A Discontinuous Sturm-Liouville Operator With Indefinite Weight

The research is financed by the Natural Science Foundation of China and the National Natural Science Foundation of Neimongo. No. 10661008 and 200711020102 (Sponsoring information) Abstract In this paper, we consider an indefinite Sturm-Liouville operator with eigenparameter-dependent boundary conditions and transmission conditions. In an appropriate space K, we define a new self-adjoint operator A such that the eigenvalues of A coincide with those of such a problem and obtain asymptotic approximation for its eigenvalues and eigenfunctions.


Introduction
In recent years, more and more researchers are interested in the discontinuous Sturm-Liouville problem for its application in physics (Demirci M., 2004, p.101-113 andBuschmann D., 1995, p.169-186).The various physics applications of this kind of problem are found in many literature, including some boundary value with transmission conditions that arise in the theory of heat and mass transfer (Aiping W., 2006, p.66-74 and Akdogan Z., 2007, p.1719-1738).
Here we consider a discontinuous Sturm-Liouville problem with the indefinite weight function r(x).By using the technics of (Kadakal M., 2005, p.229-245 and Kadakal M., 2006, p.1519-1528) and some new approaches, we define a new linear operator A associated with the problem on an appropriate Krein space K.We discuss its eigenvalues and eigenfunctions, and derive asymptotic approximation formulas for eigenvalues and eigenfunctions.
Lemma 2.1 The eigenvalues and eigenfunctions of the problem (1)-( 5) are defined as the eigenvalues and the first components of the corresponding eigenelements of the operator A, respectively.
Lemma 2.2 The domain D(A) is dense in H. where Similarly we have q = 0.So F = (0, 0, 0).Hence, D(A) is dense in H. Theorem 2.3 The linear operator A is self-adjoint in K.

Proof:
The operator A is self-adjoint in Krein space K if and only if the operator JA is self-adjoint in Hilbert space H.
For all F, G ∈ D(A).By two partial integrations we obtain where, as usual, by W( f, g; x) we denote the Wronskians f (x)g (x) − f (x)g(x).
In the following, we show that for all Hence, by standard Sturm-Liouville theory, (i) and (iv) hold.By (iv), the equation By Naimark , s Patching Lemma (Naimark M.A. (1968)) that there exists an F ∈ D(A) such that Then from ( 7), (ii) be true.Similarly (v) is proved.
Next choose function F ∈ D(A) and satisfies Then from (7), we can have Similarly we can have w (0+) = α 4 w(0−) + β 4 w (0−) Corollary 2.4 All eigenvalues of the operator JA are real, and if λ 1 and λ 2 be the two different eigenvalues of the problem (1)-( 5), then the corresponding eigenfunctions f (x) and g(x) are orthogonal in the sense of θ
Proof: Let u 0 (x) be any eigenfunction corresponding to eigenvalue λ 0 .Then the function u 0 (x) may be represented in the form where at least one of the constants c i (i = 1, 4) is not zero.
Consider the true function l v (u 0 (x)) = 0, v = 1, 4 as the homogenous system of linear equations in the variables c i (i = 1, 4) and talking into account ( 8)-( 11), it follows that the determinant of this system is Then the following integral equations hold for k = 0, 1 Proof: Regard ϕ 1λ (x) as the solution of the following non-homogeneous Cauchy problem Using the method of constant changing, ϕ 1λ (x) satisfies Then differentiating it with respect to x, we have ( 12).The proof for ( 13) is similar.
Theorem 3.6 The following asymptotic formulae hold for the real eigenvalues of the problem (1)-( 5) with r(x) = sgnx: Proof: By applying the known Rouche theorem, we can obtain these conclusions.

More accuracy asymptotic formulae for eigenvalues and eigenfunctions
In this section, for the sake of simplicity we assume that a 1 = a 2 = 1, α 4 = β 3 = 0, α 3 = β 4 and the weight function r(x) = sgnx.We will use the following method to obtain more accurate conclusions.
Similarity with the third section, we can get the following three conclusions: x+1) ) k = 0, 1.Each of this asymptotic equalities hold uniformly for x as |λ| → ∞.