Scattering properties of Sturm-Liouville equations with sign-alternating weight and transmission condition at turning point

Nimet Çoşkun; Merve Görgülü

doi:10.1515/math-2022-0566

Article Open Access

Scattering properties of Sturm-Liouville equations with sign-alternating weight and transmission condition at turning point

Nimet Çoşkun and Merve Görgülü

Published/Copyright: March 30, 2023

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$Open Mathematics$

From the journal Open Mathematics Volume 21 Issue 1

Abstract

In this article, we focus on the scattering analysis of Sturm-Liouville type singular operator including an impulsive condition and turning point. In the classical literature, there are plenty of papers considering the positive values of the weight function in both continuous and discontinuous cases. However, this article differs from the others in terms of the impulsive condition appearing at the turning point. We generate the scattering function, resolvent operator, and discrete spectrum of the operator using the hyperbolic representations of the fundamental solutions. Finally, we create an example to show the article’s primary conclusions.

Keywords: Sturm-Liouville equations; impulsive differential equations; scattering theory; discontinuous weight function

MSC 2010: 34B24; 34E20; 47A10; 47A75

1 Introduction

The early 19th century is the cradle of the Sturm-Liouville problems which rise in the field of partial differential equations for the sake of solving the heat-conduction equation at the outlet, and then gathered extensive attention worldwide in mathematics and notably in physics. When modeling a process that arise from a real-life phenomenon, we have to act according to the behavior of the endpoints within the specified interval whether finite or not. As a result, the singular problem constitutes a vast research ground for mathematicians due to its particular features [1–6].

Let us take into consideration the one-dimensional Schrödinger type operator for any x ∈ [ 0 , a ) ∪ ( a , ∞ ) ,

(1.1) − y ″ + q ( x ) y = κ 2 ρ ( x ) y ,

where ρ signifies the weight (density) function and κ denotes the eigenparameter. There is a significant number of literature on the inverse and direct problems for ρ ( x ) = 1 [7,8]. Inverse problem of the operator with equation (1.1) and discontinuous weight function

(1.2) ρ ( x ) = a 2 , 0 ≤ x < r , 1 , r ≤ x < ∞ ,

where 0 < a ≠ 1 , r ∈ R , has been handled by Mamedov [9]. Since discontinuous weight function case has an important role on modeling and solving the wave propagation problems in nonhomogeneous medium, it has attracted attention from various branches of analysis in both inverse and direct spectral theory directions [9–15].

A 1915 study [16] is the earliest substantial work that is currently thought to deal with turning points. Theoretical physicist Gans developed the Sturm-Liouville type differential equation with turning point by using Maxwell’s equations to explore how light moves through a nonhomogeneous medium. The asymptotic theory of differential operators comprises the turning point theory as an investigation subsection. It is crucial to investigate the turning points albeit they are rather unusual, to pursue guidance derived from the asymptotic behavior of the solutions of these differential equations. The study of analytic functions’ singularities is the only way to comprehend them, and the location and behavior of their critical points have a significant impact on the solutions of ordinary differential equations. In addition, many physics and engineering problems involve turning point problems in their mathematical formulation, which has been the major motivation behind most pioneer studies [16–20].

Although the turning point theory was studied mostly from the viewpoint of the asymptotic behaviors of the solutions, Gasymov and Rekheem [21] investigated the inverse scattering and inverse spectrum analyses of Sturm-Liouville type operators with sign-changing density function, which contrasted with the current literature. Due to the variety of applications, it is crucial to explore the sign-changing (also known as turning point or sign-valued weight function) case for differential operators. For instance, Bessel type singularities can be associated with a turning point in Sturm-Liouville equations. Readers who are interested might read the papers [16–23] and the references therein for more in-depth information on the case of the sign-valued weight function and its applications in other scientific fields. The primary distinction between this sort of problem and those with positive values is the new analytical difficulties that follow from the ± sign of the density function.

The events of abrupt or sudden changes appear in the dynamics of processes in the everyday world. We encounter with such situations in various branches of sciences like ecology, engineering, population dynamics, and optimum control. Impulsive differential equations are among the primary tools to model such problems. Due to their importance and variety of application fields, it is essential to examine them. For further information on the theory of impulsive differential equations and its applications, we suggest the reader to the books and recent papers listed in [24–27] and the references therein. Spectral analysis and scattering theory of Sturm-Liouville operators including impulsive (also called transmission or jump) conditions with positive valued density function have been considered in [25,26].

In this article, influenced by the prior research, the scattering analysis of the impulsive Sturm-Liouville type equation including a turning point will be taken under investigation. Due to the sign change of the density function, new analytical difficulties that does not appear in the articles [14–25] have been encountered.

We organized this article as follows: In Section 2, beside stating the problem and presenting the fundamental solutions, we construct the scattering solutions and the scattering function. Section 3 focuses on the resolvent and discrete spectrum of the operator. Finally, we state asymptotic properties of the Jost function and state an example to present our main results.

2 Construction of scattering solutions and scattering function

Consider the operator T defined in the Hilbert space L ρ 2 ( R + ) by the Sturm-Liouville differential equation:

(2.1) − y ″ + q ( x ) y = κ 2 ρ ( x ) y , x ∈ [ 0 , 1 ) ∪ ( 1 , ∞ ) ,

with the initial condition

(2.2) y ( 0 ) = 0 ,

and the transmission condition at the turning point

(2.3) y ( 1 + ) y ′ ( 1 + ) = B y ( 1 − ) y ′ ( 1 − ) , B = α β γ δ , det B ≠ 0 ,

together with the discontinuous density function

(2.4) ρ ( x ) = − 1 , 0 ≤ x < 1 , 1 , 1 ≤ x < ∞ .

Note that κ stands for the eigenparameter, and α , β , γ , and δ are assumed to be real numbers. The matrix B is called transfer matrix of the impulsive Sturm-Liouville equation [14,27]. Note that Mostafazadeh has a number of papers presented in detail in his survey [27] about the physical applications and scattering theory of the Schrödinger type operators including transfer matrix and transmission conditions.

We presume that real valued potential q ( x ) satisfies

(2.5) ∫ 0 ∞ ( 1 + x ) ∣ q ( x ) ∣ d x < ∞ .

Consider the solutions of (2.1) denoted by φ ( x , κ 2 ) and θ ( x , κ 2 ) which fulfill the initial conditions

φ ( 0 , κ 2 ) = 0 , φ ′ ( 0 , κ 2 ) = 1 , θ ( 0 , κ 2 ) = 1 , θ ′ ( 0 , κ 2 ) = 0 .

Take into account the case q ( x ) ≡ 0 . Then, (2.1) yields

y ″ = κ 2 y , 0 ≤ x < 1 .

Thus, φ ( x , κ 2 ) and θ ( x , κ 2 ) have representations in the hyperbolic type functions as follows:

φ ( x , κ 2 ) = sinh κ x κ , θ ( x , κ 2 ) = cosh κ x .

Using the results of [1,21–23] and method of variation of parameters leads one to the Volterra type integral representations of the fundamental solutions φ ( x , κ 2 ) and θ ( x , κ 2 ) as follows:

(2.6) φ ( x , κ 2 ) = sinh κ x κ + ∫ 0 x P ( x , t ) sinh κ t κ d t

and

(2.7) θ ( x , κ 2 ) = cosh κ x + ∫ 0 x Q ( x , t ) cosh κ t d t ,

where

Q ( x , x ) = 1 2 ∫ 0 x q ( t ) d t , ∂ Q ∂ t t = 0 = 0 , Q ( 0 , 0 ) = 1 , P ( x , x ) = 1 2 ∫ 0 x q ( t ) d t , P ( x , 0 ) = 0 , P ( 0 , 0 ) = 1 .

Moreover, both of the functions φ and θ are entire with regard to the variable κ . Existence and uniqueness results of the solutions φ ( x , κ 2 ) and θ ( x , κ 2 ) can also be proven analogous to [1,21–23]. The Wronskian of any solutions y 1 , y 2 of equation (2.1) is defined as follows [1]:

W [ y 1 , y 2 ] = y 1 ( x , κ 2 ) y 2 ′ ( x , κ 2 ) − y 1 ′ ( x , κ 2 ) y 2 ( x , κ 2 ) , κ ∈ C .

Therefore, the Wronskian of the solutions φ and θ is derived as follows:

W [ φ ( x , κ 2 ) , θ ( x , κ 2 ) ] = − 1 , κ ∈ C .

Let us now indicate the solution of (2.1) by e ( x , κ ) that fulfills the asymptotic criteria

lim x → ∞ e ( x , κ ) e − i κ x = 1 , x ∈ ( 1 , ∞ ) , κ ∈ C ¯ + ,

and has the representation [1]

e ( x , κ ) = e i κ x + ∫ x ∞ K ( x , t ) e i κ t d t , κ ∈ C ¯ + ,

where C ¯ + ≔ { κ ∈ C : Im κ ≥ 0 } and

K ( x , x ) = 1 2 ∫ x ∞ q ( t ) d t , x ∈ ( 1 , ∞ ) .

Also, denote the solution of (2.1) by e ( x , − κ ) which satisfies the asymptotic

lim x → ∞ e ( x , − κ ) e i κ x = 1 , x ∈ ( 1 , ∞ ) , κ ∈ C ¯ − ,

where C ¯ − ≔ { κ ∈ C : Im κ ≤ 0 } . Clearly, e ( x , − κ ) can be represented by the integral equation:

e ( x , − κ ) = e − i κ x + ∫ x ∞ K ( x , t ) e − i κ t d t , κ ∈ C ¯ − .

The Wronskian of the expressions e ( x , κ ) and e ( x , − κ ) can be computed easily [ 1 ]

W [ e ( x , κ ) , e ( x , − κ ) ] = − 2 i κ , κ ∈ R .

We define

(2.8) Ξ ( x , κ ) = Λ 1 ( κ ) φ ( x , κ 2 ) + Λ 2 ( κ ) θ ( x , κ 2 ) , x ∈ [ 0 , 1 ) , e ( x , κ ) , x ∈ ( 1 , ∞ ) .

Applying the transmission condition (2.3) and using the analytic continuity requirements of the solutions, it is easy to obtain the system of equations:

(2.9) e ( 1 , κ ) = Λ 1 ( κ ) { α φ ( 1 , κ 2 ) + β φ ′ ( 1 , κ 2 ) } + Λ 2 ( κ ) { α θ ( 1 , κ 2 ) + β θ ′ ( 1 , κ 2 ) } ,

(2.10) e ′ ( 1 , κ ) = Λ 1 ( κ ) { γ φ ( 1 , κ 2 ) + δ φ ′ ( 1 , κ 2 ) } + Λ 2 ( κ ) { γ θ ( 1 , κ 2 ) + δ θ ′ ( 1 , κ 2 ) } .

Solving equations (2.9) and (2.10) simultaneously for Λ 1 ( κ ) and Λ 2 ( κ ) , we readily obtain

(2.11) Λ 1 ( κ ) = − 1 det B [ e ( 1 , κ ) { γ θ ( 1 , κ 2 ) + δ θ ′ ( 1 , κ 2 ) } − e ′ ( 1 , κ ) { α θ ( 1 , κ 2 ) + β θ ′ ( 1 , κ 2 ) } ] ,

(2.12) Λ 2 ( κ ) = 1 det B [ e ( 1 , κ ) { γ φ ( 1 , κ 2 ) + δ φ ′ ( 1 , κ 2 ) } − e ′ ( 1 , κ ) { α φ ( 1 , κ 2 ) + β φ ′ ( 1 , κ 2 ) } ] ,

where Ξ ( x , κ ) is indicated as the Jost solution of the operator T , where the coefficients Λ 1 ( κ ) and Λ 2 ( κ ) are computed in (2.11) and (2.12), respectively. As soon as we know how to construct the Jost function from the Jost solution and the boundary condition, we define

(2.13) Ξ ( κ ) ≔ Ξ ( 0 , κ ) = Λ 2 ( κ ) ,

as the Jost function of T .

Let us also introduce another solution of (2.1)–(2.4) for κ ∈ R \ { 0 } :

(2.14) Ψ ( x , κ ) = φ ( x , κ 2 ) , x ∈ [ 0 , 1 ) , Λ 3 ( κ ) e ( x , κ ) + Λ 4 ( κ ) e ( x , − κ ) , x ∈ ( 1 , ∞ ) .

Considering the impulsive condition (2.4), it turns out that the coefficients Λ 3 ( κ ) and Λ 4 ( κ ) can be determined from the system of equations:

(2.15) Λ 3 ( κ ) e ( 1 , κ ) + Λ 4 ( κ ) e ( 1 , − κ ) = α φ ( 1 , κ 2 ) + β φ ′ ( 1 , κ 2 ) ,

(2.16) Λ 3 ( κ ) e ′ ( 1 , κ ) + Λ 4 ( κ ) e ′ ( 1 , − κ ) = γ φ ( 1 , κ 2 ) + δ φ ′ ( 1 , κ 2 ) .

By solving (2.15) and (2.16) simultaneously, it is easy to derive the values corresponding to the solution Ψ ( x , κ ) as follows:

(2.17) Λ 3 ( κ ) = det B 2 i κ [ Λ 2 ( κ ) ¯ ] ,

(2.18) Λ 4 ( κ ) = det B 2 i κ [ Λ 2 ( κ ) ] .

Theorem 2.1

Assume κ ∈ R \ { 0 } and α , β , γ , and δ are real numbers. In this case, Ξ ( κ ) ≠ 0 .

Proof

On the contrary, suppose that Ξ ( κ 0 ) = 0 for a κ 0 ∈ R \ { 0 } . In this case, (2.17) and (2.18) indicate that

Λ 3 ( κ 0 ) = Λ 4 ( κ 0 ) = 0 .

Hence, Ψ ( x , κ 0 ) turns out to be trivial solution of (2.1)–(2.4). However, this results in the contradiction to our assumption. Therefore, for every κ ∈ R \ { 0 } , Ξ ( κ ) ≠ 0 .□

Corollary 2.2

The operator T has no spectral singularities for α , β , γ , δ ∈ R .

Definition 2.3

We define the scattering function of (2.1)–(2.4) as follows:

S ( κ ) = Ξ ( κ ) ¯ Ξ ( κ ) , κ ∈ R \ { 0 } .

Given the definition of the Jost function, the expression of the scattering function S ( κ ) can be rewritten as follows:

(2.19) S ( κ ) = Ξ ( 0 , κ ) ¯ Ξ ( 0 , κ ) = Λ 2 ( κ ) ¯ Λ 2 ( κ ) , κ ∈ R \ { 0 } .

From (2.12), we arrive at

S ( κ ) = e ( 1 , − κ ) { γ φ ( 1 , κ 2 ) + δ φ ′ ( 1 , κ 2 ) } − e ′ ( 1 , − κ ) { α φ ( 1 , κ 2 ) + β φ ′ ( 1 , κ 2 ) } e ( 1 , κ ) { γ φ ( 1 , κ 2 ) + δ φ ′ ( 1 , κ 2 ) } − e ′ ( 1 , κ ) { α φ ( 1 , κ 2 ) + β φ ′ ( 1 , κ 2 ) } .

Theorem 2.4

The identity

(2.20) 2 i κ φ ( x , κ 2 ) Ξ ( 0 , κ ) = Ξ ( 0 , − κ ) − Ξ ( 0 , κ ) Ξ ( x , κ )

holds for every κ ∈ R \ { 0 } where the scattering function S ( κ ) defined by (2.19). Also, S ( κ ) holds the following relations:

S ( − κ ) ¯ = S − 1 ( − κ )

∣ S ( κ ) ∣ = 1 .

Proof

For every κ ∈ R \ { 0 } , Ξ ( x , κ ) and Ξ ( x , − κ ) form a fundamental system of solutions for equation (2.1). Therefore, one can write any solution of equation (2.1), as a linear combination of them. Let

(2.21) φ ( x , κ 2 ) = A ( κ ) Ξ ( x , κ ) + B ( κ ) Ξ ( x , − κ ) .

If we apply the initial conditions

φ ( 0 , κ 2 ) = 0 , φ ′ ( 0 , κ 2 ) = 1 ,

and solve the coefficients A ( κ ) and B ( κ ) , we obtain

A ( κ ) = Ξ ( 0 , κ ) ¯ − 2 i κ

and

B ( κ ) = Ξ ( 0 , κ ) 2 i κ .

By substituting these values in (2.21), we obtain

φ ( x , κ 2 ) = Ξ ( 0 , κ ) ¯ − 2 i κ Ξ ( x , κ ) + Ξ ( 0 , κ ) 2 i κ Ξ ( x , − κ ) .

By taking parenthesis of 2 i κ and reorganizing the last equality, the required formula (2.20) is obtained.

It is known that q ( x ) is a real valued function. Therefore, the equality Ξ ( x , − κ ) = Ξ ( x , κ ) ¯ yields that

∣ S ( κ ) ∣ = ∣ Ξ ( 0 , κ ) ¯ ∣ ∣ Ξ ( 0 , κ ) ∣ = 1 .

From (2.19), we obtain that

S ( − κ ) = Ξ ( 0 , − κ ) ¯ Ξ ( 0 , − κ ) .

Also, for every κ ∈ R \ { 0 } and α , β , γ , δ ∈ R , the desired relations on scattering function are immediate from the fact that Ξ ( 0 , − κ ) ¯ = Ξ ( 0 , κ ) .□

Lemma 2.5

The Wronskian of the expressions Ξ ( x , κ ) and Ψ ( x , κ ) for every κ ∈ R \ { 0 } is computed as in the following equation:

W [ Ξ ( x , κ ) , Ψ ( x , κ ) ] = Ξ ( κ ) , x ∈ [ 0 , 1 ) , − ( det B ) Ξ ( κ ) , x ∈ ( 1 , ∞ ) .

Proof

By using the definition of the Wronskian, we compute the Wronskian of the expressions Ξ ( x , κ ) and Ψ ( x , κ ) for x ∈ [ 0 , 1 ) as follows:

W [ Ξ ( x , κ ) , Ψ ( x , κ ) ] = Λ 2 ( κ ) = Ξ ( κ ) .

Let us also write for x ∈ ( 1 , ∞ ) . Then, we obtain

W [ Ξ ( x , κ ) , Ψ ( x , κ ) ] = Λ 4 ( κ ) W [ e ( x , κ ) , e ( x , − κ ) ] = − ( det B ) Ξ ( κ ) .

Hence, this completes the proof.□

3 Eigenvalues and the resolvent operator of T

Let us symbolize with σ d ( T ) the set of all eigenvalues of the operator T .

Theorem 3.1

σ d ( T ) = { μ = κ 2 : κ ∈ C + , Ξ ( κ ) = 0 } .

Proof

We define the unbounded solution of (2.1) as e ˜ ( x , κ ) for x ∈ ( 1 , ∞ ) satisfying equations [1]

lim x → ∞ e ˜ ( x , κ ) e i κ x = 1 , lim x → ∞ e ˜ ′ ( x , κ ) e i κ x = − i κ , κ ∈ C ¯ + .

Therefore, we can write the another solution of equation (2.1) as follows:

(3.1) Φ ( x , κ ) = φ ( x , κ 2 ) , x ∈ [ 0 , 1 ) , Λ 5 ( κ ) e ( x , κ ) + Λ 6 ( κ ) e ˜ ( x , κ ) , x ∈ ( 1 , ∞ ) .

By considering the impulsive condition (2.3) and after some algebra, we compute for C ¯ + \ { 0 }

(3.2) Λ 6 ( κ ) = − det B 2 i κ Ξ ( κ )

and

(3.3) Λ 5 ( κ ) = 1 2 i κ [ e ˜ ( 1 , κ ) { γ φ ( 1 , κ 2 ) + δ φ ′ ( 1 , κ 2 ) } − e ˜ ′ ( 1 , κ ) { α φ ( 1 , κ 2 ) + β φ ′ ( 1 , κ 2 ) } ] .

Clearly, we can deduce from (3.1) that for x ∈ [ 0 , 1 ) , Φ ( x , κ ) ∈ L ρ 2 ( 0 , 1 ) . Further, while the coefficient of the unbounded solution is identically zero, the second piece of Φ ( x , κ ) will also be in L ρ 2 ( 1 , ∞ ) . By using the classical definition of the spectrum of an operator, we obtain the discrete spectrum of the operator T :

σ d ( T ) = { μ = κ 2 : κ ∈ C + , Λ 6 ( κ ) = 0 } ,

or equivalently,

□ σ d ( T ) = { μ = κ 2 : κ ∈ C + , Ξ ( κ ) = 0 } .

Note that, the Wronskians of the solutions Ξ ( x , κ ) and Φ ( x , κ ) can also be calculated for κ ∈ C ¯ + \ { 0 } as follows:

W [ Ξ ( x , κ ) , Φ ( x , κ ) ] = Ξ ( κ ) , x ∈ [ 0 , 1 ) , ( det B ) Ξ ( κ ) , x ∈ ( 1 , ∞ ) .

Theorem 3.2

Denote with R κ ( T ) the resolvent operator of T. Then,

R κ ( T ) ψ = ∫ 0 ∞ G ( x , t ; κ ) ψ ( t ) d t ,

for the arbitrary function ψ ∈ L ρ 2 ( R + ) and the Green’s function G ( x , t ; κ ) is given by

G ( x , t ; κ ) = Ξ ( x , κ ) Ψ ( t , κ ) W [ Ξ ( x , κ ) , Ψ ( x , κ ) ] ; 0 ≤ t < x , Ψ ( x , κ ) Ξ ( t , κ ) W [ Ξ ( x , κ ) , Ψ ( x , κ ) ] ; x ≤ t < ∞ ,

for all x ≠ 1 and t ≠ 1 .

Proof

To be able to obtain the resolvent operator, we need to solve the non-homogeneous differential equation

(3.4) − y ″ + q ( x ) y − κ 2 ρ ( x ) y = ψ ( x ) , x ∈ [ 0 , 1 ) ∪ ( 1 , ∞ ) .

With the help of (2.8) and (3.1), we try to find the general solution of the nonhomogeneous equation (3.4) in the form of

y ˜ ( x , κ ) = c 1 ( x ) Ξ ( x , κ ) + c 2 ( x ) Φ ( x , κ ) .

From the variation of the parameters technique, we determine

c 1 ( x ) = k + ∫ 0 x ψ ( t ) Ψ ( t , κ ) W [ Ξ ( x , κ ) , Ψ ( x , κ ) ] d t ,

c 2 ( x ) = l + ∫ x ∞ ψ ( t ) Ξ ( t , κ ) W [ Ξ ( x , κ ) , Ψ ( x , κ ) ] d t ,

where k and l are real numbers. The proof directly follows from the fact y ˜ ( x , κ ) ∈ L ρ 2 ( R + ) and the initial condition (2.2) yielding that k = 0 .□

4 Asymptotic properties of the solutions

In this section, we will present asymptotic formulas for the fundamental solutions and Jost solution and eventually derive an asymptotic formula for the Jost function of the operator T . The asymptotic features were intimately associated to sine and cosine, which was essential in the previous studies. However, in our case, rather than sine and cosine, hyperbolic sine and hyperbolic cosine forms have to be dealt with. The studies in [12,13,25] compute their asymptotics considering the two cases κ ∈ C ¯ + and κ ∈ C ¯ − . Differently from the other studies, we need to take into consideration two cases: κ ∈ C ¯ left ≔ { κ ∈ C : Re κ ≤ 0 } and κ ∈ C ¯ right ≔ { κ ∈ C : Re κ ⩾ 0 } . Thus, the analogy given here is helpful to which is partly new and encompasses most approaches.

By using (2.6), we can write

φ ( 1 , κ 2 ) = e κ − e − κ κ + ∫ 0 1 P ( 1 , t ) e κ t − e − κ t κ d t .

After adding both sides of this equation e − κ κ and by multiplying with κ e κ , we obtain

(4.1) κ e κ φ ( 1 , κ 2 ) + e − κ κ = e 2 κ 2 + ∫ 0 1 P ( 1 , t ) e κ ( t + 1 ) − e κ ( 1 − t ) 2 d t .

By taking the modulus of the both sides of (4.1) and using triangle inequality, we have

κ e κ φ ( 1 , κ 2 ) + e − κ κ = o ( 1 ) , κ ∈ C ¯ left , ∣ κ ∣ → ∞ .

Hence, we compute the asymptotic

(4.2) φ ( 1 , κ 2 ) = e − κ κ − 1 2 + o ( 1 ) , κ ∈ C ¯ left , ∣ κ ∣ → ∞ .

By using (2.6), we can write

(4.3) φ ′ ( 1 , κ 2 ) = e κ − e − κ 2 + P ( 1 , 1 ) e κ − e − κ 2 κ + ∫ 0 1 P x ( 1 , t ) e κ t − e − κ t 2 κ d t .

If we multiply both sides with e κ and subtract 1 2 from both sides in (4.3), we compute

(4.4) φ ′ ( 1 , κ 2 ) e κ − 1 2 = e 2 κ 2 + P ( 1 , 1 ) e 2 κ − 1 2 κ + ∫ 0 1 P x ( 1 , t ) e κ ( t + 1 ) − e κ ( 1 − t ) 2 κ d t .

By taking modulus of the both sides and computing the asymptotic of (4.4), we obtain

φ ′ ( 1 , κ 2 ) e κ − 1 2 = O 1 κ , κ ∈ C ¯ left , ∣ κ ∣ → ∞ ,

which in turn results in the following equation:

(4.5) φ ′ ( 1 , κ 2 ) = e − κ 1 2 + O 1 κ , κ ∈ C ¯ left , ∣ κ ∣ → ∞ .

If we apply similar procedures for the θ ( 1 , κ 2 ) and θ ′ ( 1 , κ 2 ) , we obtain

θ ( 1 , κ 2 ) = e − κ 1 2 + o ( 1 ) , κ ∈ C ¯ left , ∣ κ ∣ → ∞ ,

θ ′ ( 1 , κ 2 ) = κ e − κ − 1 2 + O 1 κ , κ ∈ C ¯ left , ∣ κ ∣ → ∞ .

We have also classical asymptotics of the Jost solution [ 3 , 4 ]

(4.6) e ( 1 , κ ) = e i κ ( 1 + o ( 1 ) ) , κ ∈ C ¯ + , ∣ κ ∣ → ∞ ,

(4.7) e ′ ( 1 , κ ) = e i κ ( i κ + O ( 1 ) ) , κ ∈ C ¯ + , ∣ κ ∣ → ∞ .

In this step, we will compute the asymptotic of the Jost function using (2.12)–(2.13) and the asymptotic expressions (4.2), (4.5), (4.6), and (4.7) which are calculated earlier. Hence, we write for ∣ κ ∣ → ∞ and κ ∈ C ¯ + ∩ C ¯ left

Ξ ( κ ) = 1 det B e i κ ( 1 + o ( 1 ) ) γ e − κ κ − 1 2 + o ( 1 ) + δ e − κ 1 2 + O 1 κ − e i κ ( i κ + O ( 1 ) ) α e − κ κ − 1 2 + o ( 1 ) + β e − κ 1 2 + O 1 κ .

After some algebra, we obtain

Ξ ( κ ) = e i κ − κ 2 det B − i β κ 2 + ( δ + i α ) κ − γ κ + O ( 1 ) , κ ∈ C ¯ + ∩ C ¯ left , ∣ κ ∣ → ∞ .

Note that similar calculations can be done for κ ∈ C ¯ + ∩ C ¯ right and ∣ κ ∣ → ∞ .

5 An application exemplifying the main results

In this final section, an example will be constructed to help the readers better understanding of the main results. Consider the unperturbated differential equation in the Hilbert space L ρ 2 ( R + )

(5.1) − y ″ = κ 2 ρ ( x ) y , x ∈ [ 0 , 1 ) ∪ ( 1 , ∞ ) .

Also, assume that the requirements (2.2)–(2.4) are satisfied for κ is an eigenparameter and α , β , γ , δ are real numbers. We introduce the operator defined by (5.1) and the assumptions (2.2)–(2.4) by T ˜ . Clearly, the solutions take the forms

e ( x , κ ) = e i κ x , φ ( x , κ 2 ) = sinh κ x κ , θ ( x , κ 2 ) = cosh κ x .

We can construct the solution of the differential equation (5.1) with the aforementioned assumptions as follows:

(5.2) Ξ ( x , κ ) = Λ 1 ( κ ) sinh κ x κ + Λ 2 ( κ ) cosh κ x , x ∈ [ 0 , 1 ) , e i κ x , x ∈ ( 1 , ∞ ) ,

for κ ∈ C ¯ + . By using the transmission condition (2.3), we compute

Λ 1 ( κ ) = e i κ det B [ γ cosh κ + δ κ sinh κ − α κ cosh κ + β κ 2 sinh κ ] , Λ 2 ( κ ) = e i κ det B γ sinh κ κ + δ cosh κ − α sinh κ − β κ cosh κ .

By substituting the values of the Λ 1 ( κ ) and Λ 2 ( κ ) in (5.2), we obtain the Jost function of the operator T ˜ as follows:

Ξ ( κ ) = Λ 2 ( κ ) = e i κ det B γ sinh κ κ + δ cosh κ − α sinh κ − β κ cosh κ .

Hence, the scattering function of the operator T ˜ can be stated as follows:

S ( κ ) = Ξ ( κ ) ¯ Ξ ( κ ) ,

where

Ξ ( κ ) ¯ = e − i κ det B γ sinh κ μ + δ cosh κ − α sinh κ − β κ cosh κ .

We are required to calculate the roots of the function Λ 2 ( κ ) to find the eigenvalues of the operator T ˜ . Let us choose β = γ = 0 to ease the calculations. In this case, det B = α δ ≠ 0 , and our choice does not distract the nature of the problem. For this specific choices, it turns out that we have to solve the following equation:

(5.3) e i κ sinh κ − i δ + cosh κ 1 α = 0 .

We therefore see that (5.3) holds to be true if and only if

cosh κ sinh κ = i α δ .

Define i α δ ≔ U and e κ ≔ t . Thus, we may write

t + 1 t t − 1 t = U ,

which leads to the equation

e 2 κ = U + 1 U − 1 .

It is immediately seen from the last equation that

(5.4) κ k = 1 2 ln U + 1 U − 1 + i 2 Arg U + 1 U − 1 + k π i , k ∈ Z ,

are roots of (5.3).

At this point, some remarks can be made on the various cases of k and U . Clearly, eigenvalues lie on the imaginary upper half-plane. Equation (5.4) suggests that this is possible if and only if

(5.5) 1 2 A r g U + 1 U − 1 + k π > 0 .

Case 1: Since essential argument of any complex number z lies on the interval − π < A r g z ≤ π , we can infer that for any k ∈ N , κ k values of (5.4) are eigenvalues of the operator. We notice here that U values do not affect the structure of the eigenvalues for this case.

Case 2: If k = 0 , we need to determine the values of U :

(5.6) Arg U + 1 U − 1 > 0 .

By substituting i α δ = U in the expression of U + 1 U − 1 , we obtain

Re U + 1 U − 1 = δ 2 − α 2 − ( δ 2 + α 2 ) ,

and

Im U + 1 U − 1 = 2 α δ − ( δ 2 + α 2 ) .

Clearly, (5.6) holds on the condition

2 α δ − ( δ 2 + α 2 ) > 0 .

Therefore, we conclude that two cases appear for k = 0 .

If the real valued impulsive coefficients α and δ have opposite signs, then κ 0 is the eigenvalue of the operator T ˜ .

If α and δ have the same sign, then κ 0 is not the eigenvalue of the operator T ˜ .

Case 3: If k values are negative integers, then clearly (5.5) is smaller than zero. Therefore, T ˜ has no eigenvalues.

6 Conclusion

In this article, scattering analysis of the impulsive Sturm-Liouville problem with the sign-changing weight function has been considered. Turning-point theory has important applications in physics to model the propagation of the wave reflection. Most articles in applied physics investigates the turning point theory for the sign change of the potential function q . However, in this article, we particularly consider the sign change of the weight function which results in hyperbolic representations of the fundamental solutions. Also, the equations containing Bessel type singularities can be solved by Sturm-Liouville equations with turning points. Hence, this article combines two challenging cases and may lay the groundwork for future studies for both inverse and direct scattering theories.

Funding information: None declared.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state that there is no conflict of interest.
Data availability statement: Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

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Received: 2023-01-21

Revised: 2023-02-21

Accepted: 2023-02-21

Published Online: 2023-03-30

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/math-2022-0566

Keywords for this article

Sturm-Liouville equations; impulsive differential equations; scattering theory; discontinuous weight function

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