Supersymmetric method for constructing quasi-exactly solvable potentials
We propose a new method for constructing the quasi-exactly solvable (QES) potentials with two known eigenstates using supersymmetric quantum mechanics. General expression for QES potentials with explicitly known energy levels and wave functions of both ground and excited states a...
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Інститут фізики конденсованих систем НАН України
1999
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| Назва видання: | Condensed Matter Physics |
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| Цитувати: | Supersymmetric method for constructing quasi-exactly solvable potentials / V.M. Tkachuk // Condensed Matter Physics. — 1999. — Т. 2, № 2(18). — С. 197-204. — Бібліогр.: 25 назв. — англ. |
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irk-123456789-1202622017-06-12T03:04:49Z Supersymmetric method for constructing quasi-exactly solvable potentials Tkachuk, V.M. We propose a new method for constructing the quasi-exactly solvable (QES) potentials with two known eigenstates using supersymmetric quantum mechanics. General expression for QES potentials with explicitly known energy levels and wave functions of both ground and excited states are obtained. Examples of new QES potentials are considered. Використовуючи суперсиметричну квантову механіку ми запропонували новий метод для генерування квазі-точно розв’язуваних (КТР) потенціалів з двома відомими станами. Отримані загальні вирази для КТР потенціалів з явно відомими енергетичними рівнями і хвильовими функціями основного і першого збудженого станів. Розглянуті приклади нових КТР потенціалів. 1999 Supersymmetric method for constructing quasi-exactly solvable potentials / V.M. Tkachuk // Condensed Matter Physics. — 1999. — Т. 2, № 2(18). — С. 197-204. — Бібліогр.: 25 назв. — англ. 1607-324X DOI:10.5488/CMP.2.2.197 PACS: 03.65.-w; 11.30.Pb http://dspace.nbuv.gov.ua/handle/123456789/120262 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| description |
We propose a new method for constructing the quasi-exactly solvable
(QES) potentials with two known eigenstates using supersymmetric quantum mechanics. General expression for QES potentials with explicitly
known energy levels and wave functions of both ground and excited states
are obtained. Examples of new QES potentials are considered. |
| author |
Tkachuk, V.M. |
| spellingShingle |
Tkachuk, V.M. Supersymmetric method for constructing quasi-exactly solvable potentials Condensed Matter Physics |
| author_facet |
Tkachuk, V.M. |
| author_sort |
Tkachuk, V.M. |
| title |
Supersymmetric method for constructing quasi-exactly solvable potentials |
| title_short |
Supersymmetric method for constructing quasi-exactly solvable potentials |
| title_full |
Supersymmetric method for constructing quasi-exactly solvable potentials |
| title_fullStr |
Supersymmetric method for constructing quasi-exactly solvable potentials |
| title_full_unstemmed |
Supersymmetric method for constructing quasi-exactly solvable potentials |
| title_sort |
supersymmetric method for constructing quasi-exactly solvable potentials |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
1999 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/120262 |
| citation_txt |
Supersymmetric method for constructing quasi-exactly solvable potentials / V.M. Tkachuk // Condensed Matter Physics. — 1999. — Т. 2, № 2(18). — С. 197-204. — Бібліогр.: 25 назв. — англ. |
| series |
Condensed Matter Physics |
| work_keys_str_mv |
AT tkachukvm supersymmetricmethodforconstructingquasiexactlysolvablepotentials |
| first_indexed |
2025-07-08T17:33:36Z |
| last_indexed |
2025-07-08T17:33:36Z |
| _version_ |
1837100977187979264 |
| fulltext |
Condensed Matter Physics, 1999, Vol. 2, No 2(18), pp. 197–204
Supersymmetric method for
constructing quasi-exactly
solvable potentials
V.M.Tkachuk
Chair of Theoretical Physics, Ivan Franko Lviv State University,
12 Drahomanov Str., 290005 Lviv, Ukraine
Received July 11, 1998
We propose a new method for constructing the quasi-exactly solvable
(QES) potentials with two known eigenstates using supersymmetric quan-
tum mechanics. General expression for QES potentials with explicitly
known energy levels and wave functions of both ground and excited states
are obtained. Examples of new QES potentials are considered.
Key words: supersymmetry, quantum mechanics, quasi-exactly solvable
potentials
PACS: 03.65.-w; 11.30.Pb
1. Introduction
A potential is said to be quasi-exactly solvable (QES) if a finite number of eigen-
states of corresponding Schrödinger operator can be found exactly in explicit form.
The first examples of QES potentials were given in [1–4]. Subsequently several meth-
ods for generating QES potentials were worked out and as a result many QES po-
tentials were found [5–13] (see also review book [14]). Three different methods based
respectively on a polynomial ansatz for wave functions, point canonical transforma-
tion, supersymmetric (SUSY) quantum mechanics are described in [12].
Recently, an anti-isospectral transformation, also called a duality transforma-
tion, was introduced in [15]. This transformation relates the energy levels and wave
functions of two QES potentials. In [16] a new QES potential was discovered using
this anti-isospectral transformation.
The SUSY method for constructing the QES potentials was used in [10–12].
The starting point of this method is some initial QES potential with n + 1 known
eigenstates. Then applying the technique of SUSY quantum mechanics (for review
of SUSY quantum mechanics see [17, 18]) one can calculate the supersymmetric
partner of the QES potential. From the properties of the unbroken SUSY it follows
that the supersymmetric partner is a new QES potential with n known eigenstates.
c© V.M.Tkachuk 197
V.M.Tkachuk
In addition SUSY was used to develop some generalized method for construct-
ing the so-called conditionally exactly solvable (CES) potentials in [19, 20]. The
CES potential is the one for which the eigenvalues problem for the corresponding
Hamiltonian is exactly solvable only when the potential parameters obey certain
conditions. Such a class of potentials was first considered in [21].
In the present paper we develop a new SUSY method for generating QES poten-
tials which unlike the previous papers [10–12] does not require to know the initial
QES potential to generate a new one.
2. Supersymmetric quantum mechanics
In the Witten’s model of supersymmetric quantum mechanics the SUSY partner
Hamiltonians H± read
H± = B∓B± = −1
2
d2
dx2
+ V±(x), (1)
where
B± =
1√
2
(
∓ d
dx
+W (x)
)
, (2)
V±(x) =
1
2
(
W 2(x)±W ′(x)
)
, W ′(x) =
dW (x)
dx
, (3)
W (x) is referred to as a superpotential. In this paper we shall consider the systems
on the full real line −∞ < x <∞.
Consider the equation for the energy spectrum
H±ψ
±
n (x) = E±
n ψ
±
n (x), n = 0, 1, 2, ... . (4)
As a consequence of SUSY the Hamiltonians H+ and H− have the same energy
spectrum except for the zero energy ground state. The latter exists in the case
of the unbroken SUSY. Only one of the Hamiltonians H± has a square integrable
eigenfunction corresponding to zero energy. We shall use the convention that the
zero energy eigenstate belongs to H−. Due to the factorization of the Hamiltonians
H± (see (1)) the ground state forH− satisfies the equation B−ψ−
0 (x) = 0 the solution
of which is
ψ−
0 (x) = C−
0 exp
(
−
∫
W (x)dx
)
, (5)
C−
0 is the normalization constant. Hereinafter C denotes the normalization constant
of the corresponding wave function. From the condition of square integrability of
wave function ψ−
0 (x) it follows that superpotential must satisfy the condition
sign
(
W (±∞)
)
= ±1. (6)
Note that this is the condition of the existence of unbroken SUSY.
198
Quasi-exactly solvable potentials
The eigenvalues and eigenfunctions of the Hamiltonians H+ and H− are related
by SUSY transformations
E−
n+1 = E+
n , E−
0 = 0, (7)
ψ−
n+1(x) =
1
√
E+
n
B+ψ+
n (x), ψ+
n (x) =
1
√
E−
n+1
B−ψ−
n+1(x). (8)
For a detailed description of SUSY quantum mechanics and its application for the
exact calculation of eigenstates of Hamiltonians see reviews [17,18]. The properties
of the unbroken SUSY quantum mechanics reflected in SUSY transformation (7),
(8) are used to exactly calculate the energy spectrum and wave functions. In the
present paper we use these properties to generate the QES potentials with the two
known eigenstates.
3. QES potentials with the two known eigenstates
Suppose we study a Hamiltonian H−, whose ground state is given by (5). Let
us consider the SUSY partner of H−, i.e. the Hamiltonian H+. If we calculate the
ground state of H+ we immediately find the first excited state of H− using the SUSY
transformation (7), (8). In order to calculate the ground state of H+ let us rewrite
it in the following form
H+ = H
(1)
− + ǫ = B+
1 B
−
1 + ǫ, ǫ > 0, (9)
which leads to the following relation between potential energies
V+(x) = V
(1)
− (x) + ǫ, (10)
where ǫ is the energy of the ground state of H+ since we suppose that H
(1)
− has zero
energy ground state, B±
1 and V
(1)
− (x) are given by (2) and (3) with the superpotential
W1(x).
As we see from (9) the ground state wave function of H+ is also the ground state
wave function of H
(1)
− and it satisfies the equation B−
1 ψ
+
0 (x) = 0. The solution of
this equation is
ψ+
0 (x) = C+
0 exp
(
−
∫
W1(x)dx
)
, (11)
where for square integrability of this function the superpotential W1(x) satisfies the
same condition as W (x) (6). Using (7) and (8) we obtain the energy level E−
1 = ǫ
and the wave function of the first excited state ψ−
1 (x) for H−.
From (10) we obtain the following relation between W (x) and W1(x)
W 2(x) +W ′(x) =W 2
1 (x)−W ′
1(x) + 2ǫ. (12)
Previously the same equation was used in the case of the so-called shape invariant
potentials to obtain the exact solutions of Schrödinger equation [22] (see also reviews
199
V.M.Tkachuk
[17,18]). We consider a more general case and do not restrict ourselves to the shape
invariant potentials. Note, that (12) is the Riccati equation which can not generally
be solved exactly with respect to W (x) for a given W1(x) and vice versa.
The basic idea of this paper consists in finding such a pair of W (x) and W1(x)
that satisfies equation (12). Recently it has been suggested by us in [25]. For this
purpose let us rewrite equation (12) in the following form
W ′
+(x) = W−(x)W+(x) + 2ǫ, (13)
where
W+(x) = W1(x) +W (x), (14)
W−(x) = W1(x)−W (x).
This new equation (13) can be easily solved with respect to W−(x) for a given
arbitrary function W+(x) or with respect to W+(x) for a given arbitrary function
W−(x). Then from (14) we obtain superpotentials W (x) and W1(x) which satisfy
equation (12).
3.1. Solution with respect to W
−
(x)
In this subsection we construct the QES potentials using the solution of equation
(13) with respect to W−(x)
W−(x) =
(
W ′
+(x)− 2ǫ
) /
W+(x), (15)
where W+(x) is some function of x. Note, that the superpotentialsW (x) and W1(x)
must satisfy condition (6). Then as one may see from (14) W+(x) must satisfy the
same condition (6) as W (x) and W1(x) do.
Let us consider continuous functions W+(x). Because W+(x) satisfies condition
(6) the function W+(x) must pass through zeros. Then as we see from (15) W−(x),
and thus W (x), W1(x) have poles. In order to construct the superpotential free of
singularities suppose that W+(x) has only one zero at x = x0 with the following
behaviour in the vicinity of x0 W+(x) = W ′
+(x0)(x − x0). In this case the pole of
W−(x) at x = x0 can be cancelled by choosing
ǫ =W ′
+(x0)/2. (16)
Then the superpotentials free of singularities are
W (x) =
1
2
[
W+(x)−
(
W ′
+(x)−W ′
+(x0)
) /
W+(x)
]
, (17)
W1(x) =
1
2
[
W+(x) +
(
W ′
+(x)−W ′
+(x0)
) /
W+(x)
]
.
Substituting the obtained result forW (x) into (3) we obtain QES potential V−(x)
with explicitly known wave function of ground state (5) and wave function of the
first excited state. The latter can be calculated using (11) and (8)
ψ−
1 (x) = C−
1 W+(x) exp
(
−
∫
W1(x)dx
)
. (18)
200
Quasi-exactly solvable potentials
It is indeed the wave function of the first excited state because W+(x) has one zero.
We may choose various functions W+(x) and obtain as a result various QES
potentials. The functions W+(x) must be such that ψ−
0 (x) and ψ−
1 (x) are square
integrable. If the eigenfunctions ψ−
0 (x) and ψ−
1 (x) belong to the Hilbert space of
square integrable functions in which the Hamiltonian is Hermitian then these func-
tions must be orthogonal
< ψ−
0 |ψ−
1 >= −C−
0 C
−
1
[
exp
(
−
∫
dxW+(x)
)] ∣
∣
∣
∣
∞
−∞
= 0. (19)
The wave functions must also satisfy appropriate boundary conditions.
To conclude this subsection let us consider explicit examples. Choosing W+(x)=
A (sinh(αx)− sinh(αx0)) we obtain the well known QES potential derived in [8,9]
by the method elaborated in the quantum theory of spin systems. Note that the case
x0 = 0 corresponds to Razavy potential [3]. Following the proposed SUSY method
this example is considered in details in our earlier paper [25].
Consider the function W+(x) in the polynomial form
W+(x) = ax+ bx3, a > 0, b > 0, (20)
which gives a new QES potential
V−(x) =
1
8
(a2 − 12b)x2 +
ab
4
x4 +
b2
8
x6 +
3ab
8(a+ bx2)2
+
3b
8(a+ bx2)
− a
4
. (21)
The energy levels of ground and first excited states are E−
0 = 0, E−
1 = a/2. Note,
that two energy levels of this potential do not depend on the parameter b. The wave
functions of those states read
ψ−
0 (x) = C−
0 (a+ bx2)3/4e−x2(2a+bx2)/8, (22)
ψ−
1 (x) = C−
1 x(a + bx2)1/4e−x2(2a+bx2)/8. (23)
It is worth stressing that the case b = 0 corresponds to linear harmonic oscillator.
3.2. Solution with respect to W+(x)
Equation (13) is the first order differential equation with respect to W+(x). A
general solution can be written in the following form
W+(x) = exp
(
∫
dxW−(x)
) [
2ǫ
∫
dx exp
(
−
∫
dxW−(x)
)
+ λ
]
, (24)
here λ is the constant of integration.
In order to simplify solution (24) let us choose W−(x) to be of the form
W−(x) = −φ′′(x)/φ′(x), (25)
and suppose that φ′(x) > 0. Then
W+(x) = (2ǫφ(x) + λ)/φ′(x). (26)
201
V.M.Tkachuk
Note that the constant λ can be included into the function φ(x) and thus forW+(x)
we obtain
W+(x) = 2ǫφ(x)/φ′(x). (27)
Finally for superpotentials W (x) and W1(x) we have
W (x) =
(1
2
φ′′(x) + ǫφ(x)
) /
φ′(x), (28)
W1(x) =
(
−1
2
φ′′(x) + ǫφ(x)
) /
φ′(x). (29)
Using this result for wave functions of the ground state with the energy E−
0 = 0
and excited state with E−
1 = ǫ we obtain
ψ−
0 (x) = C−
0 (φ
′(x))−1/2 exp (−ǫ ∫ dxφ(x)/φ′(x)) , E−
0 = 0,
ψ−
1 (x) = C−
1 φ(x)(φ
′(x))−1/2 exp (−ǫ ∫ dxφ(x)/φ′(x)) , E−
1 = ǫ,
(30)
where function φ(x) must be such that these wave functions are square integrable.
The condition of orthogonality in this case can by written similarly to (19)
< ψ−
0 |ψ−
1 >= −C−
0 C
−
1
[
exp
(
−ǫ
∫
dxφ(x)/φ′(x)
)] ∣
∣
∣
∣
∞
−∞
= 0. (31)
QES potential V−(x) is given by (3) with superpotential (28). Choosing different
φ(x) we obtain different QES potentials with explicitly known two eigenstates. We
shall consider a nonsingular monotonic function φ(x) with one node. Then ψ−
1 (x)
has also got one node and thus corresponds to the first excited state.
In conclusion of this subsection let us consider an explicit example. Let us put
φ(x) = ax+ bx3/3, a, b > 0. (32)
Note, that the case b = 0 corresponds to a linear harmonic oscillator. The function
(32) generates the following superpotentials
W (x) = ǫx/3 + (b+ 2aǫ/3)
x
a+ bx2
, (33)
W1(x) = ǫx/3 + (−b+ 2aǫ/3)
x
a+ bx2
, (34)
which as we see satisfy condition (6).
Substituting W (x) into (3) we obtain the following QES potential V−(x) and its
SUSY partner V+(x)
V−(x) =
A−
2
x2 +
B−
a + bx2
+
D−
(a+ bx2)2
+R−, (35)
V+(x) =
A+
2
x2 +
D+
(a + bx2)2
+R+, (36)
202
Quasi-exactly solvable potentials
where
A− = A+ = ǫ2/9, B− = b+
2
3
aǫ, R− = R+ =
ǫ
18b
(3b+ 4aǫ),
D− = − 1
18b
(27ab2 + 24a2bǫ+ 4a3ǫ2), D+ =
1
18b
(9ab2 − 4a3ǫ2).
Using (30) we obtain the wave functions of the ground and first excited states
ψ−
0 (x) = C−
0 (a + bx2)−1/2−aǫ/3b exp(−ǫx2/6), (37)
ψ−
1 (x) = C−
1 (ax+ bx3/3)(a+ bx2)−1/2−aǫ/3b exp(−ǫx2/6). (38)
The same result for QES potential (35) and wave functions (37), (38) can be
obtained using the method described in section 3.1 and taking the function W+(x)
to be of the form (27).
It is interesting to stress that in the special case ǫ = 3b/2a the QES potential
V−(x) reads
V−(x) =
b2
8a2
x2 +
2b
a+ bx2
− 4ab
(a+ bx2)2
+
3b
4a
(39)
and can be solved exactly. To see this, note that in this special case the superpo-
tential W1(x) = ǫx/3 corresponds to superpotential of a linear harmonic oscillator.
Then V
(1)
− (x) and, as a result of (10), V+(x) are the potential energies of the linear
harmonic oscillator
V+(x) =
b2
8a2
x2 +
5b
4a
. (40)
The fact that V+(x) corresponds to the linear harmonic oscillator follows also directly
from (36) because the coefficientD+ in the considered case is equal to zero. Therefore
in this case H+ is the Hamiltonian of the linear harmonic oscillator and we know
all its eigenfunctions in explicit form. Using SUSY transformations (7), (8) we can
easily calculate the energy levels and the wave functions of all the excited states of
H−. Note that in this special case V−(x) can be treated as CES potential and it
corresponds to the one studied in [19,20,23,24].
As far as we know the potential in general form (35) has not been previously
discussed. This potential is interesting from that point of view that in the case of
ǫ = 3b/2a this potential is the CES potential for which the whole energy spectrum
and the corresponding eigenfunctions can be calculated in the explicit form.
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Метод суперсиметрії для генерування
квазі-точно розв’язуваних потенціалів
В.М.Ткачук
Кафедра теоретичної фізики,
Львівський державний університет ім. Івана Франка,
290005 Львів, вул. Драгоманова, 12
Отримано 11 червня 1998 р.
Використовуючи суперсиметричну квантову механіку ми запропону-
вали новий метод для генерування квазі-точно розв’язуваних (КТР)
потенціалів з двома відомими станами. Отримані загальні вирази для
КТР потенціалів з явно відомими енергетичними рівнями і хвильо-
вими функціями основного і першого збудженого станів. Розглянуті
приклади нових КТР потенціалів.
Ключові слова: суперсиметрія, квантова механіка, квазі-точно
розв’язувані потенціали.
PACS: 03.65.-w; 11.30.Pb.
204
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