Spectral properties of an impulsive Sturm–Liouville operator

This work is devoted to discuss some spectral properties and the scattering function of the impulsive operator generated by the Sturm–Liouville equation. We present a different method to investigate the spectral singularities and eigenvalues of the mentioned operator. We also obtain the finiteness of eigenvalues and spectral singularities with finite multiplicities under some certain conditions. Finally, we illustrate our results by a detailed example.


Introduction
In this paper, we consider the Sturm-Liouville equation on the semi axis -y + q(x)y = λ 2 ρ(x)y, x ∈ [0, ∞), (1.1) with boundary condition y(0) = 0, (1.2) where λ is a spectral parameter, and ρ is the density function. There is a comprehensive literature on the spectral theory of boundary value problem (1.1)-(1.2) for ρ = 1. In particular, the spectral analysis of the problem having discrete and continuous spectrum was begun by Naimark [1] for ρ = 1. He proved that some poles of the resolvent kernel are not the eigenvalues of the operator. He also showed that those poles, which are called spectral singularities by Schwartz [2], are a mathematical obstruction for the completeness of the eigenvectors and are embedded in the continuous spectrum. Pavlov [3] established the dependence of the structure of the spectral singularities of the differential operator on the behavior of the potential function at infinity. So far, a large number of problems related to the spectral analysis of differential and some other types of operators with spectral singularities have been investigated [4][5][6][7][8][9][10]. As is well known, the Sturm-Liouville equation x q(x) dx < ∞, (1.4) where K(x, t) is defined by the potential function q [11,12]. Furthermore, boundary value problems with discontinuities inside an interval have great interest in mathematical physics and quantum mechanics. To solve interior discontinuities, some extra conditions are imposed on the discontinuous point, which are often called interface conditions, point interactions, transmission conditions, and impulsive conditions. The theory of impulsive differential equations were studied in applied mathematics in detail [13,14]. A great number of authors studied the spectral theory of impulsive differential equations [15][16][17][18]. Moreover, the physical meaning and potential applications of spectral singularities of impulsive differential equations have been understood quite recently [19,20]. Especially in [21], the author provided the physical meanings of eigenvalues and spectral singularities of the Schrödinger equation at a single point. Such problems have been widely studied for impulsive differential operators on the whole axis.
In this work, we are concerned with the impulsive Sturm-Liouville operator on the semi axis. The density function ρ and impulsive condition make the spectral analysis of operator quite difficult, but by determining a transfer matrix we can obtain some spectral properties.

Statement of the problem
Let us introduce the Sturm-Liouville operator L in L 2 [0, ∞) generated by the equation with the boundary condition and the impulsive condition where α 1 , α 2 , α 3 , α 4 are complex numbers such that det B = 0, λ is a spectral parameter, x 0 is a positive real constant, and the real-valued potential function q satisfies the condition The density function ρ has the form where β ∈ R\(-1, 1).
Note that x = x 0 is the impulsive point of problem (2.1)-(2.3), and the matrix B is used to continue the solution of (2.1) from [0, x 0 ) to (x 0 , ∞).
Also, K(x, t) satisfies where c > 0 is a constant, and Now, let λ ∈ R\{0}. Using linearly independent solutions of (2.1) in the intervals [0, x 0 ) and (x 0 , ∞), we can express the general solution of (2.1) by where A ± and B ± are constant coefficients depending on λ. By (1.3), (2.6), (2.7), and (2.8) we get y -(x -0 , λ), y + (x + 0 , λ), y -(x -0 , λ), and y + (x + 0 , λ). Then, from the impulsive condition (2.3) we have transfer matrix M satisfying and N := 14) where and K x (x, t) := ∂ ∂x K(x, t). Since det N = -2iλ, we easily obtain Let us consider any two solutions of (2.1), denoting the coefficients A ± and B ± by A ± ± and B ± ± , which are expressed as and where A ± ± and B ± ± are complex coefficients. Let F and G be associated with the Jost solution of the boundary value problem (2.1)-(2.3) and the boundary condition (2.2), respectively. Then we obtain Furthermore, using the impulsive condition (2.3) and (2.11), we get uniquely for the solution F and G. Clearly, inserting these coefficients into (2.17) and (2.18), we obtain the solutions F and G satisfying the following asymptotic equations, respectively, x → ∞, and Now by (2.21) and (2.22) we can give the following lemma.

Lemma 2.1
The following equations hold for all λ ∈ R\{0}: Note that, the Wronskian of the solutions of (2.1) in the intervals [0, x 0 ) and (x 0 , ∞) are independent of x, but they are not equal because of the characteristic feature of impulsive differential equations.
Moreover, from (2.15) we understand that M 22 has an analytic continuation from the real axis to the set C + := {λ : λ ∈ C, Im λ > 0} and continuous up to the real axis because of analytic properties of solutions e(x, λ), e (x, λ), S(x, λ 2 ), S (x, λ 2 ). By Lemma 2.1 and [22], we have the following.

Corollary 2.2 A necessary and sufficient condition to investigate the eigenvalues and spectral singularities of the impulsive Sturm-Liouville operator L is to investigate the zeros of the function M 22 .
By (2.15) we have the following representation for M 22 :

Main results
We introduce the sets of spectral singularities and eigenvalues of impulsive operator L as Thus we can rewrite the sets where α 2 = 0, λ ∈ C + , and |λ| → ∞.
In this section, we assume that β ≥ 1. We give a lemma, which is necessary to discuss the properties of eigenvalues and spectral singularities of L.

Lemma 3.2 Assume (2.4).
(i) The set S 1 is bounded, and no more than a countable number of elements and its limit points can lie on a bounded subinterval of the real axis. (ii) The set S 2 is compact, and its linear Lebesgue measure is zero.
Proof Asymptotic equation (3.1) shows that M 22 cannot equal zero for sufficiently large λ ∈ C + . Thus the boundedness of the sets S 1 and S 2 follows from (3.1). Moreover, since M 22 is analytic in C + , the set S 1 has at most countable number of elements, and its limit points can lie only on a bounded subinterval of the real axis. Using the uniqueness theorem of analytic functions [23], we obtain that S 2 is a closed set and its linear Lebesgue measure is zero. Now, we can give the following theorem.

Theorem 3.3 Under condition (2.4), (i) The set of eigenvalues of L is bounded and has at most a countable number of elements, and its limit points can lie only on a bounded subinterval of the real axis. (ii) The set of spectral singularities of L is compact, and its linear Lebesgue measure is zero.
Now, we proceed by assuming an extra condition on q to assure the finiteness of the sets σ d (L) and σ ss (L).

Theorem 3.4 If
for every > 0, then there are finitely many eigenvalues and spectral singularities of the operator L, and each of them has finite multiplicity.
Proof Using (2.9), (2.10), and (3.4), we find that that is, the function M 22 has an analytic continuation from the real axis to the lower halfplane Im λ > -/2. Hence the sets σ d (L) and σ ss (L) have no limit points on the real line, and by Theorem 3.3 these sets are bounded and have a finite number of elements. Finally, using the uniqueness theorem of analytic functions [23], we see that all zeros of M 22 in C + have finite multiplicities.
Now, let us denote the set of all limit points of S 1 by S 3 and the set of all zeros of M 22 with infinite multiplicity in C + by S 4 . By the uniqueness theorem of analytic functions, we find that From the continuity of all derivatives of M 22 up to the real axis, we obtain that Next, we indicate the same result of Theorem 3.4 by using a weaker condition than (3.4).

Definition 4.3
Let α 1 , α 2 , α 3 , α 4 ∈ R and β ∈ R\(-1, 1). Then the scattering function of the operator L is defined by Since q is a real-valued potential function, it is evident from (2.17) and (2.20) that for all λ ∈ R\{0}. Then the definition of the function S turns into ) . The proof is completed.

An example
Let us consider the Sturm-Liouville operator L 0 in L 2 [0, ∞) created by the following impulsive problem: where ρ is the density function given by x > x 0 , such that β ∈ C\{0}, α 1 , α 4 ∈ C, α 1 .α 4 = 0, and x 0 ∈ R + . Using q = 0 in (2.23), we directly obtain To investigate the eigenvalues and spectral singularities of L 0 , we examine the zeros of M 22 . For this purpose, we see that e 2iλβx 0 = α 1 + βα 4 α 1βα 4 by (5.2). Using the last equation, we find where A = βα 4 α 1 . Let β = a + ib. Then we can write real and imaginary parts of λ k by and Im λ k = -1 2x 0 |β| 2 a ln respectively. It is easy to see that if then the problem has spectral singularities, and if a ln then the problem has eigenvalues. Now, we investigate some particular cases. Case1: Let A = e iθ -1 e iθ +1 for θ ∈ R. In this case, since Arg( 1+A 1-A ) = θ , we get 1a: Let β ∈ R. Thus λ k ∈ R, and then the numbers μ k = λ 2 k , k ∈ Z, are the spectral singularities of the impulsive boundary value problem (5.1).
Case2: Let Im A = 0. We investigate some subcases. 2a: Let A be purely imaginary, that is, Re A = 0. In this case, since | 1+A 1-A | = 1, from (5.3) we get which means that the numbers μ k = λ 2 k , k ∈ Z, are the spectral singularities of (5.1) for β ∈ R.
Let β ∈ C. We find If b[arg(1 + A)] < 0, then the operator L 0 has eigenvalues. Otherwise, the operator has no eigenvalues and spectral singularities.
2b: Let Re A < 0. For β ∈ R, that is, b = 0, we have If a > 0, then the numbers μ k = λ 2 k , k ∈ Z, are the eigenvalues of impulsive problem (5.1); otherwise, the operator L 0 has no eigenvalues.
3c: Let -1 < A < 0. In this case, for β ∈ R, if a > 0, then the numbers μ k = λ 2 k , k ∈ Z, are the eigenvalues of L 0 . Otherwise, the problem has no eigenvalues. Let β ∈ C. If a > 0 and b(2kπ) < 0, then the problem has eigenvalues. If a < 0 and b(2kπ) > 0, then the problem has no eigenvalues.

Conclusions
In this study, we discuss some spectral and scattering problems of an impulsive Sturm-Liouville boundary value problem on the semi axis. Although there are various studies about the spectral analysis of these problems, much of them are on the whole axis. Moreover, the method we use to investigate the eigenvalues and spectral singularities is quite different from other papers. By using a transfer matrix we introduce the sets of eigenvalues and spectral singularities, and under sufficient conditions, we guarantee the finiteness of these sets.