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Advances in Mathematical Physics
/
2011
/
Article
/
Tab 2
/
Research Article
Study of the Generalized Quantum Isotonic Nonlinear Oscillator Potential
Table 2
Conditions on the value of the parameters
𝑔
and
𝜇
for the quasipolynomial solutions in the case of
Δ
2
=
0
with different values of
𝑤
𝑎
2
and
𝑙
.
𝑛
𝑙
𝑤
𝑎
2
Conditions
𝐸
𝑤
𝑎
2
𝑛
,
𝑙
≡
𝐸
𝑤
𝑎
2
𝑛
,
𝑙
(
𝜇
,
𝑔
)
0
3/2
𝜇
=
−
1
1
/
3
+
𝐵
𝐸
3
/
2
1
,
0
=
−
1
/
6
(
3
5
−
1
2
𝐵
)
𝑤
𝑔
=
(
−
1
4
/
3
+
𝐵
)
(
−
1
7
/
3
+
𝐵
)
𝐵
=
1
/
3
ℜ
(
𝐴
1
/
3
+
4
3
𝐴
−
1
/
3
√
)
,
𝐴
=
−
9
8
+
9
𝑖
8
6
3
𝜇
=
−
1
1
/
3
−
𝐵
,
𝐸
1
1
,
0
=
−
1
/
6
(
3
5
+
1
2
𝐵
)
𝑤
𝑔
=
1
/
9
(
1
7
+
3
𝐵
)
(
1
4
+
3
𝐵
)
√
𝐵
=
1
/
6
ℜ
(
4
3
(
1
+
𝑖
3
)
𝐴
−
1
/
3
√
+
(
1
−
𝑖
3
)
𝐴
1
/
3
√
)
,
𝐴
=
−
9
8
+
9
𝑖
8
6
3
𝜇
=
−
1
1
/
3
−
𝐵
,
𝐸
1
1
,
0
=
−
1
/
6
(
3
5
+
1
2
𝐵
)
𝑤
𝑔
=
1
/
9
(
1
7
+
3
𝐵
)
(
1
4
+
3
𝐵
)
√
𝐵
=
1
/
6
ℜ
(
4
3
(
1
−
𝑖
3
)
𝐴
−
1
/
3
√
+
(
1
+
𝑖
3
)
𝐴
1
/
3
√
)
,
𝐴
=
−
9
8
+
9
𝑖
8
6
3
2
𝜇
=
−
1
3
/
3
+
𝐵
𝐸
3
/
2
1
,
0
=
−
1
/
6
(
4
3
−
1
2
𝐵
)
𝑤
𝑔
=
1
/
9
(
−
1
6
+
3
𝐵
)
(
−
1
9
+
3
𝐵
)
𝐵
=
1
/
3
ℜ
(
𝐴
1
/
3
+
5
5
𝐴
−
1
/
3
√
)
,
𝐴
=
−
5
5
+
1
6
5
𝑖
6
𝜇
=
−
1
3
/
3
−
𝐵
,
𝐸
1
1
,
0
=
−
1
/
6
(
4
3
+
1
2
𝐵
)
𝑤
𝑔
=
1
/
9
(
1
6
+
3
𝐵
)
(
1
9
+
3
𝐵
)
√
𝐵
=
1
/
6
ℜ
(
5
5
(
1
+
𝑖
3
)
𝐴
−
1
/
3
√
+
(
1
−
𝑖
3
)
𝐴
1
/
3
√
)
,
𝐴
=
−
5
5
+
1
6
5
𝑖
6
𝜇
=
−
1
3
/
3
−
𝐵
,
𝐸
1
1
,
0
=
−
1
/
6
(
4
3
+
1
2
𝐵
)
𝑤
𝑔
=
1
/
9
(
1
6
+
3
𝐵
)
(
1
9
+
3
𝐵
)
√
𝐵
=
1
/
6
ℜ
(
5
5
(
1
−
𝑖
3
)
𝐴
−
1
/
3
√
+
(
1
+
𝑖
3
)
𝐴
1
/
3
√
)
,
𝐴
=
−
5
5
+
1
6
5
𝑖
6