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

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.

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 𝑔=𝛿()𝑥,𝑦,𝑧,𝑡∣𝜉,𝜂,𝜍,𝜏𝑡−𝜏−𝑅/𝑐,4𝜋𝑅 (1) 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: ℒ𝑦∶=−𝑦^′′(𝑥)(𝑥)𝑦(𝑥)(𝑥),[)∪(]+𝑞=𝜆𝑦𝑥∈𝑎,𝑐𝑐,𝑏 (2) on [𝑎,𝑐)∪(𝑐,𝑏], with eigenparameter-dependent boundary conditions at the end points 𝑥=𝑎 and 𝑥=𝑏. One has, 𝜏₁𝑦∶=𝛼₁₀𝑦(𝑎)+𝛼₁₁𝑦^′(𝑎)𝜏=0,₂𝑦∶=𝛼₂₀𝑦(𝑏)−𝛼₂₁𝑦^′(𝑏)+𝜆𝛼^′₂₀𝑦(𝑏)−𝛼^′₂₁𝑦^′(𝑏)=0 (3)(4) and the transmission conditions at the singular interior point 𝑥=𝑐𝜏₃𝑦∶=𝛽⁻₁₁𝑦^′(𝑐−)+𝛽⁻₁₀𝑦(𝑐−)+𝛽⁺₁₁𝑦^′(𝑐+)+𝛽⁺₁₀𝑦()𝜏𝑐+=0,₄𝑦∶=𝛽⁻₂₁𝑦^′(𝑐−)+𝛽⁻₂₀𝑦(𝑐−)+𝛽⁺₂₁𝑦^′(𝑐+)+𝛽⁺₂₀𝑦(𝑐+)=0, (5) 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, 𝛼_𝑖𝑗,  𝛽^±_𝑖𝑗,(𝑖=1,2 and 𝑗=0,1),  and  𝛼^′_𝑖𝑗(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 parameter 𝜆 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 the 𝑘th and 𝑗th columns of the matrix 𝛽𝑇=⁺¹⁰𝛽⁺₁₁𝛽⁻₁₀𝛽⁻₁₁𝛽⁺₂₀𝛽⁺₂₁𝛽⁻₂₀𝛽⁻₂₁ (6) by Δ_𝑘𝑗(1≤𝑘<𝑗≤4). For self-adjoint realization in adequate Hilbert space, everywhere below we will assume that Δ₁₂>0,Δ₃₄>0. (7) 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)−𝑦^′′(𝑥)𝑦(𝑎)+𝑞𝑦=𝜆𝑦,=𝛼₁₁,𝑦^′(𝑎)=−𝛼₁₀. (8) 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 𝜑⁺()=1𝑐+,𝜆Δ₁₂Δ₂₃𝜑⁻(𝑐−,𝜆)+Δ₂₄𝜕𝜑⁻(𝑐−,𝜆),𝜕𝑥𝜕𝜑⁺(𝑐+,𝜆)=𝜕𝑥−1Δ₁₂Δ₁₃𝜑⁻(𝑐−,𝜆)+Δ₁₄𝜕𝜑⁻(𝑐−,𝜆),𝜕𝑥 (9)(10) 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 𝜑()=𝜑𝑥,𝜆⁻()[)𝜑𝑥,𝜆,𝑥∈𝑎,𝑐₊(𝑥,𝜆),𝑥∈(𝑐,𝑏] (11) 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 by𝜓()=𝜓𝑥,𝜆⁻()[)𝜓𝑥,𝜆,𝑥∈𝑎,𝑐₊(𝑥,𝜆),𝑥∈(𝑐,𝑏] (12) so that 𝜓⁺(𝑏,𝜆)=𝛼₂₁+𝜆𝛼^′₂₁,𝜕𝜓⁺(𝑏,𝜆)𝜕𝑥=𝛼₂₀+𝜆𝛼^′₂₀,𝜓⁻(𝑐−,𝜆)=−1Δ₃₄Δ₁₄𝜓⁺(𝑐+,𝜆)+Δ₂₄𝜕𝜓⁺(𝑐+,𝜆)𝜕𝑥,𝜕𝜓⁻(𝑐−,𝜆)=1𝜕𝑥Δ₃₄Δ₁₃𝜓⁺(𝑐+,𝜆)+Δ₂₃𝜕𝜓⁺(𝑐+,𝜆)𝜕𝑥. (13)(14)(15) 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 𝑤(𝜆)∶=Δ₃₄𝑤⁻(𝜆)=Δ₁₂𝑤⁺(𝜆). (16) Similar to [13], we can prove that there are infinitely many eigenvalues 𝜆_𝑛,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 𝑦^′′+𝜆−𝑞(𝑥)𝑦=𝑓(𝑥), (17) 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]): 𝑌()=Δ𝑥,𝜆³⁴𝜓⁻(𝑥,𝜆)𝜔(𝜆)×^𝑥_𝑎𝜑⁻𝑓𝑦+Δ𝑦,𝜆𝑑𝑦₃₄𝜑⁻(𝑥,𝜆)×𝜔𝜆_𝑥𝑐−𝜓⁻𝑦,𝜆𝑓𝑦𝑑𝑦+𝑑₁₁𝜑⁻(𝑥,𝜆)+𝑑₁₂𝜓⁻()[),Δ𝑥,𝜆,for𝑥∈𝑎,𝑐₁₂𝜓⁺(𝑥,𝜆)𝜔(𝜆)×^𝑥_𝑐+𝜑⁺𝑓𝑦+Δ𝑦,𝜆𝑑𝑦₁₂𝜑⁺(𝑥,𝜆)𝜔(𝜆)×^𝑏_𝑥𝜓⁺𝑦,𝜆𝑓𝑦𝑑𝑦+𝑑₂₁𝜑⁺(𝑥,𝜆)+𝑑₂₂𝜓⁺(𝑥,𝜆)for𝑥∈(𝑐,𝑏], (18) where 𝑑_𝑖𝑗(𝑖,𝑗=1,2) are arbitrary constants. Putting in (3)–(5) we have 𝑑₁₂=0,  𝑑₂₁=0,𝑑₁₁=Δ₁₂𝑤(𝜆)^𝑏_𝑐+𝑓𝑦𝜓⁺𝑑𝑦,𝜆𝑑𝑦,₂₂=Δ₃₄𝑤(𝜆)_𝑎^𝑐−𝑓𝑦𝜑⁻𝑦,𝜆𝑑𝑦. (19) 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): 𝑌₀()=Δ𝑥,𝜆³⁴𝜓⁻(𝑥,𝜆)𝜔(𝜆)×^𝑥_𝑎𝜑⁻𝑓𝑦+Δ𝑦,𝜆𝑑𝑦₃₄𝜑⁻(𝑥,𝜆)𝜔(𝜆)×_𝑥^𝑐−𝜓⁻𝑓𝑦+Δ𝑦,𝜆𝑑𝑦₁₂𝜑⁻(𝑥,𝜆)𝑤(𝜆)×^𝑏_𝑐+𝑓𝑦𝜓⁺[),Δ𝑦,𝜆𝑑𝑦,for𝑥∈𝑎,𝑐₁₂𝜓⁺(𝑥,𝜆)𝜔(𝜆)×^𝑥_𝑐+𝜑⁺𝑓𝑦+Δ𝑦,𝜆𝑑𝑦₁₂𝜑⁺(𝑥,𝜆)𝜔(𝜆)×^𝑏_𝑥𝜓⁺𝑓𝑦+Δ𝑦,𝜆𝑑𝑦₃₄𝜓⁺(𝑥,𝜆)𝑤(𝜆)×_𝑎^𝑐−𝑓𝑦𝜑⁻𝑦,𝜆𝑑𝑦for𝑥∈(𝑐,𝑏]. (20) From this formula, we find that the Green’s function of the problem (2)–(5) has the form: 𝐺₀=𝜑𝜓()𝑥,𝑦;𝜆𝑦,𝜆𝑥,𝜆𝜔(𝜆)𝜑()𝜓,for𝑎≤𝑦≤𝑥≤𝑏,𝑥,𝑦≠𝑐,𝑥,𝜆𝑦,𝜆𝜔(𝜆),for𝑎≤𝑥≤𝑦≤𝑏,𝑥,𝑦≠𝑐, (21) and the solution (20) can be rewritten in the terms of this Green’s function as 𝑌₀(𝑥,𝜆)=Δ_34𝑎^𝑐−𝐺₀𝑥,𝑦;𝜆𝑓𝑦𝑑𝑦+Δ₁₂^𝑏_𝑐+𝐺₀𝑥,𝑦;𝜆𝑓𝑦𝑑𝑦. (22)

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

In this section, we define a linear operator 𝐴 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∶=𝛼₂₁𝛼^′₂₀−𝛼₂₀𝛼^′₂₁>0 and introduce a new inner product in the Hilbert space 𝐻=(𝐿₂[𝑎,𝑐)⊕𝐿₂(𝑐,𝑏])⊕ℂ by ⟨𝐹,𝐺⟩₁∶=Δ_34𝑎^𝑐−𝑓(𝑥)𝑔(𝑥)𝑑𝑥+Δ₁₂^𝑏_𝑐+𝑓(𝑥)𝑔(𝑥)Δ𝑑𝑥+¹²Δ₀𝑓₁𝑔₁, (23) 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 𝑇_𝑏𝑓∶=𝛼₂₀𝑓(𝑏)−𝛼₂₁𝑓^′(𝑏),𝑇^′_𝑏𝑓∶=𝛼^′₂₀𝑓(𝑏)−𝛼^′₂₁𝑓^′(𝑏), (24) and define a linear operator 𝐴ℒ𝑓(𝑥),𝑇^′_𝑏𝑓=ℒ𝑓,−𝑇_𝑏𝑓 (25) 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 𝑓^′₁(𝑐∓0), ℒ𝑓∈𝐿₂[𝑎,𝑏], 𝜏₁𝑓=𝜏₃𝑓=𝜏₄𝑓=0 and 𝑓₁=𝑇^′_𝑏(𝑓).

Consequently the problems (2)–(5) can be written in the operator form as 𝑓(𝑥)𝐴𝐹=𝜆𝐹,𝐹=,𝑇^′_𝑏𝑓∈𝐷(𝐴) (26) 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 (𝜆−𝐴)𝑌=𝐹, (27) for 𝐹∈𝐻₁. This operator equation is equivalent to the nonhomogeneous differential equation 𝑦^′′+𝜆−𝑞(𝑥)𝑦=𝑓(𝑥), (28) on [𝑎,𝑐)∪(𝑐,𝑏] subject to nonhomogeneous boundary conditions and homogeneous transmission conditions 𝜏₁𝑦=𝜏₃𝑦=𝜏₄𝑦=0,𝜏₂𝑦=−𝑓₁. (29) Let 𝐼𝑚𝜆≠0. We already know that the general solution of (28) has the form (18). Putting this general solution in (29) yields 𝑑₁₁=Δ₁₂𝜔(𝜆)^𝑏_𝑐+𝜓⁺𝑓𝑦Δ𝑦,𝜆𝑑𝑦+¹²𝑓₁𝜔(𝜆),𝑑₁₂=0,𝑑₂₁=Δ₁₂𝑓₁𝜔(𝜆),𝑑₂₂=Δ₃₄𝜔(𝜆)_𝑎^𝑐−𝜑⁻𝑦,𝜆𝑓𝑦𝑑𝑦. (30) Thus, the problems (28)-(29) have a unique solution 𝑌()=Δ𝑥,𝜆³⁴𝜓⁻(𝑥,𝜆)𝜔(𝜆)×^𝑥_𝑎𝜑⁻𝑓𝑦+Δ𝑦,𝜆𝑑𝑦₃₄𝜑⁻(𝑥,𝜆)𝜔(𝜆)×_𝑥^𝑐−𝜓⁻𝑓𝑦+Δ𝑦,𝜆𝑑𝑦₁₂𝜑⁻(𝑥,𝜆)𝜔(𝜆)×^𝑏_𝑐+𝜓⁺𝑦,𝜆𝑓𝑦𝑑𝑦+𝑓₁[)Δfor𝑥∈𝑎,𝑐₁₂𝜓⁺(𝑥,𝜆)𝜔(𝜆)×^𝑥_𝑐+𝜑⁺𝑓𝑦+Δ𝑦,𝜆𝑑𝑦₁₂𝜑⁺(𝑥,𝜆)𝜔(𝜆)×^𝑏_𝑥𝜓⁺𝑓𝑦+Δ𝑦,𝜆𝑑𝑦₃₄𝜓⁺(𝑥,𝜆)𝜔(𝜆)×_𝑎^𝑐−𝜑⁻𝑓𝑦+Δ𝑦,𝜆𝑑𝑦₁₂𝑓₁𝜑⁺(𝑥,𝜆)𝜔(𝜆),for𝑥∈(𝑐,𝑏]. (31) Consequently, 𝑌(𝑥,𝜆)=𝑌₀(𝑥,𝜆)+𝑓₁Δ₁₂𝜑(𝑥,𝜆)𝜔(𝜆), (32) 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 𝐺₀(𝑥,⋅;𝜆)^′_𝛽=𝜑(𝑥,𝜆)𝜔(𝜆). (33) By using (22), (31), and (33), we deduce that 𝑌(𝑥,𝜆)=Δ_34𝑎^𝑐−𝐺₀𝑥,𝑦;𝜆𝑓𝑦𝑑𝑦+Δ₁₂^𝑏_𝑐+𝐺₀𝑥,𝑦;𝜆𝑓𝑦𝑑𝑦+𝑓₁Δ₁₂𝐺₀(𝑥,⋅;𝜆)^′_𝛽. (34) Consequently, the solution 𝑌(𝐹,𝜆) of the operator equation (27) has the form: 𝑌(𝐹,𝜆)=𝑌(𝑥,𝜆),(𝑌(⋅,𝜆))^′_𝛽. (35) From (34) and (35), it follows that 𝑌(𝐹,𝜆)=𝐺_𝑥,𝜆,𝐹₁,𝐺_𝑥,𝜆,𝐹₁^′_𝛽, (36) where under Green's vector 𝐺_𝑥,𝜆 we mean 𝐺_𝑥,𝜆∶=𝐺₀(𝑥,⋅;𝜆),𝐺₀(𝑥,⋅;𝜆)^′_𝛽. (37) Now, making use of (21), (34), (35), (36), and (37), we see that if 𝜆 not an eigenvalue of operator 𝐴, then 𝑌(𝐹,𝜆)∈𝐷(𝐴),for𝐹∈𝐻₁,𝑌))(𝐴),‖𝑌(≤||((𝜆−𝐴𝐹,𝜆=𝐹,for∈𝐷𝐹,𝜆)‖𝐼𝑚𝜆⁻¹‖𝐹‖,for𝐹∈𝐻₁,𝐼𝑚𝜆≠0. (38)(39) Hence, each nonreal 𝜆∈ℂ is a regular point of an operator 𝐴 and 𝑅(𝜆,𝐴)𝐹=𝐺_𝑥,𝜆,𝐹₁,𝐺_𝑥,𝜆,𝐹₁^′_𝛽,for𝐹∈𝐻₁. (40) Because of (38) and (40), (𝜆−𝐴)𝐷(𝐴)=𝜆−𝐴𝐷(𝐴)=𝐻₁,for𝐼𝑚𝜆≠0. (41)

Theorem 2. The Resolvent operator 𝑅(𝜆,𝐴) is compact in the Hilbert space 𝐻₁.

Proof. Let us define the operators 𝔹_𝜆∶𝐿₂[𝑎,𝑐)⊕𝐿₂(𝑐,𝑏]→𝐿₂[𝑎,𝑐)⊕𝐿₂(𝑐,𝑏], 𝔹_𝜆∶𝐻₁→𝐻₁ and 𝒞_𝜆∶𝐻₁→𝐻₁ by 𝔹_𝜆𝑓∶=Δ_34𝑎^𝑐−𝐺₀𝑥,𝑦;𝜆𝑓𝑦𝑑𝑦+Δ₁₂^𝑏_𝑐+𝐺₀𝑓𝑦𝔹𝑥,𝑦;𝜆𝑑𝑦,_𝜆𝐹∶=𝔹_𝜆𝑓,𝔹_𝜆𝑓^′_𝑏,𝒞_𝜆𝐹∶=𝑓₁Δ₁₂𝜑(𝑥,𝜆)𝜔(𝜆),𝑓₁Δ₁₂𝜑(⋅,𝜆)^′_𝑏𝜔(𝜆), (42) respectively. Then we can expressed the resolvent operator 𝑅(𝜆,𝐴) as 𝑅(𝜆,𝐴)=𝔹_𝜆+𝒞_𝜆. Since the linear operator 𝔹_𝜆 is compact in the Hilbert space 𝐿₂[𝑎,𝑐)⊕𝐿₂(𝑐,𝑏], the linear operator 𝔹_𝜆 is compact in the Hilbert space 𝐻₁. Compactness 𝒞_𝜆 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 domain 𝐷(𝐴) is dense in 𝐻₁.

Proof. Suppose that the element 𝐺=(𝑔(𝑥),𝑔₁)∈𝐻₁ is orthogonal to 𝐷(𝐴). Denote by 𝐶^∞₀[𝑎,𝑐)⊕𝐶^∞₀(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∈𝐶^∞₀[𝑎,𝑐)⊕𝐶^∞₀(𝑐,𝑏], we have Δ_34𝑎^𝑐−𝑓(𝑥)𝑔(𝑥)𝑑𝑥+Δ₁₂^𝑏_𝑐+𝑓(𝑥)𝑔(𝑥)𝑑𝑥=0, (43) for all 𝑓∈𝐶^∞₀[𝑎,𝑐)⊕𝐶^∞₀(𝑐,𝑏]. Since 𝑓 is arbitrary, _𝑎^𝑐−𝑓₁(𝑥)𝑔(𝑥)𝑑𝑥=0,_𝑏𝑐+𝑓₂(𝑥)𝑔(𝑥)𝑑𝑥=0, (44) for all 𝑓₁∈𝐶^∞₀[𝑎,𝑐]  and  𝑓₂∈𝐶^∞₀[𝑐,𝑏], respectively. Taking into account that 𝐶^∞₀[𝑎,𝑐] and 𝐶^∞₀[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),𝑇^′_𝑏𝑓₀) such that 𝑇^′_𝑏𝑓₀=1 and putting in ⟨𝐹₀,𝐺⟩₁=0, we have 𝑔₁=0. So 𝐺=(0,0). The proof is completed.

Lemma 4. The linear operator 𝐴 is symmetric in the Hilbert space 𝐻₁.

Proof. Let 𝐹=(𝑓(𝑥),𝑇^′_𝑏(𝑓)),  𝐺=(𝐺₁(𝑥),𝑇^′_𝑏(𝑓))∈𝐷(𝐴). By partial integration, we get ⟨𝐴𝐹,𝐺⟩₁=Δ_34𝑎^𝑐−ℒ𝑓(𝑥)𝑔(𝑥)𝑑𝑥+Δ₁₂^𝑏_𝑐+ℒ𝑓(𝑥)𝑔(𝑥)Δ𝑑𝑥+¹²Δ₀𝑇_𝑏𝑓𝑇^′_𝑏𝑔=⟨𝐹,𝐴𝐺⟩₁+Δ₃₄𝑊𝑓,𝑔;𝑐−0−Δ₃₄𝑊𝑓,𝑔;𝑎+Δ₁₂𝑊𝑓,𝑔;𝑏−Δ₁₂𝑊𝑓,+Δ𝑔;𝑐+0₁₂Δ₀𝑇^′_𝑏𝑓𝑇_𝑏𝑔−𝑇_𝑏𝑓𝑇^′_𝑏𝑔. (45) From the definition of domain 𝐷(𝐴), we see easily that 𝑊(𝑓,𝑔;𝑎)=0. The direct calculation gives 𝑇^′_𝑏𝑓𝑇_𝑏𝑔−𝑇_𝑏𝑓𝑇^′_𝑏𝑔=−Δ₀𝑊𝑓,,𝑊𝑔;𝑏𝑓,=Δ𝑔;𝑐−0¹²Δ₃₄𝑊𝑓,𝑔;𝑐+0. (46) Substituting these equalities in (45), we have ⟨𝐴𝐹,𝐺⟩₁=⟨𝐹,𝐴𝐺⟩₁,forevery𝐹,𝐺∈𝐷(𝐴); (47) so the operator 𝐴 is symmetric in 𝐻. The proof is completed.

Remark 5. By Lemma 4, 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. If 𝜆_𝑛 and 𝜆_𝑚 are distinct eigenvalues of the problems (2)–(5), then the corresponding eigenfunctions 𝑢_𝑛(𝑥) and 𝑢_𝑚(𝑥) are orthogonal in the sense of the following equality: Δ_34𝑎^𝑐−𝑢(𝑥)𝑣(𝑥)𝑑𝑥+Δ₁₂^𝑏_𝑐+𝑢(𝑥)𝑣(𝑥)+Δ𝑑𝑥₁₂Δ₀𝑇^′_𝑏(𝑢)𝑇^′_𝑏(𝑣)=0. (48)

Proof. The proof is immediate from the fact that the eigenelements (𝑢(𝑥),𝑇^′_𝑏(𝑢)) and (𝑣(𝑥),𝑇^′_𝑏(𝑣)) of the symmetric linear operator 𝐴 are orthogonal in the Hilbert space 𝐻₁.

Theorem 7. The operator 𝐴 is self-adjoint in 𝐻₁.

Proof. It is clear that the symmetry of a densely defined 𝐴 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^∗, ⟨𝐴𝑉,𝑈⟩₁=𝑉,𝐴^∗𝑈₁,∀𝑉∈𝐷(𝐴). (49) Let 𝜆₀ be any complex number for which 𝐼𝑚𝜆₀≠0. From this, it follows that 𝜆₀𝐼−𝐴𝑉,𝑈₁=𝑉,𝜆₀𝐼−𝐴^∗𝑈₁. (50) Since any nonreal complex number is a regular point of 𝐴, we can define the vector 𝑈₀∈𝐷(𝐴) as 𝑈₀=𝑅𝜆₀,𝐴𝜆₀𝑈−𝐴^∗𝑈. (51) Hence, 𝜆₀𝐼−𝐴𝑈₀=𝜆₀𝑈−𝐴^∗𝑈. (52) Inserting this in (50) and recalling that 𝐴 is symmetric and 𝑈₀∈𝐷(𝐴), we have 𝜆₀𝐼−𝐴𝑉,𝑈₁=𝑉,𝜆₀𝐼−𝐴𝑈₀₁=𝑉,𝜆₀𝑈₀₁−𝑉,𝐴𝑈₀₁=𝜆₀𝑉,𝑈₀₁−𝐴𝑉,𝑈₀₁=𝜆₀𝐼−𝐴𝑉,𝑈₀₁. (53) Consequently, 𝜆₀𝐼−𝐴𝑉,𝑈−𝑈₀₁=0∀𝑉∈𝐻₁. (54) 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): −𝑦^′′(𝑥)(𝑥)[)∪(],𝑦()()𝑦=𝜆𝑦,𝑥∈−1,00,1−1=0,𝜆−1^′()(1)𝑦()(),𝑦−1+𝜆𝑦=0,0−=𝑦0+_′(0−)=−𝑦^′(0+), (55) 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: ⟨𝐹,𝐺⟩₁∶=^𝑐_𝑎𝑓(𝑥)𝑔(𝑥)𝑑𝑥+^𝑏_𝑐𝑓(𝑥)𝑔(𝑥)Δ𝑑𝑥+¹²Δ₀𝑓₁𝑔₁. (56) 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.