# The Regularized Trace Formula of the Spectrum of a Dirichlet Boundary Value Problem with Turning Point

Authors: Zaki F. A. El-Raheem; A. H. Nasser
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We calculate the regularized trace formula of the infinite sequence of eigenvalues for some version of a Dirichlet boundary value problem with turning points.


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## Body

## 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. Dikiĭ [2] demonstrated a technique of using the trace of a resolvent for finding traces. Dikiĭ 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 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 ρ(x), 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(1.1)-y′′+q(x)y=λρ(x)y,0≤x≤π,(1.2)y(0)=0,y(π)=0,
where q(x) 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 ρ(x) is of the form
(1.3)ρ(x)={1;0≤x≤a<π,-1;a<x≤π.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 λn±,n=0,1,2,… where λn+ are the nonnegative eigenvalues and λn- are the negative eigenvalues which admit the asymptotic formulas
(1.4)λn+=π2a2(n-14)2+2koπa+(2k1πa)1n+O(1n2),λn-=-π2(π-a)2(n-14)2-2koππ-a-(2h1ππ-a)1n+O(1n2),
where
(1.5)ko=-18π∫0aq(t)dt,k1(x)=a2π2{[β1(a)-α1(a)][2γ1(a)+12β1(a)]-12α1(a)+α2(a)-β2(a)+γ2(a)+δ2(a)},h1(x)=π-aak1
and the constants α1(a),α2(a),β1(a),β2(a),γ1(a),γ2(a),δ1(a),andδ2(a) are given by
(1.6)α1(x)=12∫0xq(t)dt,α2(x)=14(∫0xq(t)dt)2+14[q(0)-q(x)],β1(x)=14∫0xq(t)dt,β2(x)=18(∫0xq(t)dt)2+14[q(0)-q(x)],β3(x)=124(∫0xq(t)dt)3+18[q(0)∫0x‍q(t)dt+q′(0)+q′(x)],γ1(x)=12∫xπq(t)dt,γ2(x)=-18(∫xπq(t)dt)2+14[q(x)+q(π)],δ1(x)=-18(∫xπq(t)dt)2+14[q(x)-q(π)],δ2(x)=148(∫xπq(t)dt)3+18∫xπq(t)dt[3q(x)-q(π)]+18[q′(π)+q′(x)].We introduced in [10] the function W(λ) as the Wronskian of the two solutions φ(x,λ) and ψ(x,λ) of (1.1)-(1.2). We denote by Ψ(s) the function W(λ), for λ=s2, which has the following asymptotic formula
(1.7)Ψ(s)=Zo(s)s+Z1(s)s2+Z2(s)s3+O(e|Ims|a+|Res|(π-a)|s|4),
where
(1.8)Zo(s)=-sinsacoshs(π-a)-cossasinhs(π-a),Z1(s)=-P1sinsasinhs(π-a)-P2cossacoshs(π-a),Z2(s)=-Q1sinsacoshs(π-a)-Q2cossasinhs(π-a),(1.9)P1=β1(a)+γ1(a),P2=α1(a)+γ1(a),Q1=α2(a)+β1(a)γ1(a)+δ2(a),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 q(x) and a. We also prove that the roots of Ψ(s)=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 O(1/|λ|4). We must also notice that, the formula obtained in the present work, due to the Dirichlet condition (1.2), contained q(x) 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 ρ(x) 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 thatq(x) 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
(1.10)∑k=0∞(λk+-λkο+-m+)+∑k=0∞(λk--λkο--m-)=M1+M2ln[aπ-a]+M3ln[2-π(π-2a)a(π-a)]+M4tan-1[π-aπ],
where the constants m+,m-,M1,M2,M3, and M4 are given by
(1.11)m+=-14a=∫0aq(t)dt,m-=-14(π-a)=∫aπq(t)dt,M1=916(∫0aq(t)dt)(∫0πq(t)dt)+196(∫aπq(t)dt)3-116(∫aπq(t)dt)-116a∫0aq(t)dt-116(π-a)∫aπq(t)dt+116[3q(a+0)-q(π)]∫aπq(t)dt+116[4q(0)-4q(a-0)+2q(a+0)+2q(π)+q′(π)+q′(a+0)],M2=1332π(∫0aq(t)dt)2+18π(∫0aq(t)dt)(∫aπq(t)dt)-148π(∫aπq(t)dt)3+18π[3q(a+0)-q(π)]∫aπq(t)dt+18π[4q(0)-4q(a-0)+2q(a+0)+2q(π)+q'(π)+q'(a+0)],M3=116π(∫0πq(t)dt)2,M4=34π(∫0aq(t)dt)(∫0πq(t)dt).Proof.
We use the well-known formula from the theory of functions of a complex variable(1.12)2∑k=0n[λk++λk-]=12πi∮Γn‍s2dlnΨ(s),
where Ψ(s) is given by (1.7) and the contour Γn is a quadratic contour on the s-domain as defined in [10]
(1.13)Γn={|Res|≤πa(n-14)+π2a,|Ims|≤ππ-a(n-14)+π2(π-a)}.
From (1.7) we have
(1.14)Ψ(s)=Zo(s)s[1+r(s)],wherer(s)=Z1(s)sZo(s)+Z2(s)s2Zo(s)+O(e|Ims|a+|Res|(π-a)s3Zo(s)),
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
(1.15)|Zo(s)|≥Ce|Ims|a+|Res|(π-a)Cisconstant.
which can be shown by expressing cossa,sinsa,coshsa, and sinhsa in terms of the exponential functions. So that
(1.16)r(s)=Z1(s)sZo(s)+Z2(s)s2Zo(s)+O(1|s|3).
From (1.14) the Equation (1.12) can be put in the form
(1.17)2∑k=0n[λk++λk-]=12πi∮Γn‍s2dln(Zo(s)s)+12πi∮Γn‍s2dln[1+r(s)],
further,
(1.18)12πi∮Γn‍s2dln(Zo(s)s)=2∑k=0n[λko++λko-],
where λko± are the eigenvalues of (1.1)-(1.2) for q(x)=0. Integrating by part the last term of (1.17) we obtain
(1.19)12πi∮Γn‍s2dln[1+r(s)]=-12πi∮Γn‍2s[r(s)-12r2(s)+O(1s3)]ds.
Substituting from (1.16) into (1.19) we have
(1.20)12πi∮Γn‍s2dln[1+r(s)]=-12πi∮Γn‍2s[Z1(s)sZo(s)+Z2(s)s2Zo(s)-Z12(s)2s2Zo2(s)+O(1|s|3)]ds.
From (1.18) and (1.20), (1.17) takes the form
(1.21)2∑k=0n[(λk+-λko+)]-2∑k=0n[λk--λko-]=-1πi∮Γn‍Z1(s)Zo(s)ds-1πi∮Γn‍Z2(s)sZo(s)ds+12πi∮Γn‍Z12(s)sZo2(s)ds+∮Γn‍O(1|s|2)ds.
We evaluate each term of (1.21), first of all, by direct calculation it can be easily shown that
(1.22)∮Γn‍O(1|s|2)ds→0asn→∞.
To evaluate the integration (-1/πi)∮Γn‍(Z1(s)/Zo(s))ds, notice that the function Z1(s)/Zo(s) is odd, so that from (1.8) we have
(1.23)-1πi∮Γn‍Z1(s)Zo(s)ds=-1πi∮Γn‍P1sinsasinhs(π-a)+P2cossacoshs(π-a)sinsacoshs(π-a)+cossasinhs(π-a)ds=-2π∫-ξnξn‍P1sin(σn+iξ)asinh(σn+iξ)(π-a)+P2cos(σn+iξ)acosh(σn+iξ)(π-a)sin(σn+iξ)acosh(σn+iξ)(π-a)+cos(σn+iξ)asinh(σn+iξ)(π-a)dξ+2πi∫-σnσn‍P1sin(σ+iξn)asinh(σ+iξn)(π-a)+P2cos(σ+iξn)acosh(σ+iξn)(π-a)sin(σ+iξn)acosh(σ+iξn)(π-a)+cos(σ+iξn)asinh(σ+iξn)(π-a)dσ.
From which (1.23) can be written in the form
(1.24)-1πi∮Γn‍Z1(s)Zo(s)ds=I1+I2,whereI1=-2π∫-ξnξnP1tan(σn+iξ)atanh(σn+iξ)(π-a)+P2tan(σn+iξ)a+tanh(σn+iξ)(π-a)dξ,I2=2πi∫-σnσnP1tan(σ+iξn)atanh(σ+iξn)(π-a)+P2tan(σ+iξn)a+tanh(σ+iξn)(π-a)dσ.
We notice that in the integrationI1 the function tan(σn+iξ) is bounded while
(1.25)tanh(σn+iξ)(π-a)=1+O(e-2σn(π-a)),
so that I1 becomes of the form
(1.26)I1=-2π∫-ξnξnP2+P1tan(σn+iξ)atan(σn+iξ)a-1dξ+o(1).
By the help of the relations(1.27)cos(σn+iξ)a=(-1)n2[coshξa-isinhξa]sin(σn+iξ)a=(-1)n2[coshξa+isinhξa]
(1.26) becomes
(1.28)I1=-2π-a(P1-P2)(n+14)+o(1).
To evaluateI2 we have tanh(σ+iξn)(π-a) is bounded and
(1.29)tan(σ+iξn)a=i+O(e-2ξna)
and hence I2 becomes of the form
(1.30)I2=2πi∫-σnσniP1sinh(σ+iξn)(π-a)+P2cosh(σ+iξn)(π-a)icosh(σ+iξn)(π-a)+sinh(σ+iξn)(π-a)dσ+o(1).
As before, by applying the relations(1.31)cosh(σ+iξn)(π-a)=(-1)n2[coshσ(π-a)+isinhσ(π-a)],sinh(σ+iξn)(π-a)=(-1)n2[sinhσ(π-a)+icoshσ(π-a)]
to (1.30), we have
(1.32)I2=2a(P1-P2)(n+14)+o(1).
From (1.28) and (1.32) by substitution into (1.24) we have
(1.33)-1πi∮Γn‍Z1(s)Zo(s)ds=[2(P2-P1)π-a+2(P1-P2)a](n+14)+o(1).
To evaluate(-1/πi)∮Γn‍(Z2(s)/sZo(s))ds we notice that Z2(s)/sZo(s) is odd function, then
(1.34)-1πi∮Γn‍Z2(s)sZo(s)ds=-1πi∮Γn‍Q1sinsacoshs(π-a)+Q2cossasinhs(π-a)s[sinsacoshs(π-a)+cossasinhs(π-a)]ds=-2π∫-ξnξnQ1sin(σn+iξ)acosh(σn+iξ)(π-a)+Q2cos(σn+iξ)asinh(σn+iξ)(π-a)(σn+iξ)[sin(σn+iξ)acosh(σn+iξ)(π-a)+cos(σn+iξ)asinh(σn+iξ)(π-a)]dξ+2πi∫-σnσnA2sin(σ+iξn)acosh(σ+iξn)(π-a)+B2cos(σ+iξn)asinh(σ+iξn)(π-a)(σ+iξn)[sin(σ+iξn)acosh(σ+iξn)(π-a)+cos(σ+iξn)asinh(σ+iξn)(π-a)]dσ.
From (1.34) we have
(1.35)-1πi∮Γn‍Z2(s)sZo(s)ds=I1*+I2*,whereI1*=-2π∫-ξnξnQ1tan(σn+iξ)a+Q2tanh(σn+iξ)(π-a)(σn+iξ)[tan(σn+iξ)a+tanh(σn+iξ)(π-a)]dξ,I2*=2πi∫-σnσnQ1tan(σ+iξn)a+Q2tanh(σ+iξn)(π-a)(σ+iξn)[tan(σ+iξn)a+tanh(σ+iξn)(π-a)]dσ.
By using (1.25) and keeping in mind that tan(σ+iξn)a is bounded we have
(1.36)I1*=-2π∫-ξnξnQ1sin(σn+iξ)a+Q2cos(σn+iξ)a(σn+iξ)[cos(σn+iξ)a+sin(σn+iξ)a]dξ+o(1).
By the help of (1.27), after some elementary calculation, I1* takes the form
(1.37)I1*=2(Q1+Q2)πtan-1(aπ-a)+(Q2-Q1)πln[1+a2(π-a)2]+o(1).
Now we evaluateI2*, by using (1.29) and the roundedness of tanh(σ+iξn)(π-a) we have from (1.35)
(1.38)I2*=-2πi∫-σnσnQ2sinh(σ+iξn)(π-a)+Q1cosh(σ+iξn)(π-a)(σ+iξn)[icosh(σ+iξn)(π-a)+sinh(σ+iξn)(π-a)]dσ+o(1).
From (1.38) and (1.31) after calculation we have
(1.39)I2*=2(Q1+Q2)πtan-1(π-aa)-(Q2-Q1)πln[1+(π-a)2a2]+o(1).
By substitution from (1.39) and (1.36) into (1.35), we have
(1.40)-1πi∮ΓnZ2(s)sZo(s)ds=Q1+Q2+Q2-Q1πlna2(π-a)2.
We evaluate the third integral of (1.21)
(1.41)-12πi∮ΓnZ12(s)sZo2(s)ds=-12πi∮Γn[P1sinsasinhs(π-a)+P2cossacoshs(π-a)]2s[sinsacoshs(π-a)+cossasinhs(π-a)]2ds=1π∫-ξnξn[P1sin(σn+iξ)asinh(σn+iξ)(π-a)+P2cos(σn+iξ)acosh(σn+iξ)(π-a)]2(σn+iξ)[sin(σn+iξ)acosh(σn+iξ)(π-a)+cos(σn+iξ)asinh(σn+iξ)(π-a)]2dξ+1πi∫-σnσn[P1sin(σ+iξn)asinh(σ+iξn)(π-a)+Q2cos(σ+iξn)acosh(σ+iξn)(π-a)]2(σ+iξn)[sin(σ+iξn)acosh(σ+iξn)(π-a)+cos(σ+iξn)asinh(σ+iξn)(π-a)]2dσ.
After simplification (1.41) becomes
(1.42)12πi∮ΓnZ12(s)sZo2(s)ds=I1**+I2**,whereI1**=1π∫-ξnξn[P1-P22+P1+P22icoshξasinhξa]21σn+iξdξ,I2**=1πi∫-σnσn[i(P1+P2)-2+P2-P1-2coshσ(π-a)sinhσ(π-a)]21σ+iξndσ.
ForI1** we have
(1.43)I1**=14π∫-ξnξn(P1-P2)2σn+iξdξ-12πi∫-ξnξn(P12-P22)σn+iξcoshξasinhξadξ-18π∫-ξnξn(P1-P2)2σn+iξcosh2ξasinh2ξadξ
by calculating the integrations in (1.43) we have
(1.44)I1**=-2P1P2πtan-1aπ-a+(P12+P22)2πln(1+a2(π-a)2)+o(1).
On the Other hand,(1.45)I2**=-14πi∫-σnσn(P1+P2)2σ+iξndσ--12πi∫-σnσn(P22-P12)σ+iξncoshσ(π-a)sinhσ(π-a)dσ-14πi∫-σnσn(P2-P1)2σ+iξncosh2σ(π-a)sinh2σ(π-a)dσ
from which after calculation we have
(1.46)I2**=2P1P2πtan-1π-aa+(P22-P12)2πln(1+(π-a)2a2)+o(1).
Substituting from (1.44) and (1.46) into (1.42), we have
(1.47)12πi∮ΓnZ12(s)sZo2(s)ds=P12πlnaπ-a+P222πln(2-π(π-2a)a(π-a))+4P1P2πtan-1π-aa-P1P2.
Substituting from (1.47), (1.40), (1.33), and (1.22) into (1.21) we have
(1.48)2∑k=0n[(λk+-λkο+)]+2∑k=0n[(λk--λkο-)]=2(P2-P1)(π-a)(n+14)+2(P1-P2)a(n+14)+Q1+Q2-P1P2+[2π(Q2-Q1)+P12π]lnaπ-a+P222πln[2-π(π-2a)a(π-a)]+4P1P2πtan-1[π-aπ]+o(1).
Passing to the limit asn→∞ 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 pointx=a approaches the point “π”, because we do not put the condition y(a)=0 at the point x=a.Due to the presence of the turning point (1.3), as it is well known for all unbounded operator, the convergent series ∑n‍(λn-λn(o)) is actually the sum of two infinite siriases, each of which cannot be summed separately.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].



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