Electronic Journal of Differential Equations, Vol. 2008(2008), No. 64, pp. 1–15.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu (login: ftp)
MATRIX ELEMENTS FOR SUM OF POWER-LAW POTENTIALS
IN QUANTUM MECHANIC USING GENERALIZED
HYPERGEOMETRIC FUNCTIONS
QUTAIBEH D. KATATBEH, MA’ZOOZEH E. ABU-AMRA
Abstract. In this paper we derive close form for the matrix elements for
Ĥ = −∆ + V , where V is a pure power-law potential. We use trial functions
of the form
s
√
p q
2β γ/2 (γ)n γ−1/2 − β rq
ψn (r) =
r
e 2
βr ),
p F1 (−n, a2 , . . . , ap ; γ;
n!Γ(γ)
for β, q, γ > 0 to obtain the matrix elements for Ĥ. These formulas are then
optimized with respect to variational parameters β, q and γ to obtain accurate upper bounds for the given nonsolvable eigenvalue problem in quantum
mechanics. Moreover, we write the matrix elements in terms of the generalized hypergeomtric functions. These results are generalization of those found
earlier in [2], [8]–[16] for power-law potentials. Applications and comparisons
with earlier work are presented.
1. Introduction
In 1998 Hall et al [15] found a closed form expressions for the singular-potential
integrals hm | r−α | ni that obtained with respect to the Gol’dman and Krivchenkov
eigenfunctions for the singular Hamiltonian with ~2 = 2m = 1,
d2
A
+ βr2 + 2 , (β > 0, A ≥ 0).
(1.1)
dr2
r
We present a variational analysis of the generalized spiked harmonic oscillator
Hamiltonian [2, 3, 4], [7]–[19] of the form
Ĥ = −
d2
A
λ
+ βr2 + 2 + α , 0 ≤ r < ∞,
(1.2)
dr2
r
r
where α and λ are positive real numbers. By writing Ĥ = H (0) + λV with H (0)
standing for the generalized spiked harmonic-oscillator Hamiltonian, and V (r) =
r−α , Hall et al [8]–[16] used the basis set
s
γ
√
p
− β 2
2β 2 (γ)n
ψn (r) = Tn rγ−1/2 e 2 r 1 F1 (−n; γ; βr2 ), Tn =
,
n!Γ(γ)
Ĥ = −
2000 Mathematics Subject Classification. 34L15, 34L16, 81Q10, 35P15.
Key words and phrases. Schrödinger equation; variational technique; eigenvalues;
upper bounds; analytical computations.
c
2008
Texas State University - San Marcos.
Submitted February 19, 2008 Published April 28, 2008.
1
2
Q. D. KATATBEH, M. E. ABU-AMRA,
EJDE-2008/64
n = 0, 1, 2, 3, . . ., constructed from the normalized solutions of H (0) ψn = En ψn to
evaluate the matrix elements of Ĥ. They found that
p
Hmn = hm | Ĥ | ni = 2 β(2n + γ)δmn + λhm | r−α | ni m, n = 0, 1, 2, . . . , N − 1
where
r
(γ)n (γ)m Γ(γ − α2 )( α2 )n
n!m!
(γ)n Γ(γ)
α
α
α
×3 F2 (−m, γ − , 1 − ; γ, 1 − − n; 1).
2
2
2
Now, by considering the problem in a finite dimensional subspace and choosing a
suitable trial functions as a linear combination of the form
s
γ
√
p
− β q
2β 2 (γ)n
γ−1/2
r
q
2
βr ), Tn =
ψn (r) = Tn r
e
(1.3)
p F1 (−n, a2 , . . . , ap ; γ;
n!Γ(γ)
hm | r
−α
m+n
| ni = (−1)
β
α
4
This basis is more general than the basis used in [2] and [14], because the number
of variational parameters for the exponential function and the generalized hypergeometric function play an important rule for improving the numerical results for our
eigenvalue problem. Herein, p F1 stands for the hypergeometric function defined by
p Fq (α1 , α2 , . . . , αp ; β1 , β2 , . . . , βq ; x) =
∞
X
Πpi=1 (αi )k xk
;
Πqj=1 (βj )k k!
(1.4)
k=0
where p and q are non-negative integers and βj (j = 1, 2, . . . , q) cannot be a nonpositive integer [1]. The expression (a)n is the Pochhammer symbol defined by the
relations
Γ(a + n)
(a)n = a(a + 1) . . . (a + n − 1) =
Γ(a)
(a)0 = 1
(a)−n =
(−1)n
,
(1 − a)n
where n is an integer (positive, negative or zero).
2. Variational Technique
In this section we review the well known variational technique for eigenvalue
problem in quantum mechanics [5]. We consider a non solvable eigenvalue problem
of the form Ĥ = −4 + V (r) in N -dimensional space over a finite subspace in the
domain of its angular momentum subspace. Our variational technique is based on
forming trial wave function from a linear combination of some of orthonormal basis
function ψn (r), n = 1, 2, . . . , D such that,
Ψ(r) =
D
X
cn ψn (r)
(2.1)
n=1
where the cn are variational parameters. The variational energy EΨ we obtain with
this trial function is given by,
R
R ∗
PD PD
c∗n cm ψn∗ Ĥψm dτ
Ψ ĤΨdτ
n=1
m=1
= PD PD
(2.2)
EΨ = R ∗
R
∗
∗
Ψ Ψdτ
n=1
m=1 cn cm ψn ψm dτ
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MATRIX ELEMENTS
Now, defining the matrix element of Ĥ as
Z
Hnm = ψn∗ Ĥψm dτ
and using the orthonormal property of the ψ functions, we obtain that
PD PD
c∗ cm Hnm
EΨ = n=1PDm=1 n
2
n=1 | cn |
or
D
D X
D
X
X
EΨ
| cn |2 =
c∗n cm Hnm
n=1
3
(2.3)
(2.4)
n=1 m=1
We seek the minimum value of EΨ as a function of all the cn . By differentiating
with respect to ci , we obtain
D
D
X
∂E X
2
∗
c∗n Hni
| cn | + Eci =
∂ci n=1
n=1
and setting
∂E
∂ci
= 0, we have
D
X
c∗n Hni − Ec∗i = 0.
(2.5)
n=1
We can derive one equation similar to (2.5) for each possible value of i, i = 1, . . . , D,
and we may take (2.5) to represent a set of linear equations with each equation
characterized by a different value of i. According to the theory of linear algebraic
equations, there is a nontrivial solution, if and only if the determinant of the coefficients vanishes or if and only if
H11 − E
H12
H13 · · ·
H1D H21
H22 − E H23 · · ·
H2D =0
..
..
..
..
.
.
.
.
HD1
HD2
HD3 · · · HDD − E This determinant is called a secular determinant. If we use a basis set that is not
orthonormal, we define the matrix element Snm by
Z
Snm = ψn ψm dτ ,
in this case the equation corresponding to (2.5) and the secular determinant are
D
X
cn Hin − ESin cn = 0
n=1
and the secular determinant
H11 − ES11
H21 − ES21
..
.
HD1 − ESD1
can be written as
H12 − ES12
H22 − ES22
..
.
···
···
HD2 − ESD2
···
= 0.
HDD − ESDD H1D − ES1D
H2D − ES2D
..
.
(2.6)
4
Q. D. KATATBEH, M. E. ABU-AMRA,
EJDE-2008/64
3. Derivation of the Matrix Elements
We divide the problem of computing the matrix elements for Schrödinger equation into simple parts. In the first lemma we compute the matrix elements for
singular potential in three-spatial dimensions, it will be used to compute the matrix elements for the kinetic energy.
Lemma 3.1. For the generalized spiked harmonic oscillator Hamiltonian (1.2), the
expectation values of the operator V (x) = r−α with respect to a trial basis (1.3) are
given by
m
hψm | r−α | ψn i =
α−2γ
1
2γ − α X (−m)k (a2 )k · · · (ap )k (
Tm Tn β 2q Γ(
)
q
q
(γ)k k!
2γ−α
q )k
k=0
2γ − α
×p+1 F1 (−n, a2 , . . . , ap ,
+ k; γ; 1)
q
(3.1)
Proof. Using (1.3), it follows that
hψm | r−α | ψn i =
Z
∞
√
r2γ−α−1 e−
βr q
p F1 (−m, a2 , . . . , ap ; γ;
p q
βr )
0
×p F1 (−n, a2 , . . . , ap ; γ;
(3.2)
p q
βr )dr
Using the series representation (1.4) of the hypergeometric function p F1 , yields
−α
rmn
= Tm Tn
×β
n
m X
X
(−m)k (a2 )k · · · (ap )k (−n)l (a2 )l · · · (ap )l
(γ)k (γ)l k!l!
k=0 l=0
Z ∞
√
l+k
2γ−α+qk+ql−1 − βr q
2
r
e
(3.3)
dr
0
By restoring to the integral representation of gamma function, we obtain under the
condition 2γ−α
q + k + l > 0, that
−α
rmn
m
=
n
l+k
X X (−m)k (a2 )k · · · (ap )k (−n)l (a2 )l · · · (ap )l β 2
1
2γ − α
Tm Tn
Γ(
+ k + l)
q
(γ)k (γ)l
k!l!
q
k=0 l=0
=
m hX
n
i
α−2γ X
1
(−n)l (a2 )l · · · (ap )l 2γ − α
Tm Tn β 2q
Γ(
+ k + l)
q
(γ)l l!
q
k=0
l=0
(−m)k (a2 )k · · · (ap )k
×
(γ)k k!
(3.4)
On the other hand, using the definition of the Pochhammer symbols and the series representation of the hypergeometric functions (1.4), the finite sum inside the
EJDE-2008/64
MATRIX ELEMENTS
5
bracket collapses to
n
X
(−n)l (a2 )l · · · (ap )l
(γ)l l!
l=0
=
Γ(
2γ − α
+ k + l)
q
n
X
+ k)l
(−n)l (a2 )l · · · (ap )l ( 2γ−α
q
(γ)l l!
l=0
=p+1 F1 (−n, a2 , . . . , ap ,
Γ(
2γ − α
+ k)
q
(3.5)
2γ − α
2γ − α
+ k; γ; 1)Γ(
+ k)
q
q
Consequently, we arrive at
−α
rmn
=
m
α−2γ X
1
2γ − α
(−m)k (a2 )k · · · (ap )k
Tm Tn β 2q
Γ(
+ k)
q
q
(γ)k k!
k=0
2γ − α
+ k; γ; 1)
q
2γ−α
m
α−2γ
2γ − α X (−m)k (a2 )k · · · (ap )k ( q )k
1
2q
Γ(
)
= Tm Tn β
q
q
(γ)k k!
×p+1 F1 (−n, a2 , . . . , ap ,
k=0
2γ − α
×p+1 F1 (−n, a2 , . . . , ap ,
+ k; γ; 1).
q
This completes the proof.
For the case of γ > 1 and α = 2, we have
−2
rmn
=
m
1−γ X (−m)k (a2 )k · · · (ap )k
2(γ − 1)
1
Tm Tn β q
Γ(
+ k)
q
(γ)k k!
q
k=0
(3.6)
2(γ − 1)
×p+1 F1 (−n, a2 , . . . , ap ,
+ k; γ; 1).
q
3.1. Matrix Elements for V (r) = rα . We now use the suggested basis (1.3) to
compute the matrix elements for the power-law potential operators rα , α > 0.
This kind of computation is important for may problems in the literature, such as
Kratzer potential [6]. Whose calculation is achieved by means of following lemma.
Lemma 3.2. For the generalized spiked harmonic oscillator Hamiltonian (1.2), the
expectation values of the operator V (x) = rα with respect to a trial basis (1.3) are
given by
m
hψm | rα | ψn i =
α+2γ
1
2γ + α X (−m)k (a2 )k · · · (ap )k (
Tm Tn β − 2q Γ(
)
q
q
(γ)k (k!
2γ+α
q )k
k=0
2γ + α
×p+1 F1 (−n, a2 , . . . , ap ,
+ k; γ; 1)
q
(3.7)
Proof. Using (1.3), it immediately follows that
Z ∞
√ q
p
hψm | rα | ψn i =
r2γ+α−1 e− βr p F1 (−m, a2 , . . . , ap ; γ; βrq )
0
p
×p F1 (−n, a2 , . . . , ap ; γ; βrq )dr
(3.8)
6
Q. D. KATATBEH, M. E. ABU-AMRA,
EJDE-2008/64
Using the series representation (1.4) of the hypergeometric function p F1 , yields
α
rmn
= Tm Tn
×β
m X
n
X
(−m)k (a2 )k · · · (ap )k (−n)l (a2 )l · · · (ap )l
(γ)k (γ)l k!l!
k=0 l=0
Z ∞
√
l+k
2γ+α+qk+ql−1 − βr q
2
r
e
(3.9)
dr.
0
Using the integral representation of gamma function, we obtain under the condition
2γ+α
q + k + l > 0 that
α
rmn
= Tm Tn
β−
α+2γ
2q
q
m X
n
X
(−m)k (a2 )k · · · (ap )k (−n)l (a2 )l · · · (ap )l 1
(γ)k (γ)l
k!l!
k=0 l=0
2γ + α
× Γ(
+ k + l)
q
α+2γ
m
n
i
β − 2q X h X (−n)l (a2 )l · · · (ap )l 2γ + α
= Tm Tn
Γ(
+ k + l)
q
(γ)l l!
q
k=0
(3.10)
l=0
(−m)k (a2 )k · · · (ap )k
×
(γ)k k!
On the other hand, by using the definition of the Pochhammer symbols and the
series representation of the hypergeometric functions (1.4), the finite sum inside
the bracket collapses to
n
X
(−n)l (a2 )l · · · (ap )l
(γ)l l!
l=0
=
Γ(
2γ + α
+ k + l)
q
n
X
(−n)l (a2 )l · · · (ap )l ( 2γ+α
+ k)l
q
(γ)l l!
l=0
=p+1 F1 (−n, a2 , . . . , ap ,
Γ(
2γ + α + 2
+ k)
q
(3.11)
2γ + α
2γ + α
+ k; γ; 1)Γ(
+ k).
q
q
Finally, we arrive at
α
rmn
= Tm Tn
β−
α+2γ
2q
q
m
X
(−m)k (a2 )k · · · (ap )k
k=0
(γ)k k!
Γ(
2γ + α
+ k)
q
2γ + α
+ k; γ; 1)
q
2γ+α
m
2γ + α X (−m)k (a2 )k · · · (ap )k ( q )k
Γ(
)
q
(γ)k k!
×p+1 F1 (−n, a2 , . . . , ap ,
= Tm Tn
β−
α+2γ
2q
q
k=0
2γ + α
×p+1 F1 (−n, a2 , . . . , ap ,
+ k; γ; 1).
q
The proof is complete.
EJDE-2008/64
MATRIX ELEMENTS
7
On the other hand, for the case of γ > 0 and α = 2 > 0, we have that
m
1 1+γ X (−m)k (a2 )k · · · (ap )k 2(γ + 1)
hψm | r2 | ψn i = Tm Tn β − q
Γ(
+ k)
q
(γ)k k!
q
k=0
(3.12)
2(γ + 1)
+ k; γ; 1).
×p+1 F1 (−n, a2 , . . . , ap ,
q
Lemma 3.3. For γ > 0, and m, n = 0, 1, 2, . . . ,
Smn = hψm | ψn i
2γ
m
β − q 2γ X (−m)k (a2 )k · · · (ap )k ( q )k
Γ( )
= Tm Tn
q
q
(γ)k k!
γ
k=0
×p+1 F1 (−n, a2 , . . . , ap ,
(3.13)
2γ
+ k; γ; 1)
q
Proof. Using (1.3), it follows that
Z ∞
√ q
p
Smn =
r2γ−1 e− βr p F1 (−m, a2 , . . . , ap ; γ; βrq )
0
p
×p F1 (−n, a2 , . . . , ap ; γ; βrq )dr
(3.14)
By means of the series representation (1.4) of the hypergeometric function p F1 ,
that
Smn = Tm Tn
Z
×
l+k
n
m X
X
(−m)k (a2 )k · · · (ap )k (−n)l (a2 )l · · · (ap )l β 2
k=0 l=0
∞
√
2γ+qk+ql−1 − βr q
r
e
(γ)k (γ)l
k!l!
(3.15)
dr.
0
Further, after restoring the integral representation of the gamma function and a
change of variables, we obtain for 2γ
q + k + l > 0, that
Smn = Tm Tn
γ
2γ
m n
β − q X X (−m)k (a2 )k · · · (ap )k (−n)l (a2 )l · · · (ap )l Γ( q + k + l)
q
(γ)k (γ)l
k!l!
− γq
k=0 l=0
m hX
n
X
i (−m) (a ) · · · (a )
(−n)l (a2 )l · · · (ap )l 2γ
k 2 k
p k
Γ(
+ k + l)
q
(γ)l l!
q
(γ)k k!
k=0 l=0
(3.16)
On the other hand, by using the definition of the pochhammer symbols and the
series representation of the hypergeometric functions (1.4), the finite sum inside
the bracket collapses to
= Tm Tn
β
n
X
(−n)l (a2 )l · · · (ap )l
l=0
=
(γ)l l!
Γ(
2γ
+ k + l)
q
n
X
(−n)l (a2 )l · · · (ap )l ( 2γ
q + k)l
l=0
(γ)l l!
=p+1 F1 (−n, a2 , . . . , ap ,
Γ(
2γ
+ k)
q
2γ
2γ
+ k; γ; 1)Γ(
+ k)
q
q
(3.17)
8
Q. D. KATATBEH, M. E. ABU-AMRA,
EJDE-2008/64
Consequently,
Smn
γ
m
β − q X 2γ
(−m)k (a2 )k · · · (ap )k
Γ(
= Tm Tn
+ k)
q
q
(γ)k k!
k=0
2γ
+ k; γ; 1)
q
γ
2γ
m
β − q 2γ X (−m)k (a2 )k · · · (ap )k ( q )k
= Tm Tn
Γ( )
q
q
(γ)k k!
×p+1 F1 (−n, a2 , . . . , ap ,
k=0
×p+1 F1 (−n, a2 , . . . , ap ,
2γ
+ k; γ; 1)
q
This completes the proof.
Lemma 3.4. For γ > 1, and m, n = 1, 2, . . . , the matrix elements of the generalized
spiked harmonic oscillator Hamiltonians (1.2) can be written as
Hmn
= Tm Tn [β
1−γ
q
m
(A − γ 2 + 2γ − 43 ) X (−m)k (a2 )k · · · (ap )k 2γ − 2
Γ(
+ k)
q
(γ)k k!
q
k=0
2γ − 2
+ k; γ; 1)
×p+1 F1 (−n, a2 , . . . , ap ,
q
m
1 X (−m)k (a2 )k · · · (ap )k 2γ + 2
− 1+γ
q
+B
Γ(
+ k)
q
(γ)k k!
q
k=0
×p+1 F1 (−n, a2 , . . . , ap ,
−β
1−γ
q
2γ + 2
+ k; γ; 1)
q
m
q X (−m)k (a2 )k · · · (ap )k 2γ − 2
Γ(
+ 2 + k)
4
(γ)k k!
q
k=0
×p+1 F1 (−n, a2 , . . . , ap ,
+β
1−γ
q
2γ − 2
+ 2 + k; γ; 1)
q
m
q X (−m)k (a2 )k · · · (ap )k 2γ − 2
(γ − 1 + )
Γ(
+ 1 + k)
2
(γ)k k!
q
k=0
2γ − 2
+ 1 + k; γ; 1)
q
m
na2 · · · ap X (−m)k (a2 )k · · · (ap )k
×p+1 F1 (−n, a2 , . . . , ap ,
+β
1−γ
q
(q + 2(γ − 1))
γ
k=0
×p+1 F1 (1 − n, 1 + a2 , . . . , 1 + ap ,
−β
1−γ
q
q
(γ)k k!
Γ(
2γ − 2
+ 1 + k)
q
2γ − 2
+ 1 + k; 1 + γ; 1)
q
m
na2 · · · ap X (−m)k (a2 )k · · · (ap )k 2γ − 2
Γ(
+ 2 + k)
γ
(γ)k k!
q
k=0
2γ − 2
+ 2 + k; 1 + γ; 1)
q
m
1−γ
n(−1 + n)a2 (1 + a2 ) · · · ap (1 + ap ) X (−m)k (a2 )k · · · (ap )k
−β q q
γ(1 + γ)
(γ)k k!
×p+1 F1 (1 − n, 1 + a2 , . . . , 1 + ap ,
k=0
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MATRIX ELEMENTS
9
2γ − 2
2γ − 2
+ 2 + k)p+1 F1 (2 − n, 2 + a2 , . . . , 2 + ap ,
+ 2 + k; 2 + γ; 1)
q
q
2γ−α
m
α−2γ 1
2γ − α X (−m)k (a2 )k · · · (ap )k ( q )k
)
+ λβ 2q Γ(
q
q
(γ)k k!
× Γ(
k=0
×p+1 F1 (−n, a2 , . . . , ap ,
2γ − α
+ k; γ; 1)
q
(3.18)
Proof. By writing the Hamiltonian H in compactified form as
d2
2
−2
−α
| ψn +βrmn
+ Armn
+ λrmn
dr2
Hmn =≺ ψm | −
2
−2
−α
where rmn
is given by (3.12) rmn
is given by (3.6), and rmn
is given by (3.1). By
using (1.3) it is follows that
Z ∞
√
√
B q
1
Hmn = −
rγ− 2 e− 2 r p F1 (−m, a2 , . . . , ap ; γ; Brq )
0
√
√
B q
1
d2
× 2 (rγ− 2 e− 2 r p F1 (−n, a2 , . . . , ap ; γ; Brq ))dr
dr
2
−2
−α
+ βrmn
+ Armn
+ λrmn
(3.19)
We denote the first term on the right-hand side of (3.19) by Imn and have by means
of the series representation (1.4) of the hypergeometric function p F1 , that
m
n
l+k
3 X X (−m)k (a2 )k · · · (ap )k (−n)l (a2 )l · · · (ap )l B 2
Imn = (−γ 2 + 2γ − )
4
(γ)k (γ)l
k!l!
k=0 l=0
Z ∞
√ q
×
r2γ−3+qk+ql e− Br dr
0
−B
l+k
m n
q 2 X X (−m)k (a2 )k · · · (ap )k (−n)l (a2 )l · · · (ap )l B 2
4
(γ)k (γ)l
k!l!
k=0 l=0
∞
Z
√
r2γ−3+2q+qk+ql e−
×
Br q
dr
0
m
n
l+k
√
q X X (−m)k (a2 )k · · · (ap )k (−n)l (a2 )l · · · (ap )l B 2
Bq(γ − 1 + )
2
(γ)k (γ)l
k!l!
Z
∞
+
k=0 l=0
×
0
n
X
√
r2γ−3+q+qk+ql e−
Br q
√
dr +
Bq(2(γ − 1) + q)
m
na2 · · · ap X
γ
k=0
(−m)k (a2 )k · · · (ap )k (1 − n)l (1 + a2 )l · · · (1 + ap )l
(γ)k (1 + γ)l
l=0
l+k Z ∞
√ q
B 2
×
r2γ−3+q+qk+ql e− Br dr
k!l! 0
m n
na2 · · · ap X X (−m)k (a2 )k · · · (ap )k (1 − n)l (1 + a2 )l · · · (1 + ap )l
− Bq 2
γ
(γ)k (1 + γ)l
k=0 l=0
l+k Z ∞
√ q
B 2
×
r2γ−3+2q+qk+ql e− Br dr
k!l! 0
×
10
Q. D. KATATBEH, M. E. ABU-AMRA,
EJDE-2008/64
n(−1 + n)a2 (1 + a2 ) · · · ap (1 + ap )
γ(1 + γ)
m X
n
X
(−m)k (a2 )k · · · (ap )k (2 − n)l (2 + a2 )l · · · (2 + ap )l
− Bq 2
×
(γ)k (2 + γ)l
k=0 l=0
l+k
×
B 2
k!l!
∞
Z
√
r2γ−3+2q+qk+ql e−
Br q
dr
0
Further, after restoring the integral representation of the gamma function and a
change of variables, we obtain for 2(γ−1)
+ k + l > 0, 2(γ−1)
+ 1 + k + l > 0 and
q
q
2(γ−1)
+
q
2 + k + l > 0 that
Imn
=B
1−γ
q
m
n
(−γ 2 + 2γ − 34 ) X X (−m)k (a2 )k · · · (ap )k (−n)l (a2 )l · · · (ap )l
q
(γ)k (γ)l
k=0 l=0
×
Γ( 2(γ−1)
q
−B
+B
l−γ
q
k!l!
2(γ−1)
m n
q X X (−m)k (a2 )k · · · (ap )k (−n)l (a2 )l · · · (ap )l Γ( q + 2 + k + l)
4
l−γ
q
+ k + l)
(γ)k (γ)l
k=0 l=0
m
k!l!
n
q X X (−m)k (a2 )k · · · (ap )k (−n)l (a2 )l · · · (ap )l
(γ − 1 + )
2
(γ)k (γ)l
k=0 l=0
×
×
+ 1 + k + l)
Γ( 2(γ−1)
q
k!l!
+B
1−γ
q
m
na2 · · · ap X
(2(γ − 1) + q)
γ
k=0
2(γ−1)
n
X
(−m)k (a2 )k · · · (ap )k (1 − n)l (1 + a2 )l · · · (1 + ap )l Γ( q + 1 + k + l)
(γ)k (1 + γ)l
l=0
−B
1−γ
q
q
k!l!
m n
na2 · · · ap X X (−m)k (a2 )k · · · (ap )k (1 − n)l (1 + a2 )l · · · (1 + ap )l
γ
(γ)k (1 + γ)l
k=0 l=0
×
×
×
Γ( 2(γ−1)
q
+ 2 + k + l)
k!l!
−B
1−γ
q
q
n(−1 + n)a2 (1 + a2 ) · · · ap (1 + ap )
γ(1 + γ)
m X
n
X
(−m)k (a2 )k · · · (ap )k (2 − n)l (2 + a2 )l · · · (2 + ap )l
k=0 l=0
Γ( 2(γ−1)
q
= −B
(γ)k (2 + γ)l
+ 2 + k + l)
k!l!
1−γ
q
m
n
k=0
l=0
(−γ 2 + 2γ − 34 ) X h X (−n)l (a2 )l · · · (ap )l Γ(
q
(γ)k (γ)l
2(γ−1)
q
l!
(−m)k (a2 )k · · · (ap )k
k!
2(γ−1)
n
m hX
X
l−γ q
(−n)l (a2 )l · · · (ap )l Γ( q + 2 + k + l) i
q
−B
4
(γ)k (γ)l
l!
×
k=0
l=0
+ k + l) i
EJDE-2008/64
MATRIX ELEMENTS
11
m
×
l−γ
(−m)k (a2 )k · · · (ap )k
q X
+ B q (γ − 1 + )
k!
2
k=0
2(γ−1)
n
hX
(−m)k (a2 )k · · · (ap )k (−n)l (a2 )l · · · (ap )l Γ( q + 1 + k + l) i
×
(γ)k (γ)l
l!
l=0
(−m)k (a2 )k · · · (ap )k
×
k!
m
n
1−γ
na2 · · · ap X h X (1 − n)l (1 + a2 )l · · · (1 + ap )l
+ B q (2(γ − 1) + q)
γ
(γ)k (1 + γ)l
k=0
×
−B
×
×
1−γ
q
q
+ 1 + k + l) i (−m)k (a2 )k · · · (ap )k
k!
k!
2(γ−1)
m hX
n
X
na2 · · · ap
(1 − n)l (1 + a2 )l · · · (1 + ap )l Γ( q + 2 + k + l) i
γ
k=0
(γ)k (1 + γ)l
l=0
l!
1−γ
(−m)k (a2 )k · · · (ap )k
n(−1 + n)a2 (1 + a2 ) · · · ap (1 + ap )
−B q q
k!
γ(1 + γ)
2(γ−1)
m
n
X h X (2 − n) (2 + a ) · · · (2 + a ) Γ(
+ 2 + k + l) i
l
k=0
×
l=0
Γ( 2(γ−1)
q
2 l
p l
q
(γ)k (2 + γ)l
l=0
l!
(−m)k (a2 )k · · · (ap )k
k!
(3.20)
We denote the finite sum inside the brackets in the expression above as I1mn , I2mn ,
. . . , I6mn . On the other hand, by using of the definition of the pochhammer symbols
and the series representation of the hypergeometric functions (1.4), the finite sum
inside the bracket collapses to
I1mn =
n
X
(−n)l (a2 )l · · · (ap )l
l=0
=
(γ)l l!
(γ)l l!
=p+1 F1 (−n, a2 , . . . , ap ,
n
X
(−n)l (a2 )l · · · (ap )l
l=0
=
2(γ − 1)
+ k + l)
q
n
X
(−n)l (a2 )l · · · (ap )l ( 2(γ−1)
+ k)l
q
l=0
I2mn =
Γ(
(γ)l l!
Γ(
l=0
=p+1 F1 (−n, a2 , . . . , ap ,
2(γ − 1)
+ k)
q
(3.21)
2(γ − 1)
2(γ − 1)
+ k; γ; 1)Γ(
+ k)
q
q
2(γ − 1)
+ 1 + k + l)
q
n
X
(−n)l (a2 )l · · · (ap )l ( 2(γ−1)
+ 1 + k)l
q
(γ)l l!
Γ(
Γ(
2(γ − 1)
+ 1 + k)
q
2(γ − 1)
2(γ − 1)
+ 1 + k; γ; 1)Γ(
+ 1 + k)
q
q
(3.22)
12
Q. D. KATATBEH, M. E. ABU-AMRA,
I3mn =
n
X
(−n)l (a2 )l · · · (ap )l
l=0
=
(γ)l l!
(γ)l l!
=p+1 F1 (−n, a2 , . . . , ap ,
Γ(
2(γ − 1)
+ 2 + k)
q
(1 + γ)l l!
Γ(
2(γ − 1)
+ 1 + k + l)
q
n
X
+ 1 + k)l
(1 − n)l (1 + a2 )l · · · (1 + ap )l ( 2(γ−1)
q
(1 + γ)l l!
l=0
=p+1 F1 (1 − n, 1 + a2 , . . . , 1 + ap ,
× Γ(
(3.23)
2(γ − 1)
2(γ − 1)
+ 2 + k; γ; 1)Γ(
+ 2 + k)
q
q
n
X
(1 − n)l (1 + a2 )l · · · (1 + ap )l
l=0
=
2(γ − 1)
+ 2 + k + l)
q
n
X
+ 2 + k)l
(−n)l (a2 )l · · · (ap )l ( 2(γ−1)
q
l=0
I4mn =
Γ(
EJDE-2008/64
Γ(
2(γ − 1)
+ 1 + k)
q
2(γ − 1)
+ 1 + k; 1 + γ; 1)
q
2(γ − 1)
+ 1 + k)
q
(3.24)
I5mn =
n
X
(1 − n)l (1 + a2 )l · · · (1 + ap )l
l=0
=
(1 + γ)l l!
Γ(
2(γ − 1)
+ 2 + k + l)
q
n
X
(1 − n)l (1 + a2 )l · · · (1 + ap )l ( 2(γ−1)
+ 2 + k)l
q
(1 + γ)l l!
l=0
=p+1 F1 (1 − n, 1 + a2 , . . . , 1 + ap ,
× Γ(
Γ(
2(γ − 1)
+ 2 + k)
q
2(γ − 1)
+ 2 + k; 1 + γ; 1)
q
2(γ − 1)
+ 2 + k)
q
(3.25)
I6mn =
2(γ−1)
n
X
(2 − n)l (2 + a2 )l · · · (2 + ap )l Γ( q + 2 + k + l)
l=0
=
(2 + γ)l
l!
n
X
+ 2 + k)l
(2 − n)l (2 + a2 )l · · · (2 + ap )l ( 2(γ−1)
q
(2 + γ)l l!
l=0
=p+1 F1 (2 − n, 2 + a2 , . . . , 2 + ap ,
× Γ(
Γ(
2(γ − 1)
+ 2 + k)
q
2γ − 2
+ 2 + k; 2 + γ; 1)
q
2(γ − 1)
+ 2 + k)
q
(3.26)
Substituting (3.21), (3.22), (3.23), (3.24), (3.25) and (3.26) into (3.20) we obtain
m
3 X
2
1−γ (γ − 2γ +
(−m)k (a2 )k · · · (ap )k
4)
Imn = −B q
I1mn
q
(γ)k k!
k=0
EJDE-2008/64
−B
MATRIX ELEMENTS
13
m
q X (−m)k (a2 )k · · · (ap )k
I2mn
4
(γ)k k!
1−γ
q
k=0
+B
1−γ
q
m
q X (−m)k (a2 )k · · · (ap )k
(γ − 1 + )
I3mn
2
(γ)k k!
k=0
+B
1−γ
q
(q + 2(γ − 1))
m
na2 · · · ap X (−m)k (a2 )k · · · (ap )k
I4mn
γ
(γ)k k!
k=0
m
1−γ
na2 · · · ap X (−m)k (a2 )k · · · (ap )k
I5mn
−B q q
γ
(γ)k k!
k=0
−B
1−γ
q
m
n(−1 + n)a2 (1 + a2 ) · · · ap (1 + ap ) X (−m)k (a2 )k · · · (ap )k
q
I6mn
γ(1 + γ)
(γ)k k!
k=0
This completes the proof.
In the following tables, we used the derived formulas to compute upper bounds
for many problems of interest in the literature. We compare our results with other
authors who analyze the spectrum for this kind of problems. One of the advantage
and the purpose of such formulas is to provide the researcher with simple method
to compute the spectrum for sum of power-law potentials in the literature other
than the saving in the time in these computations.
d2
λ
2
Table 1 shows the eigenvalues of H = − dr
+ r2.5
2 + V (r), where V (r) = r
anharmonic oscillator potential for different values of λ using
s
γ
p
2β 2 (γ)n γ− 1 − √βrq
r 2 e 2 3 F1 (−n, 0.2, 0.07; γ; βrq )
ψn (r) =
n!Γ(γ)
obtained using the computed matrix elements (3.18) and minimizing with respect
to q, γ, andβ. The exponent (n) refer to the dimensions of the matrix used for the
variational computations.
λ
100
10
1
0.5
0.01
0.001
q
1.824
1.811
1.801
1.807
1.95
1.993
γ
11.117
4.797
2.553
2.229
1.55
1.506
β
2.4115
2.154
1.917
1.8169
1.132
1.017
E (3)
17.541 916
7.735 317
4.318 485
3.850 244
3.037 422
3.004 046 2
E (4)
17.541 912
7.735 256
4.318 181
3.850 028
3.037 421
3.004 046 2
E (5)
17.541 912
7.735 254
4.318 180
3.849 862
3.037 415
3.004 046 1
E
17.541 890
7.735 111
4.317 312
3.848 553
3.036 665
3.004 014
Table 1.
Table 2 Shows a comparison between the results ofE HS and the results of the
present workE U using
s
γ
p
2β 2 (γ)n γ− 1 − √βrq
ψn (r) =
r 2 e 2 3 F1 (−n, 0.2, 0.07; γ; βrq )
n!Γ(γ)
2
λ
d
2
and H = − dr
+ r2.5
anharmonic oscillator potential
2 + V (r) where V (r) = r
for different values of λ, obtained using the computed matrix elements (3.18) and
minimizing with respect to q, γ, andβ. The exponent (n) refer to the dimensions
of the matrix used for the variational computations.
14
λ
100
10
1
0.5
0.01
0.001
Q. D. KATATBEH, M. E. ABU-AMRA,
q
1.824
1.811
1.801
1.807
1.95
1.993
γ
11.117
4.797
2.553
2.229
1.55
1.506
β
2.4115
2.154
1.917
1.8169
1.132
1.017
E HS
17.541 890(30)
7.735 637(30)
4.323 263(30)
3.869 547(30)
3.039 244(30)
3.004 074(30)
HS
Em
17.542 040(5)
7.735 596(5)
4.318 963(5)
3.850 823(5)
3.037 474(5)
3.004 047(5)
EJDE-2008/64
EU
17.541 912(5)
7.735 254(5)
4.318 141(7)
3.849 759(7)
3.037 399(8)
3.004 046(8)
E
17.541 890
7.735 111
4.317 312
3.848 553
3.036 665
3.004 014
HS
Table 2. E HS from [14]. Em
from [14] after minimizing with
respect to one parameter.
Table 3 shows a comparison between the results of E B and the results of the
present work E U using
s
γ
p
2β 2 (γ)n γ− 1 − √βrq
r 2 e 2 3 F1 (−n, 0.2, 0.07; γ; βrq )
ψn (r) =
n!Γ(γ)
2
d
4
2
and H = − dr
anharmonic oscillator potential for
2 + V (r)where V (r) = r − λr
different values of λ, obtained using the computed matrix elements (3.18) and
minimizing with respect to q, γ, and β. The exponent (n) refer to the dimensions
of the matrix used for the variational computations.
λ
2.0
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
q
2.928
2.023
2.616
2.595
2.574
2.536
2.536
2.518
2.500
2.484
2.468
γ
1.472
1.469
1.469
1.470
1.470
1.740
1.470
1.471
1.471
1.472
1.472
β
0.425
0.927
0.989
1.052
1.117
1.184
1.253
1.323
1.394
1.467
1.542
EB
1.726 29(10)
2.838 91(10)
2.941 23(10)
3.042 10(10)
3.141 55(10)
3.239 62(10)
3.336 36(10)
3.431 79(10)
3.525 96(10)
3.618 90(10)
3.710 64(10)
EU
1.713 36(9)
2.835 34(8)
2.937 73(9)
3.039 06(8)
3.138 86(8)
3.237 24(9)
3.334 12(9)
3.429 94(9)
3.524 02(9)
3.617 47(8)
3.709 39(8)
E
1.713
2.834
2.937
3.038
3.138
3.236
3.333
3.429
3.523
3.617
3.708
03
54
30
56
37
76
78
47
87
01
93
Table 3. E B from [4].
Conclusion. In this paper we compute closed formula for the matrix elements
for Schrödinger equation using generalized hypergeometric function. We used general basis using special kind of functions obtained from the exact solutions of the
Gol’dman and Krivchenkov eigenvalue problem [19]. In fact, for specific values of
the constants p and q in (1.3) we retrieve to the old results found in the literature
[8, 12, 13, 14, 15]. One of the advantages of our results is to obtain analytical formulas that provide us with accurate bounds for the non-solvable singular eigenvalues
problems as well as saves time of the computations using the derived formulas.
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15
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Qutaibeh D. Katatbeh
Department of Mathematics and Statistics, Faculty of Science and Arts, Jordan University of Science and Technology, Irbid 22110, Jordan
E-mail address: qutaibeh@yahoo.com
Ma’zoozeh E. Abu-Amra
Department of Mathematics and Statistics, Faculty of Science and Arts, Jordan University of Science and Technology, Irbid 22110, Jordan