The Modified Parseval Equality of Sturm-Liouville Problems with Transmission Conditions

We consider the Sturm-Liouville (S-L) problems with very general transmission conditions on a finite interval. Firstly, we obtain the sufficient and necessary condition for λ being an eigenvalue of the S-L problems by constructing the fundamental solutions of the problems and prove that the eigenvalues of the S-L problems are bounded below and are countably infinite. Furthermore, the asymptotic formulas of the eigenvalues and eigenfunctions of the S-L problems are obtained. Finally, we derive the eigenfunction expansion for Green’s function of the S-L problems with transmission conditions and establish the modified Parseval equality in the associated Hilbert space.


Introduction
The Sturm-Liouville (S-L) theory, as an active area of research in pure and applied mathematics, plays an important role in solving many problems in mathematical physics and is concerned in many publications [1][2][3][4][5][6][7].It is well known that for the classical S-L problems, the solutions or the derivatives of the solutions are continuous on the interval, but these conditions cannot be satisfied in many practical physical problems.So, a class of S-L operators with "discontinuity, " that is, the S-L problems with transmission conditions at an interior point, are concerned by many mathematical and physical researchers [8][9][10].Such conditions are known by various names including transmission conditions [11,12], interface conditions [13][14][15], jump conditions [16], and discontinuous conditions [17,18].
In this paper, we consider the following Sturm-Liouville equation: with boundary conditions: and transmission conditions: where  is a complex eigenparameter and  ∈ (,R); notice that the potential function () guarantee (0±) and   (0±) in (4) makes sense (see Theorem 1); all coefficients of the boundary and transmission conditions are real numbers.Throughout this paper, we assume that  2 1 + 2 2 ̸ = 0, We derive the eigenfunction expansion for Green's function of the S-L problem (1)-( 4) and establish the modified Parseval equality of the S-L problem with very general transmission conditions at one inner point 0 of the finite interval [−1 , 1].The organization of this paper is as follows.After the Introduction, we construct the basic solutions of S-L equation (1) with transmission conditions (4) and obtain the sufficient and necessary condition for  being an eigenvalue of the S-L problem in Section 2. In Section 3, the asymptotic formulas for eigenvalues and eigenfunctions of the S-L problem are

The Basic Solutions and Eigenvalues
We construct the basic solutions of S-L equation (1) with transmission conditions (4) and obtain the sufficient and necessary condition for  being an eigenvalue of the S-L problem in this section.At first, we prove the existence of finite limits for all solution  of (1) and its derivative at both sides of 0 point in the following theorem.
In order to prove Theorem 1, we need the following lemma.
Let us define a new inner product in  2 () as follows, which is associated with transmission conditions (4) and useful to investigate the S-L problem (1)-( 4): where . It is easy to verify that ( 2 (), ⟨⋅, ⋅⟩) is a Hilbert space.For simplicity, it is denoted by .The norm induced by the inner product is denoted by ‖ ⋅ ‖  .We consider the S-L problem (1)-( 4) in the associated Hilbert space .

Asymptotic Formulas for Eigenvalues and Eigenfunctions
In this section, the asymptotic formulas for eigenvalues and eigenfunctions of the S-L problem (1)-( 4) are obtained by using the asymptotic expressions of the solutions.At first, we calculate the asymptotic expressions of the solutions.
Proof.The asymptotic formulas for  1 (, ) follow from the similar formulas of Lemma 1.7 in [20].
The asymptotic formulas for  2 (, ) are as follows.
By the asymptotic formulas for the eigenvalues and eigenfunctions of the S-L problem (1)-(4) in the above theorem, we know that the series ∑ ∞ =1 (  ()  ()/  ) converges absolutely and uniformly where   () is the normalized eigenfunction.