Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 492576, 12 pages
doi:10.1155/2012/492576
Research Article
The Regularized Trace Formula of the
Spectrum of a Dirichlet Boundary Value Problem
with Turning Point
Zaki F. A. El-Raheem1 and A. H. Nasser2
1
2
Department of Mathematics, Faculty of Education, Alexandria University, Alexandria 21526, Egypt
Faculty of Industrial Education, Helwan Unversity, Cairo, Egypt
Correspondence should be addressed to Zaki F. A. El-Raheem, zaki55@alex-sci.edu.eg
Received 12 May 2012; Revised 17 September 2012; Accepted 17 September 2012
Academic Editor: Toka Diagana
Copyright q 2012 Z. F. A. El-Raheem and A. H. Nasser. This is an open access article distributed
under the Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
We calculate the regularized trace formula of the infinite sequence of eigenvalues for some version
of a Dirichlet boundary value problem with turning points.
1. Introduction
The study of regularized traces of ordinary differential operators has a long history and there
are a large number of papers and books studying this issue. The trace formulae for the scalar
differential operators have been found by Gelfand and Levitan 1. The formula obtained
there gave rise to a large and very important theory, which started from the investigation
of specific operators and further embraced the analysis of regularized traces of discrete
operators in general form. In a short time, a number of authors turned their attention to
trace theory and obtained interesting results. Dikiı̆ 2 demonstrated a technique of using the
trace of a resolvent for finding traces. Dikiı̆ provided a proof of the Gelfand-Levitan formula
in 2 on the basis of direct methods of perturbation theory, and in 3, he derived trace
formulas of all orders for the Sturm-Liouville operator by constructing the fractional powers
of the operator in closed form and by computing an analytic extension for its zeta function.
Later, Levitan 4 suggested one more method for computing the traces of the Sturm-Liouville
operator: by matching the expressions for the characteristic determinant via the solution of
an appropriate Cauchy problem and via the corresponding infinite product, he found and
compared the coefficients of the asymptotic expansions of these expressions thus obtaining
trace formulas. The investigation carried out in 1957 by Faddeev 5 linked the trace theory
2
Abstract and Applied Analysis
to a substantially new class of problems, singular differential operators. Gasymov’s paper
6 was the first paper in which a singular differential operator with discrete spectrum was
considered. Afterwards these investigations were continued in many directions, such as Dirac
operators, differential operators with abstract operator-valued coefficients, and the case of
matrix-valued Sturm-Liouville operators see, 7.
Beyond their aesthetic appeal, trace formulas play an important role in the inverse
spectral theory 8, 9. Equation 1.1 is called differential equation with turning points if the
weight function ρx, which is given by 1.3, changes sign. The turning points appear in
elasticity, optics, geophysics, and other branches of natural sciences. The inverse problems
for equations with turning points and singularities help to study blow-up solutions for
some nonlinear integrable evolution equations of mathematical physics. The turning points
cause analytical difficulties, not only in calculating the eigenvalues asymptotes, but also in
calculating the trace formula. In 10–12, the authors studied the spectral analysis of problem
1.1-1.2. They investigated the asymptotic relations of both eigenvalues and eigenfunctions
also studied the eigenfunction expansion formula and proved the equiconvergence formula
of that eigenfunctions. To complement the picture of spectral analysis of problem 1.1-1.2,
we crown the series of papers 10–12 with the present work. In the present work, we evaluate
the regularized trace formula for the problem 1.1-1.2 by using contour integration method
as in 13. It should be noted here that in 13 the author studied such formula for continuous
spectrum in the whole line, while the present work contains point spectrum in a finite
interval.
Consider the following Dirichlet problem
−y qxy λρxy,
y0 0,
0 ≤ x ≤ π,
yπ 0,
1.1
1.2
where qx is a nonnegative real function, which has a second piecewise integrable
derivatives of the second order in 0, π, λ is a spectral parameter and the weight function or
the explosive factor ρx is of the form
1;
ρx −1;
0 ≤ x ≤ a < π,
a < x ≤ π.
1.3
Following 10, we state the basic notations and results that are needed in the
subsequent calculation. In 10, the author proved that the Dirichlet problem 1.1-1.2 has a
countable number of eigenvalues λ±n , n 0, 1, 2, . . . where λn are the nonnegative eigenvalues
and λ−n are the negative eigenvalues which admit the asymptotic formulas
2k1 π 1
1
π2
1 2 2ko π
O 2 ,
2 n−
4
a
a
n
a
n
2
2h1 π 1
1
π2
2ko π
1
−
O
−
n
−
,
λ−n −
2
4
π −a
π −a n
n2
π − a
λn
1.4
Abstract and Applied Analysis
3
where
ko −
k1 x a
1
8π
qtdt,
0
a 1
1
β
α
β
−
α
2γ
−
α
−
β
γ
δ
,
a
a
a
a
a
a
a
a
a
1
1
1
1
1
2
2
2
2
2
2
2π 2
π −a
k1
a
h1 x 1.5
and the constants α1 a, α2 a, β1 a, β2 a, γ1 a, γ2 a, δ1 a, and δ2 a are given by
α1 x 1
α2 x 4
x
1
2
qtdt,
x
0
2
qtdt
0
β1 x 1
4
x
1
q0 − qx ,
4
qtdt,
0
2
x
1
1
qtdt q0 − qx ,
β2 x 8 0
4
x
x
3
1
1
qtdt q0 qtdt q 0 q x ,
β3 x 24 0
8
0
π
1
qtdt,
γ1 x 2 x
π
2
1
1
qtdt qx qπ ,
γ2 x −
8 x
4
2
π
1
1
qtdt qx − qπ ,
δ1 x −
8 x
4
π
π
3
1
1
1
qtdt qtdt 3qx − qπ q π q x .
δ2 x 48 x
8 x
8
1.6
We introduced in 10 the function Wλ as the Wronskian of the two solutions ϕx, λ
and ψx, λ of 1.1-1.2. We denote by Ψs the function Wλ, for λ s2 , which has the
following asymptotic formula
Zo s Z1 s Z2 s
O
Ψs s
s2
s3
e| Im s|a| Re s|π−a
|s|4
,
1.7
4
Abstract and Applied Analysis
where
Zo s − sin sa cosh sπ − a − cos sa sinh sπ − a,
Z1 s −P1 sin sa sinh sπ − a − P2 cos sa cosh sπ − a,
1.8
Z2 s −Q1 sin sa cosh sπ − a − Q2 cos sa sinh sπ − a,
P1 β1 a γ1 a,
P 2 α1 a γ1 a,
Q1 α2 a β1 aγ1 a δ2 a,
1.9
Q2 α1 a γ1 a β2 a γ2 a,
where α1 , α2 , β1 , β2 , γ1 , γ2 , and δ2 are expressed, in 1.6, in terms of qx and a. We also
prove that the roots of Ψs 0 coincide with the eigenvalues of the Dirichlet problem
1.1-1.2 and these eigenvalues are simple. Our aim is to calculate the summation of these
eigenvalues which we call the trace formula or more precisely the regularized trace formula.
During the calculation of the eigenvalues and the eigenfunctions, the condition 1.2 forced
us to evaluate up to the term containing O1/|λ|4 . We must also notice that, the formula
obtained in the present work, due to the Dirichlet condition 1.2, contained qx together
with its first derivatives on 0, π. We used the methodology as in 4, 14, by Levitan, but
our problem contains more difficulties because of the presence of the ρx as ± sign. In the
following theorem we calculate the summation of these eigenvalues in a certain form called
the regularized trace formula for the Dirichlet problem 1.1-1.2.
Theorem 1.1. Suppose that qx has a second-order piecewise integrable derivatives on 0, π then,
in view of the introduced notations, 1.4 and 1.7, the following regularized trace formula takes place
∞
k0
∞
−
−
−
m
λk − λo
λk − λo−
k
k −m
k0
a
ππ − 2a
π −a
M1 M2 ln
M3 ln 2 −
M4 tan−1
,
π −a
aπ − a
π
where the constants m , m− , M1 , M2 , M3 , and M4 are given by
−1
m 4a
a
qtdt,
0
−1
m 4π − a
−
π
qtdt,
a
1.10
Abstract and Applied Analysis
π
3
π
π
1
1
qtdt
qtdt qtdt −
qtdt
96 a
16 a
0
0
a
π
π
1
1
1
−
qtdt −
qtdt qtdt
3q a 0 − qπ
16a 0
16π − a a
16
a
9
M1 16
a
5
1
4q0 − 4qa − 0 2qa 0 2qπ q π q a 0 ,
16
2
π
a
a
13
1
qtdt qtdt
qtdt
M2 32π
8π
0
0
a
π
3
π
1 1
qtdt qtdt
−
3q a 0 − qπ
48π
8π
a
a
1 4q 0 − 4q a − 0 2q a 0 2q π q π q a 0 ,
8π
π
a
2
π
1
3
qtdt ,
M4 qtdt
qtdt .
M3 16π
4π
0
0
0
1.11
Proof. We use the well-known formula from the theory of functions of a complex variable
n
1
s2 d ln Ψs,
2
λk λ−k 2πi
Γn
k0
1.12
where Ψs is given by 1.7 and the contour Γn is a quadratic contour on the s-domain as
defined in 10
Γn π
π
1
π
1
π
, |Im s| ≤
.
n−
n−
|Re s| ≤
a
4
2a
π −a
4
2π − a
1.13
From 1.7 we have
Zo s
Ψs 1 rs,
s
Z2 s
Z1 s
O
where rs sZo s s2 Zo s
e| Im s|a| Re s|π−a
s3 Zo s
, 1.14
where, Zo s, Z1 s, and Z2 s are given by 1.8. It is clear that, on the contour Γn , the term
Zo s satisfies the following inequality from below
|Zo s| ≥ C e| Im s|a| Re s|π−a
C is constant.
1.15
6
Abstract and Applied Analysis
which can be shown by expressing cos sa, sin sa, cosh sa, and sinh sa in terms of the
exponential functions. So that
Z2 s
Z1 s
O
rs sZo s s2 Zo s
1
|s|3
1.16
.
From 1.14 the Equation 1.12 can be put in the form
n
1
Zo s
1
2
s2 d ln
s2 d ln1 rs,
λk λ−k 2πi
s
2πi
Γ
Γ
n
n
k0
1.17
further,
1
2πi
n
o
Zo s
2
s2 d ln
λk λo−
k ,
s
Γn
k0
1.18
are the eigenvalues of 1.1-1.2 for qx 0. Integrating by part the last term of
where λo±
k
1.17 we obtain
1
2πi
−1
s d ln1 rs 2πi
Γn
2
1 2
1
2s rs − r s O 3 ds.
2
s
Γn
1.19
Substituting from 1.16 into 1.19 we have
1
2πi
−1
s d ln1 rs 2πi
Γn
2
Z12 s
Z2 s
Z1 s
2
−
2s
O
sZo s s Zo s 2s2 Zo2 s
Γn
1
ds. 1.20
|s|3
From 1.18 and 1.20, 1.17 takes the form
n
n
−
2
λk − λo
−2
λk − λo−
k
k
k0
k0
−1
πi
Z1 s
1
ds −
Z
πi
s
Γn o
Z2 s
1
ds sZ
2πi
s
o
Γn
Z12 s
Γn
sZo2 s
ds Γn
O
1
|s|2
1.21
ds.
We evaluate each term of 1.21, first of all, by direct calculation it can be easily shown
that
Γn
O
1
|s|2
ds −→ 0
as n −→ ∞.
1.22
Abstract and Applied Analysis
7
To evaluate the integration −1/πi
is odd, so that from 1.8 we have
−1
πi
Γn
Z1 s/Zo s ds, notice that the function Z1 s/Zo s
Z1 s
ds
Z
Γn o s
P1 sin sa sinh sπ − a P2 cos sa cosh sπ − a
−1
ds
πi Γn sin sa cosh sπ − a cos sa sinh sπ − a
−2 ξn P1 sinσn iξa sinhσn iξπ − a P2 cosσn iξa coshσn iξπ − a
dξ
π −ξn
sinσn iξa coshσn iξπ − a cosσn iξa sinhσn iξπ − a
2 σn P1 sinσ iξn a sinhσ iξn π −a P2 cosσ iξn a coshσ iξn π −a
dσ.
πi −σn
sinσ iξn a coshσ iξn π −a cosσ iξn a sinhσ iξn π −a
1.23
From which 1.23 can be written in the form
−1
πi
Z1 s
ds I1 I2 ,
Γn Zo s
−2 ξn P1 tanσn iξa tanhσn iξπ − a P2
where I1 dξ,
π −ξn
tanσn iξa tanhσn iξπ − a
2 σn P1 tanσ iξn a tanhσ iξn π − a P2
I2 dσ.
πi −σn
tanσ iξn a tanhσ iξn π − a
1.24
We notice that in the integration I1 the function tanσn iξ is bounded while
tanhσn iξπ − a 1 O e−2σn π−a ,
1.25
so that I1 becomes of the form
I1 −2
π
ξn
−ξn
P2 P1 tanσn iξa
dξ o1.
tanσn iξa − 1
1.26
By the help of the relations
−1n
cosσn iξa √ cosh ξa − i sinh ξa
2
−1n
sinσn iξa √ cosh ξa i sinh ξa
2
1.27
8
Abstract and Applied Analysis
1.26 becomes
−2
1
o1.
I1 P1 − P2 n π −a
4
1.28
To evaluate I2 we have tanhσ iξn π − a is bounded and
tanσ iξn a i O e−2ξn a
1.29
and hence I2 becomes of the form
I2 2
πi
σn
−σn
iP1 sinhσ iξn π − a P2 coshσ iξn π − a
dσ o1.
i coshσ iξn π − a sinhσ iξn π − a
1.30
As before, by applying the relations
−1n
coshσ iξn π − a √ cosh σ π − a i sinh σ π − a,
2
−1n
sinhσ iξn π − a √ sinh σπ − a i cosh σπ − a
2
1.31
to 1.30, we have
2
1
I2 P1 − P2 n o1.
a
4
1.32
From 1.28 and 1.32 by substitution into 1.24 we have
−1
πi
Z1 s
2P2 − P1 2P1 − P2 1
ds n
o1.
π −a
a
4
Γn Zo s
1.33
Z2 s/sZo sds we notice that Z2 s/sZo s is odd function,
Γn
then
−1
Z2 s
ds
πi Γn sZo s
−1
Q1 sin sa cosh sπ − a Q2 cos sa sinh sπ − a
ds
πi Γn ssin sa cosh sπ − a cos sa sinh sπ − a
−2 ξn Q1 sinσn iξa coshσn iξπ − a Q2 cosσn iξa sinhσn iξπ − a
dξ
π −ξn σn iξsinσn iξa coshσn iξπ − a cosσn iξa sinhσn iξπ − a
2 σn
A2 sinσ iξn a coshσ iξn π −aB2 cosσ iξn a sinhσ iξn π −a
dσ.
πi −σn σ iξn sinσ iξn a coshσ iξn π −a cosσ iξn a sinhσ iξn π −a
1.34
To evaluate −1/πi
Abstract and Applied Analysis
9
From 1.34 we have
−1
πi
Z2 s
ds I1∗ I2∗ ,
sZ
s
o
Γn
Q1 tanσn iξa Q2 tanhσn iξπ − a
−2 ξn
where I1∗ dξ,
π −ξn σn iξtanσn iξa tanhσn iξπ − a
Q1 tanσ iξn a Q2 tanhσ iξn π − a
2 σn
∗
I2 dσ.
πi −σn σ iξn tanσ iξn a tanhσ iξn π − a
1.35
By using 1.25 and keeping in mind that tanσ iξn a is bounded we have
I1∗ −2
π
ξn
−ξn
Q1 sinσn iξa Q2 cosσn iξa
dξ o1.
σn iξcosσn iξa sinσn iξa
1.36
By the help of 1.27, after some elementary calculation, I1∗ takes the form
I1∗
a2
2Q1 Q2 a
Q2 − Q1 −1
o1.
tan
ln 1 π
π −a
π
π − a2
1.37
Now we evaluate I2∗ , by using 1.29 and the roundedness of tanhσ iξn π − a we
have from 1.35
I2∗
−2
πi
σn
−σn
Q2 sinhσ iξn π − a Q1 coshσ iξn π − a
dσ o1.
σ iξn i coshσ iξn π − a sinhσ iξn π − a
1.38
From 1.38 and 1.31 after calculation we have
I2∗
2Q1 Q2 π − a2
Q2 − Q1 −1 π − a
o1.
tan
ln 1 −
π
a
π
a2
1.39
By substitution from 1.39 and 1.36 into 1.35, we have
−1
πi
Z2 s
a2
Q2 − Q1
ln
.
ds Q1 Q2 π
π − a2
Γn sZo s
1.40
10
Abstract and Applied Analysis
We evaluate the third integral of 1.21
−1
2πi
Z12 s
ds
sZo2 s
−1
P1 sin sa sinh sπ − a P2 cos sa cosh sπ − a2
ds
2πi Γn ssin sa cosh sπ − a cos sa sinh sπ − a2
1
π
Γn
ξn
−ξn
1
πi
P1 sinσn iξa sinhσn iξπ − a P2 cosσn iξa coshσn iξπ − a2
σn iξsinσn iξa coshσn iξπ − a cosσn iξa sinhσn iξπ − a2
σn
−σn
P1 sinσ iξn a sinhσ iξn π −aQ2 cosσ iξn a coshσ iξn π −a2
σ iξn sinσ iξn a coshσ iξn π −acosσ iξn a sinhσ iξn π −a2
dξ
dσ.
1.41
After simplification 1.41 becomes
1
2πi
where I1∗∗
Z12 s
ds I1∗∗ I2∗∗ ,
sZo2 s
1 ξn P1 − P2 P1 P2 cosh ξa 2 1
dξ,
π −ξn
2
2i sinh ξa σn iξ
I2∗∗ Γn
σn 1
πi
−σn
iP1 P2 P2 − P1 cosh σπ − a
−2
−2 sinh σπ − a
2
1.42
1
dσ.
σ iξn
.
For I1∗∗ we have
I1∗∗
1
4π
ξn
−ξn
1
−
8π
1
P1 − P2 2
dξ −
σn iξ
2πi
ξn
−ξn
ξn −ξn
P12 − P22 cosh ξa
dξ
σn iξ sinh ξa
P1 − P2 2 cosh2 ξa
dξ
σn iξ sinh2 ξa
1.43
by calculating the integrations in 1.43 we have
I1∗∗
2
P1 P22
a2
a
−2P1 P2
−1
o1.
tan
ln 1 π
π −a
2π
π − a2
1.44
Abstract and Applied Analysis
11
On the Other hand,
−1
4πi
I2∗∗
σn
−σn
1
−
4πi
−1
P1 P2 2
dσ −
σ iξn
2πi
σn
−σn
σn −σn
P22 − P12 cosh σπ − a
dσ
σ iξn sinh σπ − a
2
2
P2 − P1 cosh σπ − a
dσ
σ iξn sinh2 σπ − a
1.45
from which after calculation we have
I2∗∗
2
P2 − P 12
2P1 P2
π − a2
−1 π − a
o1.
tan
ln 1 π
a
2π
a2
1.46
Substituting from 1.44 and 1.46 into 1.42, we have
1
2πi
P12
P22
a
ππ − 2a
4P1 P2
π −a
ln
ln 2 −
tan−1
− P1 P2 . 1.47
ds 2
π
π
−
a
2π
aπ
−
a
π
a
Γn sZo s
Z12 s
Substituting from 1.47, 1.40, 1.33, and 1.22 into 1.21 we have
n
n
−
2
λk − λo
2
λk − λo−
k
k
k0
k0
2P2 − P1 π − a
n
1
4
1
2P1 − P2 n
Q1 Q2 − P1 P2
a
4
P12
a
2
ln
Q2 − Q1 π
π
π −a
1.48
P22
ππ − 2a
4P1 P2
π −a
ln 2 −
tan−1
o1.
2π
aπ − a
π
π
Passing to the limit as n → ∞ and by using 1.9 and 1.6 we reach to the required
formula 1.10.
2. Conclusion and Comment
It should be noted here that, we cannot expect that the trace formula must become the
Gelfand-Levitan trace formula as the point x a approaches the point “π”, because we do
not put the condition ya 0 at the point x a.
Due to the presence of the turning point 1.3, as it is well known for all unbounded
o
operator, the convergent series n λn − λn is actually the sum of two infinite siriases, each
of which cannot be summed separately.
12
Abstract and Applied Analysis
Moreover the presence of the turning point helps in the solution of inverse problem
for different densities medium 15. The present work belongs to the school of Gasymov 6.
Conflict of Interests
The authors declare that they have no competing interests.
Acknowledgments
The authors are very grateful to the referees for their fruitful comments and advice, also
thanks for Professor Gussein G. Sh. for his advice.
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