R E S E A R C H
Open Access
Asymptotic behavior of eigenvalues of
hydrogen atom equation
Etibar S Panakhov
1and Ismail Ulusoy
2**Correspondence:
iulusoy@adiyaman.edu.tr
2Department of Mathematics,
Adiyaman University, Adiyaman, 02040, Turkey
Full list of author information is available at the end of the article
Abstract
We consider a spectral problem for a class of singular Sturm-Liouville operators on the unit interval with explicit singularity 2/x-2/x2, related to the Schrödinger operator with radially symmetric potential. In particular, we give the asymptotic behavior of the eigenvalues of the hydrogen atom equation.
Keywords: hydrogen atom equation; analytic solutions; method of successive
approximations
1 Introduction
The distribution of eigenvalues in differential operator’s spectral theory has an important place. This classic issue was first examined in a finite interval for second order operators in the th century by Sturm and Liouville. Later, where the regular boundary conditions were satisfied, the distribution of eigenvalues of differential operators in a finite interval in arbitrary order was also examined by Birkhoff in [].
Especially, the distribution of eigenvalues of the operators with a discrete spectrum defined in the whole of space for quantum mechanics has great importance. Firstly, the formula for the distribution of the eigenvalues of the single-dimensional Sturm opera-tor defined in the whole of the straight-line axis with increasing potential at infinity was given by Titchmarsh in []. Titchmarsh also has shown the distribution formula for the Schrödinger operator. In later years, Levitan and Gasymov improved the Titchmarsh method and found important asymptotic formulas for the eigenvalues of different differ-ential operators [, ].
Two important methods have been dealt with to examine the asymptotic formula for eigenvalues. The first method, the variation method, is due to Courant and Hilbert []. Birman and Solomyak have improved this method in recent years []. The second method that is related with the resolvent of the operator in question was suggested by Carleman []. Another important method for examining the asymptotic of the eigenvalues in singu-lar condition was suggested by Fedoryuk []. This method is very useful in that it ensures that the distribution of the eigenvalues of the operators with partial derivation are such that the coefficients are analytic functions. Later, many studies have been conducted to ex-amine the eigenvalues [–]. Many mathematicians have exex-amined the eigenvalues so far. The spectral problem for the Sturm-Liouville operator with Dirichlet boundary condi-tion is given in detail in [] by Poeschel and Trubowitz. Guillot and Ralston have extended
these results to the singular Sturm-Liouville operator
L= –d
dx +
x +q(x)
with domain{y∈L[, ] :y,y, absolutely continuous on (, ],Ly∈L[, ] andy() = } [].
Later, this work was generalized by Carlson []. For real numbersb and real valued functionsq(x)∈L[, ], Carlson dealt with the operator
L(m,q) = – d
dx+
m(m+ )
x +q(x), m= , , , . . .
with domain {y∈ L[, ] : y,y, absolutely continuous on (, ], L(m,q)y ∈ L[, ],
limx↓y(x) = andy() +by() = }. Similar features of the Sturm-Liouville operator were
studied in [–].
Consider the Schrödinger equation for two particles in dimensionless variables,
–∂
ψ
∂x –
∂ψ ∂y –
∂ψ
∂z +V(x,y,z)ψ=k
ψ. (.)
If the potential functionV(x,y,z) depends only onr= (x+y+z)/,i.e. V(x,y,z) =V(r),
then the variables in (.) can be separated by putting
ψ(x,y,z) = ψ(r)
r/ Y
l
m(θ,ϕ), l= , , , . . . ,
wherex=rsinθcosϕ,y=rsinθsinψ,z=rcosθ, andYl
m(θ,ϕ) are the spherical
harmon-ics. This gives a differential equation of the form
dϕ
∂r +
r dϕ
∂r – λ
r –V(r)ϕ+k
ϕ= (.)
for the functionϕ(r), whereλ=l+ / (l= , , , . . .). If the potential functionV(r) satisfies the condition∞r|V(r)|dr<∞, then, for a solution of (.), which is regular at zero and normalized, the following asymptotic formula is satisfied:
r/ϕ(r,k,λ) =A(k,λ)sin
kr–π
λ–
+δ(k,λ)
+o()
for fixedλ, andk, andr→ ∞.
In this formula,A(k,λ) is called the scattering amplitude andδ(k,λ) the scattering phase or phase shift [].
In quantum mechanics the study of the energy levels of the hydrogen atom leads to the equation [–]
–d
R
dr +
a r
dR dr –
l(l+ )
r R+
E+a
r
R= ( <r<∞).
The substitutionR=y/rreduces this equation to the form
y+
E+
r– l(l+ )
r
Our aim here is to find the asymptotic behavior of the eigenvalues of the problem
–y+
x –
x+q(x)
y=λy ( <x≤),
y() = ,
with domain{y∈L[, ] :y,ythat are absolutely continuous on (, ],Ly∈L[, ]}. Here
we haveλ=√–E,E< .
Spectral problems for the hydrogen atom equation were considered by many mathe-maticians. Particularly, the inverse problem was examined in Panakhov and Yilmazer’s papers [, ].
2 Basic properties
We consider the singular Sturm-Liouville equation
–y+
x –
x+q(x)
y=λy, x∈(, ], (.)
where the functionq(x)∈L[, ]. Let us give the solutions of this equation by integral equation representations.
Lemma The solutions of(.)have the following form:
ϕ(x,λ,q) =x+
x
x
t – t
x
q(t) –
t –λ
ϕ(t,λ,q)dt (.)
and
ψ(x,λ,q) =cx+cx––
x
x
t – t
x
q(t) –
t –λ
ψ(t,λ,q)dt, (.)
where q(x)∈L[, ].
Proof Let us show that (.) is satisfied. The general solution of the equation
–y+
xy=
is
y=cx+cx–.
Let us apply the method of variation of parameters of (.),
–y+
xy=
λ+
x–q(x)
y.
Taking the second derivative of the equation
and substituting this into (.), we obtain
u(x)x+u(x)x–= , –u(x)x+u(x)x–=
λ+
x–q(x)
y(x).
If we multiply the first equation by –/xand combine with the second equation we have
–u(x)x=
λ+
x–q(x)
y(x). (.)
Take the integral of this equation from tox:
u(x) =
x
t
q(t) –λ–
t
y(t)dt.
If we multiply the first equation by and the second equation byxand combine these equations we have
xu
(x) =x
λ+
x–q(x)
y(x). (.)
Take the integral of this equation from tox:
u(x) = –
x
t
q(t) –λ–
t
y(t)dt.
Then we get the equation
y=
x
x t –
t x
q(t) –λ–
t
y(t)dt.
We use the above method to show (.). Take the integral of (.) fromxto :
u(x) = –
x
t
q(t) –λ–
t
y(t)dt.
Take the integral of (.) fromxto :
u(x) =
x
t
q(t) –λ–
t
y(t)dt.
Then we get the equation
y= –
x
x
t – t
x
q(t) –λ–
t
y(t)dt.
Now we will show that these solutions are analytic by using the method of successive approximations. Addressing (.) first, let
y(x) =x, yn+(x) =y(x) +
x
x
t – t
x
q(t) –λ–
t
yn(t)dt.
Theorem The sequence yn(x)converges uniformly to a functionϕ(x,λ,q)satisfying(.)
and(.).Moreover,limx↓x–ϕ(x,λ,q) = and the mapping(λ,q)→ϕ(x,λ,q)is analytic
fromC×L[, ]→C[, ].
Proof Let us show that
yn(x) –yn–(x)≤x
n
xn
n m=m
/
q–λ n
+
n–
k=
xn–k
(n–k)nm–=m
/
q–λ n–k+x
n
n!
,
by using the method of induction. Fork= ,
y(x) –y(x)=
xxt –t
x
q(t) –λ–
t
y(t)dt
=
x
x t –
t x
q(t) –λ–
t
tdt
≤x
x
t– t
x
q(t) –λ–
t
dt.
We have|t–t x| ≤t, y(x) –y(x)≤x
xt
q(t) –λ–
t
dt
≤x
x
tq(t) –λdt+x
.
By the Cauchy-Schwarz inequality, we get
y(x) –y(x)≤x
x
tdt
/ x
q(t) –λdt
/
+x
≤x
x
/
q–λ +x
.
Fork= ,
y(x) –y(x)=
xxt –t
x
q(t) –λ–
t
y(t) –y(t)
dt
=
x
x t –
t x
q(t) –λ–
t
t
t
/
q–λ +t
dt
≤x
x
t–t
x
q(t) –λ–
t
t
/
q–λ +t
Since|t–tx| ≤t,
y(x) –y(x)≤x
x
t
q(t) –λ–
t t /
q–λ +t
dt
≤x
x
t
q(t) –λ+ t /
q–λ +t
dt
≤x
x t /
q(t) –λ q–λ dt+
x t /
q–λ dt
+x
x
tq(t) –λdt+t
.
By the Cauchy-Schwarz inequality, we get
y(x) –y(x)≤x
x . /
q–λ +
x
. /
q–λ +
x
!
.
Assume that the inequality is true fork=n. Now we will show that the inequality holds fork=n+ ,
yn+(x) –yn(x)=
xxt–t
x
q(t) –λ–
t
yn(t) –yn–(t)
dt = x t x t – t x
q(t) –λ–
t
n
xn
n m=m
/
q–λ n
+
n–
k=
xn–k
(n–k)nm–=m
/
q–λ n–k+x
n n! dt = x t
q(t) –λ+
n
xn
n m=m
/
q–λ n
+
n–
k=
xn–k
(n–k)nm–=m
/
q–λ n–k+x
n n! dt
≤x
n+
x(n+)
n+
m=m
/
q–λ n+
+
n
k=
x(n+)–k
((n+ ) –k)nm=m
/
q–λ n+–k+ x
n+
(n+ )!
.
By the ratio test, the series converges. Thenyn(x) converges uniformly by the Weierstrass
sufficiency theorem.
Differentiation of (.) gives the formula
ϕ(x,λ,q) = x+ x x t + t x
q(t) –
t –λ
ϕ(t,λ,q)dt
and
lim
x↓xϕ
Turning to (.), let
y(x) =cx+cx–,
yn+(x) =y(x) –
x
x
t – t
x
q(t) –
t –λ
yn(t)dt.
Theorem The sequence xyn(x)converges uniformly for x∈(, ]to a function xψ(x,λ,q)
whereψ(x,λ,q)satisfies(.)and(.).Moreover,limx↓xψ(x,λ,q)exists and the mapping
(λ,q)→xψ(x,λ,q)is analytic fromC×L[, ]→C[, ].
Proof Let us show that
yn(x) –yn–(x)≤
x
n
( –x)n
n! /
q–λ n|cn|
+
n–
k=
( –x)k
k! /
q–λ k|ck|+|cn+|
by using the method of induction. Fork= ,
y(x) –y(x)=
x
x
t – t
x
q(t) –λ–
t
y(t)dt
=
x
xt –t
x
q(t) –λ–
t
ct+ct–dt.
Sincet≥xwe get|xt –tx|=|tx(xt – )| ≤t
x and
y(x) –y(x)≤
x
t x
q(t) –λ–
t
ct+ct–dt
≤ x
x
q(t) –λct+ctdt+
x
ct
+c dt
.
Because oft≤, we have
y(x) –y(x)≤
x
x
q(t) –λ|c|dt+|c|
.
By the Cauchy-Schwarz inequality, we get
y(x) –y(x)≤
x
( –x)/ q–λ |c|+|c|
.
Assume that the inequality is true fork=n. Now we will show that the inequality holds fork=n+ ,
yn+(x) –yn(x)=
x
x
t – t
x
q(t) –λ–
t
yn(t) –yn–(t)
Sincet≥xwe get|xt –tx|=|tx(xt – )| ≤t
x. We have
yn+(x) –yn(x)≤
x
t
x
q(t) –λ–
t
t
q–λ
n
( –t)n
n! /n
|cn|
+
n–
k=
( –t)k
k! /
q–λ k|ck|+|cn+|
dt
≤
x
t
xq(t) –λ+
q–λ
n
( –t)n
n! /n
|cn|
+
n–
k=
( –t)k
k! /
q–λ k|ck|+|cn+|
dt.
Because oft≤, we have
yn+(x) –yn(x)≤
x
x
q(t) –λ+
q–λ
n
( –t)n
n! /n
|cn|
+
n–
k=
( –t)k
k! /
q–λ k|ck|+|cn+|
dt.
By the Cauchy-Schwarz inequality, we get
yn+(x) –yn(x)≤
x
n+
( –x)n+
(n+ )! /
q–λ n+|cn+|
+
n
k=
( –x)k
k! /
q–λ k|ck|+|cn+|
.
By the ratio test, the series converges. Thenyn(x) converges uniformly by the Weierstrass
sufficiency theorem.
3 Asymptotic behavior of eigenvalues
The main result of the paper is given by the following theorem. Assume that <x<x≤.
Theorem If y is a nontrivial solution of the equation
–y+
x –
x+q(x)
y=λy (.)
with y(x) =y(x) +by(x) = ,then
λ≥–
x
+
|b|+ x
x
|q|dx
,
Proof Multiplying (.) byyand integrating of this equation fromxtoxgives the formula
x
x
–yy+q(x)y–λy+
x –
x
y
dx= .
Sincex
x
xydx≥ the remaining term will be negative or zero, x
x
–yy+q(x)y–λy–
xy
dx≤. (.)
Integrating the first term by parts gives
x
x
–yy dx= –y(x)y(x) +y(x)y(x) +
x
x
ydx
=by(x) +
x
x
ydx.
So (.) is equal to
by(x) +
x
x
y+q(x)y–λy–
xy
dx≤.
Moreover, we findx
x q(x)y
(x)dxand that equals
x
x
y(x)y(x) x
x
q(t)dt dx
=
y(x) x
x
q(t)dt
x
x –
x
x
y(x)y(x) x
x
q(t)dt dx– x
x
y(x)q(x)dx
=y(x)
x
x
q(t)dt– x
x
y(x)q(x)dx.
Then we have x
x
q(x)y(x)dx=y(x)
x
x
q(t)dt– x
x
y(x)y(x) x
x
q(t)dt dx. (.)
Integratingx
x y(x)y
(x)dxby parts gives
x
x
y(x)y(x)dx=y(x)x
x– x
x
y(x)y(x)dx
= y(x) –
x
x
y(x)y(x)dx.
So we get the formula
y(x) =
x
x
Adding (.) toby(x
) and using the triangle inequality gives the formula
by(x)+
xx
q(x)y(x)dx
≤ |b|y(x) +
y(x)
x
x
q(t)dt+ x
x
y(x)y(x) x
x
q(t)dt dx
≤ |b|
x
x
yydx+ x
x
|q|dx
x
x
yydx+ x
x yy
x
x
q(t)dt dx.
Sincex≤xwe get
by(x)+
x
x
q(x)y(x)dx≤
|b|+ x
x
|q|dx
x
x
yydx.
By the Cauchy-Schwarz inequality, we get
by(x)+
xx
q(x)y(x)dx
≤
|b|+ x
x
|q|dx
x
x
ydx
/ x
x
ydx/
and, from the inequality of A/B/≤εA+ (/ε)B, we get
by(x)+
x
x
q(x)y(x)dx
≤
|b|+ x
x
|q|dx
ε
x
x
ydx
+ /ε
x
x
ydx
.
For anyε>|b|+ x
x |q|dx,
by(x) +
x
x
y+q(x)y–λy–
xy
dx
≥
x
x
y–λy–
xy
dx–|b|y(x) –
x
x
qydx ≥
x
x
y–λydx–
x
x
x
ydx
–
|b|+ x
x
|q|dx
ε
x
x
ydx
+ /ε
x
x
ydx
≥
–
|b|+ x
x
|q|dxε
x
x
ydx
+
–λ–
x
–ε
|b|+ x
x
|q|dx
x
x
ydx
=C(ε) +
–λ–
x
–ε
|b|+ x
x
|q|dx
x
x
whereC(ε) > . If we assume that
λ< –
x
+
|b|+ x
x
|q|dx
,
the equationx
x[–y
y+q(x)y–λy–
xy
]dxshould be positive. It contradicts our
as-sumption. So we proved the theorem.
Theorem If y is a nontrivial solution of the equation
–y+
x –
x+q(x)
y=λy
with y(x) =y(x) = ,then
λ≥–
x
+ x
x
|q|dx
,
where q(x)∈L[, ].
Proof Multiplying the equation byyand integrating of this equation fromxtoxgives
the formula x
x
–yy+q(x)y–λy+
x –
x
y
dx= .
Sincex
x
xydx≥ the remaining term will be negative or zero, x
x
–yy+q(x)y–λy–
xy
dx≤. (.)
Integrating the first term by parts gives
x
x
–yy dx= –y(x)y(x) +y(x)y(x) +
x
x
ydx= x
x
ydx.
So (.) is equal to
by(x) +
x
x
y+q(x)y–λy–
xy
dx≤.
Moreover, we findx
x q(x)y
(x)dxand that equals
x
x
y(x)y(x) x
x
q(t)dt dx
=
y(x) x
x
q(t)dt
x
x –
x
x
y(x)y(x) x
x
q(t)dt dx– x
x
y(x)q(x)dx
= – x
x
Then we have x
x
q(x)y(x)dx= – x
x
y(x)y(x) x
x
q(t)dt dx,
sincex≤xwe get
xx
q(x)y(x)dx≤
x
x
|q|dx
x
x
yydx.
By the Cauchy-Schwarz inequality, we get
xx
q(x)y(x)dx≤ x
x
|q|dx
x
x
ydx
/ x
x
ydx
/
,
and from the inequality of A/B/≤εA+ (/ε)B, we get
xx
q(x)y(x)dx≤ x
x
|q|dx
ε
x
x
ydx
+ /ε
x
x
ydx
.
For anyε> x
x |q|dx, x
x
y+q(x)y–λy–
xy
dx
≥
x
x
y–λy–
xy
dx– x
x
qydx
≥
x
x
y–λydx–
x
x
x
ydx
– x
x
|q|dx
ε
x
x
ydx
+ /ε
x
x
ydx
≥
–
ε
x
x
|q|dx
x
x
ydx
+
–λ–
x
– ε
x
x
|q|dx
x
x
ydx
=C(ε) +
–λ–
x
–ε
|b|+ x
x
|q|dx
x
x
ydx,
whereC(ε) > . If we assume that
λ< –
x
+ x
x
|q|dx
,
the equationx
x[–y
y+q(x)y–λy–
xy]dxshould be positive. It contradicts our
as-sumption. So we get our result.
Theorem If y is a nontrivial solution of the equation
–y+
x –
x+q(x)
where q(x)∈L[, ],and if y(x
) =y(x) = ,where≤x≤x≤,then
x +
x
≤ x
x
|q|dx
+
(x–x)
x
x
|q|dx.
Proof Multiplying (.) byyand integrating of this equation fromxtoxgives the formula
x
x
–yy+q(x)y+
x –
x
y
dx= .
Since x
x
y(x)y(x) x
x
q(t)dt dx=y(x)
x
x
q(t)dt– x
x
q(x)y(x)dx
we get
x
x
q(x)y(x)dx=y(x)
x
x
q(t)dt– x
x
y(x)y(x) x
x
q(t)dt dx. (.)
From the mean value theorem we can write
y(x) =
/(x–x)
x
x
ydx, (.)
wherex∈[x,x]. From (.) and integrating
x
x y(x)y
(x)dxby parts we have
y(x) =y(x) +
x
x
yydx=
x–x
x
x
ydx+ x
x
yydx.
Adding (.) tox
x
y
x dxwe have
x
x
q–
x
ydx=
x–x
x
x
ydx+ x
x
yydx
x
x
q dx
– x
x yy
x
x
q(t)dt dx– x
x y
x dx.
From the triangle inequality we have the formula
x
x
q–
x
ydx≤
x–x
x
x
ydx+ x
x
yydx
x
x
q dx
+ x
x yy
x
x
q dt dx+ x
x y
x dx
≤
x–x
x
x
|q|dx+
x
x
x
ydx+ x
x yy
x
x
q dt dx.
By using the inequality
we get xx
q–
x
ydx≤
x–x
x
x
|q|dx+
x
x
x
ydx
+
ε
x
x
ydx+
ε
x
x
ydx
x
x
|q|dx.
Integratingx
x –y
y dxby parts gives
x
x
–yy dx= –y(x)y(x) +y(x)y(x) +
x
x
ydx= x
x
ydx, (.)
x
x
xy dx≥
x
x
x
ydx. (.)
From (.) and (.) and for anyε>|b|+ x
x |q|dx, there exists a numberC(ε) > such that
x
x
–yy+q(x)y+
x –
x y dx = x x
ydx+ x
x
q–
x
ydx+ x
x
xy dx
≥
x
x
ydx–
x–x
x
x
|q|dx+
x
x
x
ydx
–
ε
x
x
ydx+
ε
x
x
ydx
x
x
|q|dx+ x
x
x
ydx
= – ε x x
|q|dx
x
x
ydx
+
x+
x
–
x–x
+ ε
x
x
|q|dx
x
x
ydx
=C(ε) + x + x –
x–x
+ ε
x
x
|q|dx
x
x
ydx.
Let us assume that
x +
x
–
x–x
+ ε
x
x
|q|dx> .
In this case, we get
x + x >ε x x
|q|dx+ (x–x)
x
x
|q|dx
> x
x
|q|dx
+
(x–x)
x
x
|q|dx.
Then we have x
x
–yy+q(x)y+
x –
x
y
This is a contradiction. So we proved the theorem.
Conclusion In the Carlson case, the potentials are inL[, ], but in our paper, the
po-tentials are not inL[, ].
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
IU carried out the design of the study and performed the analysis. EP (adviser) participated in its design and coordination. All authors read and approved the final manuscript.
Author details
1Department of Mathematics, Firat University, Elazig, 23119, Turkey.2Department of Mathematics, Adiyaman University,
Adiyaman, 02040, Turkey.
Acknowledgements
The authors would like to thank the referees for valuable comments in improving the original paper.
Received: 23 November 2014 Accepted: 11 May 2015 References
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