# Green's Function Method for Self-Adjoint Realization of Boundary-Value Problems with Interior Singularities

Authors: K. Aydemir; O. Sh. Mukhtarov
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

The purpose of this paper is to investigate some spectral properties of Sturm-Liouville type problems with interior singularities. Some of the mathematical
aspects necessary for developing our own technique are presented. By applying
this technique we construct some special solutions of the homogeneous equation and present a formula and the existence conditions of Green's function.
Furthermore, based on these results and introducing operator treatment in adequate
Hilbert space, we derive the resolvent operator and prove self-adjointness of
the considered problem.


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## Body

## 1. Introduction

For inhomogeneous linear systems, the basic superposition principle says that the response to a combination of external forces is the self-same combination of responses to the individual forces. In a finite-dimensional system, any forcing function can be decomposed into a linear combination of unit impulse forces, each applied to a single component of the system, and so the full solution can be written as a linear combination of the solutions to the impulse problems. This simple idea will be adapted to boundary value problems governed by differential equations, where the response of the system to a concentrated impulse force is known as Green's function. With Green's function in hand, the solution to the inhomogeneous system with a general forcing function can be reconstructed by superimposing the effects of suitably scaled impulses. Green's function method provides a powerful tool to solve linear problems consisting of a differential equation (partial or ordinary, with, possibly, an inhomogeneous term) and enough initial and/or boundary conditions (also possibly inhomogeneous) so that this problem has a unique solution. The history of Green's function dates back to 1828, when Green [1] published work in which he sought solutions of Poisson’s equation ∇2u=f for the electric potential u defined inside a bounded volume with specified boundary conditions on the surface of the volume. He introduced a function now identified as what Riemann later coined Green's function. In 1877, Neumann [2] embraced the concept of Green's function in his study of Laplace's equation, particularly in the plane. He found that the two-dimensional equivalent of Green's function was not described by singularity of the form 1/|r-r0| as in the three-dimensional case but by a singularity of the form log(1/|r-r0|). With the function’s success in solving Laplace's equation, other equations began to be solved using Green's function. The heat equation and Green's function have a long association with each other. After discussing heat conduction in free space, the classic solutions of the heat equation in rectangular, cylindrical, and spherical coordinates are offered. In the case of the heat equation, Hobson [3] derived the free-space Green's function for one, two and three dimensions, and the French mathematician Appell [4] recognized that there was a formula similar to Green's for the one-dimensional heat equation. Green's function is particularly well suited for wave problems with the detailed analysis of electromagnetic waves in surface wave guides and water waves. The leading figure in the development of Green's function for the wave equation was Kirchhoff [5], who used it during his study of the three-dimensional wave. Starting with Green's second formula, he was able to show that the three-dimensional Green's function is
(1)g(x,y,z,t∣ξ,η,ς,τ)=δ(t-τ-R/c)4πR,
where R=(x-ξ)2+(y-η)2+(z-ς)2.The application of Green's function to ordinary differential equations involving boundary-value problems began with the work of Burkhardt [6]. Determination of Green's function is also possible using Sturm-Liouville theory. This leads to series representation of Green's function. Sturm-Liouville problems which contained spectral parameter in boundary conditions form an important part of the spectral theory of boundary value problems. This type of problems has a lot of applications in mechanics and physics (see [7–9] and references cited therein). In the recent years, there has been increasing interest in this kind of problems which also may have discontinuities in the solution or its derivative at interior points (see [10–18]). In this study, we will investigate some spectral properties of the Sturm-Liouville differential equation on two intervals:
(2)ℒy:=-y′′(x)+q(x)y(x)=λy(x),x∈[a,c)∪(c,b]
on [a,c)∪(c,b], with eigenparameter-dependent boundary conditions at the end points x=a and x=b. One has,
(3)τ1(y):=α10y(a)+α11y′(a)=0,(4)τ2(y):=α20y(b)-α21y′(b)+λ(α20′y(b)-α21′y′(b))=0
and the transmission conditions at the singular interior point x=c(5)τ3(y):=β11-y′(c-)+β10-y(c-)+β11+y′(c+)+β10+y(c+)=0,τ4(y):=β21-y′(c-)+β20-y(c-)+β21+y′(c+)+β20+y(c+)=0,
where the potential q(x) is real continuous function in each of the intervals [a,c) and (c,b] and has finite limits q(c∓0), λ  is a complex spectral parameter, αij,  βij±,(i=1,2 and j=0,1),  and  αij′(i=2 and j=0,1) are real numbers.Our problem differs from the usual regular Sturm-Liouville problem in the sense that the eigenvalue parameterλ is contained in both differential equation and boundary conditions, and two supplementary transmission conditions at one interior point are added to boundary conditions. Such problems are connected with discontinuous material properties, such as heat and mass transfer, vibrating string problems when the string loaded additionally with points masses, diffraction problems [8, 9], and varied assortment of physical transfer problems. We develop our own technique for the investigation of some spectral properties of this problem. In particular, we construct the Green’s function and adequate Hilbert space for self-adjoint realization of the considered problem.


## 2. Some Basic Solutions and Green's Function

Denote the determinant of thekth and jth columns of the matrix
(6)T=[β10+β11+β10-β11-β20+β21+β20-β21-]
by Δkj(1≤k<j≤4). For self-adjoint realization in adequate Hilbert space, everywhere below we will assume that
(7)Δ12>0,Δ34>0.
With a view to construct the Green's function we will define two special solutions of (2) by our own technique as follows. At first, consider the next initial-value problem on the left interval [a,c)(8)-y′′+q(x)y=λy,y(a)=α11,y′(a)=-α10.
It is known that this problem has an unique solution  u=φ-(x,λ)  which is an entire function of λ∈ℂ for each fixed x∈[a,c) (see, e.g., [19]). By applying the similar method of [13], we can prove that (2) on the right interval (c,b] has an unique solution u=φ+(x,λ) satisfying the equalities
(9)φ+(c+,λ)=1Δ12(Δ23φ-(c-,λ)+Δ24∂φ-(c-,λ)∂x),(10)∂φ+(c+,λ)∂x=-1Δ12(Δ13φ-(c-,λ)+Δ14∂φ-(c-,λ)∂x),
which is also an entire function of the parameter λ for each fixed x∈[c,b]. Consequently, the solution u=φ(x,λ) defined by
(11)φ(x,λ)={φ-(x,λ),x∈[a,c)φ+(x,λ),x∈(c,b]
satisfies (2) on whole [a,c)∪(c,b], the first boundary condition of (3), and both transmission conditions (5).By the same technique, we can define the solution by(12)ψ(x,λ)={ψ-(x,λ),x∈[a,c)ψ+(x,λ),x∈(c,b]
so that
(13)ψ+(b,λ)=α21+λα21′,∂ψ+(b,λ)∂x=α20+λα20′,(14)ψ-(c-,λ)=-1Δ34(Δ14ψ+(c+,λ)+Δ24∂ψ+(c+,λ)∂x),(15)∂ψ-(c-,λ)∂x=1Δ34(Δ13ψ+(c+,λ)+Δ23∂ψ+(c+,λ)∂x).
Consequently, ψ(x,λ) satisfies (2) on whole [a,c)∪(c,b], the second boundary condition (4), and both transmission condition (5). By using (9), (10), (14), and (15) and the well-known fact that the Wronskians w-(λ):=W[φ-(x,λ),ψ-(x,λ)] and w+(λ):=W[φ+(x,λ),ψ+(x,λ)] are independent of variable x, it is easy to show that Δ12w+(λ)=Δ34w-(λ). We will introduce the characteristic function for the problems (2)–(5) as
(16)w(λ):=Δ34w-(λ)=Δ12w+(λ).
Similar to [13], we can prove that there are infinitely many eigenvalues λn,n=1,2,… of the BVTP (2)–(5) which coincide with the zeros of characteristic function  w(λ).Now, let us consider the nonhomogenous differential equation(17)y′′+(λ-q(x))y=f(x),
on [a,c)∪(c,b] together with the same boundary and transmission conditions (2)–(5), when w(λ)≠0. We will search the solution of this problem in the form (see, for example, [13]):
(18)Y(x,λ)={Δ34ψ-(x,λ)ω(λ)×∫axφ-(y,λ)f(y)dy+Δ34φ-(x,λ)ωλ×∫xc-ψ-(y,λ)f(y)dy+d11φ-(x,λ)+d12ψ-(x,λ),forx∈[a,c),Δ12ψ+(x,λ)ω(λ)×∫c+xφ+(y,λ)f(y)dy+Δ12φ+(x,λ)ω(λ)×∫xbψ+(y,λ)f(y)dy+d21φ+(x,λ)+d22ψ+(x,λ)forx∈(c,b],
where dij(i,j=1,2) are arbitrary constants. Putting in (3)–(5) we have d12=0,  d21=0,(19)d11=Δ12w(λ)∫c+bf(y)ψ+(y,λ)dy,d22=Δ34w(λ)∫ac-f(y)φ-(y,λ)dy.
Now, by substituting these equalities in (18), the following formula is obtained for the solution Y=Y0(x,λ) of (17) under boundary and transmission conditions (3)–(5):
(20)Y0(x,λ)={Δ34ψ-(x,λ)ω(λ)×∫axφ-(y,λ)f(y)dy+Δ34φ-(x,λ)ω(λ)×∫xc-ψ-(y,λ)f(y)dy+Δ12φ-(x,λ)w(λ)×∫c+bf(y)ψ+(y,λ)dy,forx∈[a,c),Δ12ψ+(x,λ)ω(λ)×∫c+xφ+(y,λ)f(y)dy+Δ12φ+(x,λ)ω(λ)×∫xbψ+(y,λ)f(y)dy+Δ34ψ+(x,λ)w(λ)×∫ac-f(y)φ-(y,λ)dyforx∈(c,b].
From this formula, we find that the Green’s function of the problem (2)–(5) has the form:
(21)G0(x,y;λ)={φ(y,λ)ψ(x,λ)ω(λ),fora≤y≤x≤b,x,y≠c,φ(x,λ)ψ(y,λ)ω(λ),fora≤x≤y≤b,x,y≠c,
and the solution (20) can be rewritten in the terms of this Green’s function as
(22)Y0(x,λ)=Δ34∫ac-G0(x,y;λ)f(y)dy+Δ12∫c+bG0(x,y;λ)f(y)dy.


## 3. Construction of the Resolvent Operator by means of Green's Function in the Adequate Hilbert Space

In this section, we define a linear operatorA in suitable Hilbert space in such a way that the considered problem can be interpreted as the eigenvalue problem of this operator. For this, we assume that Δ0:=α21α20′-α20α21′>0 and introduce a new inner product in the Hilbert space H=(L2[a,c)⊕L2(c,b])⊕ℂ by
(23)〈F,G〉1:=Δ34∫ac-f(x)g(x)¯dx+Δ12∫c+bf(x)g(x)¯dx+Δ12Δ0f1g1¯,
for F=(f(x),f1),  G=(g(x),g1)∈H.Remark 1.
Note that this modified inner product is equivalent to standard inner product of(L2[a,c)⊕L2(c,b])⊕ℂ; so H1=(L2[a,c)⊕L2(c,b]⊕ℂ,〈·,·〉1) is also Hilbert space.For convenience, denote(24)Tb(f):=α20f(b)-α21f′(b),Tb′(f):=α20′f(b)-α21′f′(b),
and define a linear operator
(25)A(ℒf(x),Tb′(f))=(ℒf,-Tb(f))
with the domain D(A) consisting of all elements (f(x),f1)∈H1 such that f(x)  and  f′(x) are absolutely continuous in each interval [a,c) and (c,b] and has a finite limit f(c∓0) and f1′(c∓0), ℒf∈L2[a,b], τ1f=τ3f=τ4f=0 and f1=Tb′(f).Consequently the problems (2)–(5) can be written in the operator form as
(26)AF=λF,F=(f(x),Tb′(f))∈D(A)
in the Hilbert space H1. It is easy to see that the operator A is well defined in H1. Let A be defined as above and let λ not be an eigenvalue of this operator. For construction of the resolvent operator R(λ,A):=(λ-A)-1, we will solve the operator equation
(27)(λ-A)Y=F,
for F∈H1. This operator equation is equivalent to the nonhomogeneous differential equation
(28)y′′+(λ-q(x))y=f(x),
on [a,c)∪(c,b] subject to nonhomogeneous boundary conditions and homogeneous transmission conditions
(29)τ1(y)=τ3(y)=τ4(y)=0,τ2(y)=-f1.
Let Imλ≠0. We already know that the general solution of (28) has the form (18). Putting this general solution in (29) yields
(30)d11=Δ12ω(λ)∫c+bψ+(y,λ)f(y)dy+Δ12f1ω(λ),d12=0,d21=Δ12f1ω(λ),d22=Δ34ω(λ)∫ac-φ-(y,λ)f(y)dy.
Thus, the problems (28)-(29) have a unique solution
(31)Y(x,λ)={Δ34ψ-(x,λ)ω(λ)×∫axφ-(y,λ)f(y)dy+Δ34φ-(x,λ)ω(λ)×∫xc-ψ-(y,λ)f(y)dy+Δ12φ-(x,λ)ω(λ)×(∫c+bψ+(y,λ)f(y)dy+f1∫c+bψ+(y,λ)f(y))forx∈[a,c)Δ12ψ+(x,λ)ω(λ)×∫c+xφ+(y,λ)f(y)dy+Δ12φ+(x,λ)ω(λ)×∫xbψ+(y,λ)f(y)dy+Δ34ψ+(x,λ)ω(λ)×∫ac-φ-(y,λ)f(y)dy+Δ12f1φ+(x,λ)ω(λ),forx∈(c,b].
Consequently,
(32)Y(x,λ)=Y0(x,λ)+f1Δ12φ(x,λ)ω(λ),
where G0(x,λ) and Y0(x,λ) are the same with (21) and (22), respectively. From the equalities (13) and (21), it follows that
(33)(G0(x,·;λ))β′=φ(x,λ)ω(λ).
By using (22), (31), and (33), we deduce that
(34)Y(x,λ)=Δ34∫ac-G0(x,y;λ)f(y)dy+Δ12∫c+bG0(x,y;λ)f(y)dy+f1Δ12(G0(x,·;λ))β′.
Consequently, the solution Y(F,λ) of the operator equation (27) has the form:
(35)Y(F,λ)=(Y(x,λ),(Y(·,λ))β′).
From (34) and (35), it follows that
(36)Y(F,λ)=(〈Gx,λ,F¯〉1,(〈Gx,λ,F¯〉1)β′),
where under Green's vector Gx,λ we mean
(37)Gx,λ:=(G0(x,·;λ),(G0(x,·;λ))β′).
Now, making use of (21), (34), (35), (36), and (37), we see that if λ not an eigenvalue of operator A, then
(38)Y(F,λ)∈D(A),forF∈H1,(39)Y((λ-A)F,λ)=F,for∈D(A),∥Y(F,λ)∥≤|Imλ|-1∥F∥,forF∈H1,Imλ≠0.
Hence, each nonreal λ∈ℂ is a regular point of an operator A and
(40)R(λ,A)F=(〈Gx,λ,F¯〉1,(〈Gx,λ,F¯〉1)β′),forF∈H1.
Because of (38) and (40),
(41)(λ-A)D(A)=(λ¯-A)D(A)=H1,forImλ≠0.Theorem 2.
The Resolvent operatorR(λ,A) is compact in the Hilbert space H1.Proof.
Let us define the operators𝔹λ:L2[a,c)⊕L2(c,b]→L2[a,c)⊕L2(c,b], 𝔹λ~:H1→H1 and 𝒞λ:H1→H1 by
(42)𝔹λf:=Δ34∫ac-G0(x,y;λ)f(y)dy+Δ12∫c+bG0(x,y;λ)f(y)dy,𝔹λ~F:=(𝔹λf,(𝔹λf)b′),𝒞λF:=(f1Δ12φ(x,λ)ω(λ),f1Δ12(φ(·,λ))b′ω(λ)),
respectively. Then we can expressed the resolvent operator R(λ,A) as R(λ,A)=𝔹λ~+𝒞λ. Since the linear operator 𝔹λ is compact in the Hilbert space L2[a,c)⊕L2(c,b], the linear operator 𝔹λ~ is compact in the Hilbert space H1. Compactness 𝒞λ in H1 is obvious. Therefore, the resolvent operator R(λ,A) is also compact in H1.


## 4. Self-Adjoint Realization of the Problem

At first, we will prove the following lemmas.Lemma 3.
The domainD(A) is dense in H1.Proof.
Suppose that the elementG=(g(x),g1)∈H1 is orthogonal to D(A). Denote by C0∞[a,c)⊕C0∞(c,b] the set of infinitely differentiable functions on [a,c)∪(c,b], each of which vanishes on some neighborhoods of the end-points x=a,x=c, and  x=b. Since (f(·),0)∈D(A) for f∈C0∞[a,c)⊕C0∞(c,b], we have
(43)Δ34∫ac-f(x)g(x)¯dx+Δ12∫c+bf(x)g(x)¯dx=0,
for all f∈C0∞[a,c)⊕C0∞(c,b]. Since f is arbitrary,
(44)∫ac-f1(x)g(x)¯dx=0,∫c+bf2(x)g(x)¯dx=0,
for all f1∈C0∞[a,c]  and  f2∈C0∞[c,b], respectively. Taking into account that C0∞[a,c] and C0∞[c,b] are dense in L2[a,c] and L2[c,b], respectively, we get that the function g(x) is equal to zero as element of L2[a,c]⊕L2[c,b]. By choosing an element F0=(f0(x),Tb′f0) such that Tb′f0=1 and putting in 〈F0,G〉1=0, we have g1=0. So G=(0,0). The proof is completed.Lemma 4.
The linear operatorA is symmetric in the Hilbert space H1.Proof.
LetF=(f(x),Tb′(f)),  G=(G1(x),Tb′(f))∈D(A). By partial integration, we get
(45)〈AF,G〉1=Δ34∫ac-(ℒf)(x)g(x)¯dx+Δ12∫c+b(ℒf)(x)g(x)¯dx+Δ12Δ0Tb(f)Tb′(g)¯=〈F,AG〉1+Δ34W(f,g¯;c-0)-Δ34W(f,g¯;a)+Δ12W(f,g¯;b)-Δ12W(f,g¯;c+0)+Δ12Δ0(Tb′(f)Tb(g)¯-Tb(f)Tb′(g)¯).
From the definition of domain D(A), we see easily that W(f,g¯;a)=0. The direct calculation gives
(46)Tb′(f)Tb(g)¯-Tb(f)Tb′(g)¯=-Δ0W(f,g¯;b),W(f,g¯;c-0)=Δ12Δ34W(f,g¯;c+0).
Substituting these equalities in (45), we have
(47)〈AF,G〉1=〈F,AG〉1,foreveryF,G∈D(A);
so the operator A is symmetric in H. The proof is completed.Remark 5.
By Lemma4, all eigenvalues of the problems (2)–(5) are real. Therefore, it is enough to investigate only real-valued eigenfunctions. Taking in view this fact, we can assume that the eigenfunctions are real-valued.Corollary 6.
Ifλn and λm are distinct eigenvalues of the problems (2)–(5), then the corresponding eigenfunctions un(x)  and  um(x)  are orthogonal in the sense of the following equality:
(48)Δ34∫ac-u(x)v(x)dx+Δ12∫c+bu(x)v(x)dx+Δ12Δ0Tb′(u)Tb′(v)=0.Proof.
The proof is immediate from the fact that the eigenelements(u(x),Tb′(u)) and (v(x),Tb′(v)) of the symmetric linear operator A are orthogonal in the Hilbert space H1.Theorem 7.
The operatorA is self-adjoint in H1.Proof.
It is clear that the symmetry of a densely definedA is equivalent to the condition 〈AF,G〉1=〈F,AG〉1 for all F,G∈D(A). Notice that this implies that A*⊃A. If, in addition, we also have that D(A*)=D(A), then A is self-adjoint. Let U∈D(A*). Then, by definition of A*,
(49)〈AV,U〉1=〈V,A*U〉1,∀V∈D(A).
Let λ0 be any complex number for which Imλ0≠0. From this, it follows that
(50)〈(λ0I-A)V,U〉1=〈V,(λ0¯I-A*)U〉1.
Since any nonreal complex number is a regular point of A, we can define the vector U0∈D(A) as
(51)U0=R(λ0¯,A)(λ0¯U-A*U).
Hence,
(52)(λ0¯I-A)U0=λ0¯U-A*U.
Inserting this in (50) and recalling that A is symmetric and U0∈D(A), we have
(53)〈(λ0I-A)V,U〉1=〈V,(λ0¯I-A)U0〉1=〈V,λ0¯U0〉1-〈V,AU0〉1=〈λ0V,U0〉1-〈AV,U0〉1=〈(λ0I-A)V,U0〉1.
Consequently,
(54)〈(λ0I-A)V,U-U0〉1=0∀V∈H1.
Since λ0 is regular point of A, we can choose V=R(λ0,A)(U-U0). Inserting this in the last equality yields ∥U-U0∥1=0, and so U=U0, and therefore, U∈D(A). The proof is completed.Remark 8.
The main results of this study are derived in modified Hilbert space under simple condition (7). We can show that these conditions cannot be omitted. Indeed, let us consider the next special case of the problems (2)–(5):
(55)-y′′(x)=λy(x),x∈[-1,0)∪(0,1],y(-1)=0,(λ-1)y′(-1)+λy(1)=0,y(0-)=y(0+),y′(0-)=-y′(0+),
for which the condition (7) is not valid (Δ12<0). It is easy to verify that the operator A corresponding to this problem is not symmetric in the classic Hilbert space L2[a,c)⊕L2(c,b]⊕ℂ under standard inner-product. Consider the following:
(56)〈F,G〉1:=∫acf(x)g(x)¯dx+∫cbf(x)g(x)¯dx+Δ12Δ0f1g1¯.
Moreover, it is well known that the standard Sturm-liouville problems have infinitely many real eigenvalues. But it can be shown by direct calculation that the problem (55) has only one eigenvalue λ=1.



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