+

1
3

(2a− 2b1 + 1 −p)(b1 − 2b2)
4a + 3

)
. (5.20)

According to its value, the wavefunction ψ1,p(x) may correspond to a ground or second-excited state for p = 0
and to a first- or third-excited state for p = 1.

6. Conclusion
In the present paper, we have reconsidered the general conditions that have to be satisfied by the coefficients of
a second-order differential equation with at most k + 1 singular points in order that the equation has particular
solutions that are nth-degree polynomials yn(z) and we have expressed them in terms of symmetric polynomials
in the polynomial solution roots. This has been done in two different ways.

125



Christiane Quesne Acta Polytechnica

In the first one, we have shown that these conditions involve k − 2 integration constants, which satisfy a
system of linear equations whose coefficients can be expressed in terms of elementary symmetric polynomials in
the polynomial solution roots whenever such roots are all real and distinct. In the second approach, we have
considered the solution of the FBA method in its most general form under the same assumption.

Comparing the outcomes of both descriptions, we have proved that the above-mentioned k − 2 integration
constants can be expressed as linear combinations of monomial symmetric polynomials in the polynomial solution
roots, corresponding to partitions into no more than two parts. As far as the author knows, this property and
the general solution of the FBA method in terms of such symmetric polynomials are new results.

In addition, their practical usefulness has been demonstrated by solving a QES extension of the Mathews-
Lakshmanan nonlinear oscillator, corresponding to k = 4.

7. Appendix
The purpose of this appendix is to solve equation (2.20) for r = 3, 4, 5 and to show that the resulting expressions
of C1,n, C2,n, and C3,n agree with equation (2.22).

For r = 3, equation (2.20) directly leads to

C1,n
(k − 3)!

= −[2(n− 1)ak + bk−1]e1 −
2n(n− 1)

k
ak−1 −

n

k − 1
bk−2, (7.1)

which corresponds to equation (2.22) for q = 1 because e1 = m(1,0̇).
For r = 4, equation (2.20) becomes

C2,n
(k − 4)!

−
C1,n

(k − 3)!
e1 = 2[(2n− 3)ak + bk−1]e2 +

[2
k

(n− 1)(n−k)ak−1 +
1

k − 1
(n−k + 1)bk−2

]
e1

−
2n(n− 1)
k(k − 1)

(2k − 3)ak−2 −
2n
k − 1

bk−3. (7.2)

On inserting (7.1) in (7.2) and using the identities e2 = m(12,0̇), e21 = m(2,0̇) + 2m(12,0̇), we get

C2,n
(k − 4)!

= −[2(n− 1)ak + bk−1]m(2,0̇) − [2(n− 1)ak−1 + bk−2]m(1,0̇)

− 2akm(12,0̇) −
2n(n− 1)
k(k − 1)

(2k − 3)ak−2 −
2n
k − 1

bk−3, (7.3)

which agrees with equation (2.22) for q = 2.
On setting now r = 5 in equation (2.20), we obtain

C3,n
(k − 5)!

−
C2,n

(k − 4)!
e1 +

C1,n
(k − 3)!

e2 = −3[2(n− 2)ak + bk−1]e3

−
{2
k

[n2 − (2k + 1)n + 3k]ak−1 +
1

k − 1
(n− 2k + 2)bk−2

}
e2

+
{ 2
k(k − 1)

(n− 1)[(2k − 3)n−k(k − 1)]ak−2 +
1

k − 1
(2n−k + 1)bk−3

}
e1

−
6n(n− 1)
k(k − 1)

(k − 2)ak−3 −
3n
k − 1

bk−4. (7.4)

Here, let us employ equations (7.1) and (7.3), as well as the identities e3 = m(13,0̇), m(2,0̇)m(1,0̇) = m(3,0̇) +
m(2,1,0̇), and m(12,0̇)m(1,0̇) = m(2,1,0̇) + 3m(13,0̇). For the coefficient of m(13,0̇) in C3,n/(k − 5)!, we obtain
−3[2(n− 2)ak + bk−1] from the right-hand side of (7.4), −6ak from C2,ne1/(k − 4)!, and 3[2(n− 1)ak + bk−1]
from −C1,ne2/(k−3)!, respectively. We conclude that m(13,0̇) does not occur in C3,n/(k−5)!, which is given by

C3,n
(k − 5)!

= −[2(n− 1)ak + bk−1]m(3,0̇) − [2(n− 1)ak−1 + bk−2]m(2,0̇) − [2(n− 1)ak−2 + bk−3]m(1,0̇)

− 2akm(2,1,0̇) − 2ak−1m(12,0̇) −
6n(n− 1)
k(k − 1)

(k − 2)ak−3 −
3n
k − 1

bk−4, (7.5)

in agreement with equation (2.22) for q = 3.

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vol. 58 no. 2/2018 Quasi-Exactly Solvable Schrödinger Equations

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127


	Acta Polytechnica 58(2):118–127, 2018
	1 Introduction
	2 Second-order differential equations with polynomial solutions and integration constants
	3 Functional bethe ansatz method
	4 Comparison between the two approaches
	5 Example: QES extension of the Mathews-Lakshmanan nonlinear oscillator
	6 Conclusion
	7 Appendix
	References