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Advances in Mathematical Physics
/
2011
/
Article
/
Tab 3
/
Research Article
Study of the Generalized Quantum Isotonic Nonlinear Oscillator Potential
Table 3
Exact eigenvalues for different values of
𝑙
and
𝑤
𝑎
2
in the case
Δ
3
=
0
.
𝑛
𝑙
𝑤
𝑎
2
Conditions
𝐸
𝑛
,
𝑙
≡
𝐸
𝑤
𝑎
2
𝑛
,
𝑙
(
𝜇
,
𝑔
)
2
−1
1
/
2
𝜇
1
=
−
6
.
3
0
1
8
7
0
8
7
8
9
9
4
1
9
8
𝐸
1
/
2
2
,
−
1
=
−
6
.
0
5
1
8
7
0
8
7
8
9
9
4
1
9
8
𝑔
1
=
7
7
.
2
2
2
9
3
0
9
7
0
4
8
6
0
9
𝜇
2
=
−
2
.
4
8
5
5
3
6
5
0
8
2
1
0
8
5
9
4
𝐸
1
/
2
2
,
−
1
=
−
2
.
2
3
5
5
3
6
5
0
8
2
1
0
8
5
9
4
𝑔
2
=
2
4
.
6
0
5
5
7
4
2
7
4
7
0
3
3
3
3
1
𝜇
1
=
−
7
.
3
9
8
1
8
2
9
8
4
3
2
6
8
7
6
𝐸
1
2
,
−
1
=
−
7
.
1
4
8
1
8
2
9
8
4
3
2
6
8
7
6
𝑔
1
=
9
7
.
7
2
4
0
2
6
3
9
1
2
1
8
1
𝜇
2
=
−
3
.
3
5
5
0
5
7
9
0
1
4
9
6
8
1
9
4
𝐸
1
2
,
−
1
=
−
3
.
1
0
5
0
5
7
9
0
1
4
9
6
8
1
9
4
𝑔
2
=
3
4
.
0
3
1
7
0
3
0
2
9
8
8
0
3
3
𝜇
3
=
0
.
9
4
9
8
1
0
5
4
1
7
5
7
4
7
5
6
𝐸
2
2
,
−
1
=
1
.
1
9
9
8
1
0
5
4
1
7
5
7
4
7
5
6
𝑔
3
=
2
.
1
5
3
0
8
7
3
5
6
4
4
6
2
5
1
4
3/2
𝜇
1
=
−
8
.
4
6
9
6
2
3
3
4
1
1
2
4
4
1
4
𝐸
3
/
2
2
,
−
1
=
−
8
.
2
1
9
6
2
3
3
4
1
1
2
4
4
1
4
𝑔
1
=
1
2
0
.
0
8
2
6
3
6
2
4
6
1
4
1
5
6
𝜇
2
=
−
4
.
2
7
7
5
0
5
2
1
2
1
6
5
0
4
𝐸
1
2
,
−
1
=
−
4
.
0
2
7
5
0
5
2
1
2
1
6
5
0
4
𝑔
2
=
4
5
.
6
8
4
5
7
6
9
0
0
9
2
4
2
8
4
𝜇
3
=
0
.
9
2
8
2
6
5
3
6
0
1
7
5
7
6
1
3
𝐸
1
2
,
−
1
=
1
.
1
7
8
2
6
5
3
6
0
1
7
5
7
6
1
3
𝑔
3
=
2
.
2
2
0
3
4
9
7
7
8
0
2
3
4
2
9
4
2
𝜇
1
=
−
9
.
5
2
5
1
2
2
1
1
5
0
6
5
3
8
6
𝐸
2
2
,
−
1
=
−
9
.
2
7
5
1
2
2
1
1
5
0
6
5
3
8
3
𝑔
1
=
1
4
4
.
3
5
3
5
6
1
8
8
2
2
3
4
6
3
𝜇
2
=
−
5
.
2
2
6
9
4
2
1
7
9
9
1
1
1
4
5
𝐸
2
2
,
−
1
=
−
4
.
9
7
6
9
4
2
1
7
9
9
1
1
1
4
5
𝑔
2
=
5
9
.
4
5
5
6
3
5
4
5
1
6
8
9
9
9
𝜇
3
=
0
.
9
1
8
6
5
0
8
1
6
9
8
5
9
2
4
4
𝐸
2
2
,
−
1
=
1
.
1
6
8
6
5
0
8
1
6
9
8
5
9
2
4
4
𝑔
3
=
2
.
2
5
0
6
6
5
2
3
8
6
1
9
2
8
4
2
0
1/2
𝜇
1
=
−
8
.
0
3
2
2
4
3
0
2
3
4
3
8
4
6
3
𝐸
2
2
,
−
1
=
−
7
.
2
8
2
2
4
3
0
2
3
4
3
8
4
6
3
𝑔
1
=
1
1
0
.
6
7
8
1
4
3
1
0
4
7
6
8
1
8
𝜇
2
=
−
4
.
3
2
8
2
5
4
7
0
6
1
2
1
8
2
𝐸
2
,
−
1
=
−
3
.
5
7
8
2
5
4
7
0
6
1
2
1
8
2
𝑔
2
=
4
6
.
3
7
5
0
6
2
3
3
1
6
7
4
7
8
2
𝜇
1
=
−
1
1
.
3
0
7
7
3
7
2
5
9
7
7
3
4
6
1
𝐸
2
2
,
−
1
=
−
1
0
.
5
5
7
7
3
7
2
5
9
7
7
3
4
6
1
𝑔
1
=
1
9
0
.
4
0
3
6
0
8
2
3
4
9
3
6
3
𝜇
2
=
−
7
.
1
8
0
5
6
4
9
0
5
7
0
3
8
6
7
𝐸
2
2
,
−
1
=
−
6
.
4
3
0
5
6
4
9
0
5
7
0
3
8
6
7
𝑔
2
=
9
3
.
4
6
3
3
3
6
8
9
3
5
4
5
3
3
𝜇
3
=
0
.
9
4
7
2
0
0
9
1
0
1
3
9
3
0
3
3
𝐸
2
2
,
−
1
=
1
.
6
9
7
2
0
0
9
1
0
1
3
9
3
0
3
3
𝑔
3
=
2
.
1
6
1
1
8
5
0
1
3
4
7
2
2
0
8
4