EQUIVALENCE OF STURM-LIOUVILLE PROBLEM WITH FINITELY MANY δ-INTERACTIONS AND MATRIX EIGENVALUE PROBLEMS

The purpuse of this article is to show the matrix representations of Sturm-Liouville operators with finitely many δ-interactions. We show that a Sturm-Liouville problem with finitely many δ-interactions can be represented as a finite dimensional matrix eigenvalue problem which has the same eigenvalue with the former Sturm-Liouville operator. Moreover an example is also presented.

and boundary conditions has infinite spectrum under some assumptions.Atkinson in his book [1] suggested that if the coefficients of SLP satisfy some conditions, the problem may have finite eigenvalues.Then in [2], Kong, Wu and Zettl obtained the following result: For every positive integer n, we can construct a class of regular self-adjoint and nonself-adjoint SLP with exactly n eigenvalues by choosing p and w such that 1/p and w are alternatively zero on consecutive subintervals.
Recently, there has been much attention paid to the SLPs with finite spectrum.For a comprehensive treatment of the subject we refer the reader to the book by Zettl [3], and the papers by Kong, Wu and Zettl [2], Ao, Sun, and Zhang [4], [5] and Ao, Bo and Sun [6], [7].In 2009, the equivalence of SLP with a matrix eigenvalue problem was first constructed by Volkmer and Zettl in [8].By equivalance of matrix eigenvalue problems for the SLPs with finite spectrum we mean to construct a matrix eigenvalue problem with exactly the same eigenvalues as the corresponding SLP.Then, the matrix representations of SLPs with finite spectrum are extended to various problems.For the SLPs see [8]- [11] and for fourth order boundary value problems see [12]- [16].
The goal of this paper is to find the matrix representation of the following Sturm-Liouville problem with finitely many δ-interactions: where is the Dirac delta function and λ ∈ C is a spectral parameter.Sturm-Liouville equations with Dirac delta function potentials often appear in quantum mechanics.For example, such an equations had been used for modelling of atomic and molecular systems including atomic lattices, quantum heterostructures, semiconductors, organic fluorescent materials, solar cells etc. (see [17], [18], [19] and citations of them).
Recently, we generalize the finite spectrum result to the problem (1.1) in [20].The equation (1.1) is equivalent to the many-point boundary value problem, (see [19]).So we can understand problem (1.1) as studying the equation and n transmission conditions where x j 's are inner discontinuity points and Additionally, let us consider the boundary conditions of the form where A = (a ij ) 2×2 , B = (b ij ) 2×2 are complex valued 2 × 2 matrices and M 2 (C) denotes the set of square matrices of order 2 over C. Here, the coefficients fulfill the following minimal conditions: where L(J, C) denotes the complex valued functions which are Lebesgue integrable on J.
The BC (1.3) is said to be self-adjoint if the following two conditions are satisfied: It is well known that under the condition (1.5), the BCs (1.The real coupled boundary conditions have the canonical representation: Let u = y and v = (py ).Then we have the system representation of equation (1.2) 3) is said to be of Atkinson type if, for some integers m j ≥ 1, j = 0, 1, ..., n, there exists a partition of the interval J a = x 00 < x 01 < x 02 < ... < x 0,2m0+1 = x 1 , (2.1) . . .
Definition 2.2.A SLP with finitely many δ-Interactions of Atkinson type is said to be equivalent to a matrix eigenvalue problem if the former has exactly the same eigenvalues as the latter.
On the other hand, if u jk , v jk satisfy (2.8) and (2.9), then we define u(x) and v(x) according to (2.6) and (2.7), and then extend them continuously to the whole interval J as a solution of (1.9) by integrating the equations in (1.9) over subintervals.
Proof.If we divide the first and the last rows of system (2.10) by sin α and sin β respectively, then we obtain (2.22).
Theorem 2.1 and its Corollary show that the problem (

) of Atkinson type have represen-
tations by tridiagonal matrix eigenvalue problems.Now, we will show that the problem ( Atkinson type also have representations.

Example
In this section, we give an example to illustrate that a SLP with finitely many δ-interactions and it's equivalent matrix eigenvalue problem, we will construct it, have same eigenvalues.

Theorem 2 . 2 .
Consider the boundary condition (1.8) with k 12 = 0. Define the m × m matrix which is tridiagonal except for the (1, m) and (m, 1) entries