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

and Applied Analysis 3


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 ∇ 2  =  for the electric potential  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/| −  0 | as in the three-dimensional case but by a singularity of the form log(1/| −  0 |).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 freespace 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 where 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][8][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][11][12][13][14][15][16][17][18]).In this study, we will investigate some spectral properties of the Sturm-Liouville differential equation on two intervals: where the potential () is real continuous function in each of the intervals [, ) and (, ] and has finite limits ( ∓ 0),  is a complex spectral parameter,   ,  ±  , ( = 1, 2 and  = 0, 1), and    ( = 2 and  = 0, 1) are real numbers.Our problem differs from the usual regular Sturm-Liouville problem in the sense that the eigenvalue parameter  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.

Some Basic Solutions and Green's Function
Denote the determinant of the th and th columns of the matrix by Δ  (1 ≤  <  ≤ 4).For self-adjoint realization in adequate Hilbert space, everywhere below we will assume that With a view to construct the Green's function we will define two special solutions of (2) by our own technique as follows.

Construction of the Resolvent Operator by means of Green's Function in the Adequate Hilbert Space
In this section, we define a linear operator  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 Δ 0 :=  21   20 −  20   21 > 0 and introduce a new inner product in the Hilbert space  = ( 2 [, ) ⊕  2 (, ]) ⊕ C by for  = ((),  1 ),  = ((),  1 ) ∈ .
Consequently the problems (2)-( 5) can be written in the operator form as in the Hilbert space  1 .It is easy to see that the operator  is well defined in  1 .Let  be defined as above and let  not be an eigenvalue of this operator.For construction of the resolvent operator (, ) := ( − ) −1 , we will solve the operator equation for  ∈  1 .This operator equation is equivalent to the nonhomogeneous differential equation on [, ) ∪ (, ] subject to nonhomogeneous boundary conditions and homogeneous transmission conditions Let  ̸ = 0. We already know that the general solution of (28) has the form (18) (30) Thus, the problems (28)-( 29) have a unique solution Consequently, the solution (, ) of the operator equation ( 27) has the form: From ( 34) and ( 35), it follows that where under Green's vector  , we mean  , := ( 0 (, ⋅; ) , ( 0 (, ⋅; )) Now, making use of ( 21), (34), ( 35), (36), and (37), we see that if  not an eigenvalue of operator , then respectively.Then we can expressed the resolvent operator (, ) as (, ) = B + C  .Since the linear operator B  is compact in the Hilbert space  2 [, ) ⊕  2 (, ], the linear operator B is compact in the Hilbert space  1 .Compactness C  in  1 is obvious.Therefore, the resolvent operator (, ) is also compact in  1 .

Self-Adjoint Realization of the Problem
At first, we will prove the following lemmas.
Proof.The proof is immediate from the fact that the eigenelements ((),    ()) and (V(),    (V)) of the symmetric linear operator  are orthogonal in the Hilbert space  1 .
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