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# On the spectral investigation of the scattering problem for some version of one-dimensional Schrödinger equation with turning point

*Boundary Value Problems*
**volume 2014**, Article number: 97 (2014)

## Abstract

In this paper we introduce and investigate the eigenvalues and the normalizing numbers as well as the scattering function for some version of the one-dimensional Schrödinger equation with turning point on the half line.

**MSC:**58C40, 34L25.

## 1 Introduction

The solution of many problems of mathematical physics are reduced to the spectral investigation of a differential operator. The differential operator is called regular if its domain is finite and its coefficients are continuous, otherwise it is called a singular differential operator. The Sturm-Liouville theory occupies a central position in the spectral theory of regular operator. During the development of quantum mechanics there was an increase in the interest of spectral theory of singular operators, on which we will restrict our attention. The first basic role in the development of the spectral theory of singular operators dates back to Titchmarsh [1]. He gave a new approach in the spectral theory of singular differential operator of the second order by using contour integration. Also Levitan [2] gave a new method, he obtained the eigenfunction expansion in an infinite interval by taking the limit of a regular case. In the last 35 or so years, due to the needs of mathematical physics, in particular, quantum mechanics, the question of solving various spectral problems with explosive factor has appeared in the study of geophysics and electromagnetic fields; see [3, 4]. The spectral theory of differential operators with explosive factor is studied by Tikhonov [5], Gasymov [6]. For earlier results on various aspects of solvability theory of boundary value problems and spectral theory in the half line case, the situation closely related to the principal topic of this paper, we refer, for instance, to [7–10]. Notice that the paper [11] presented an approximate construction of the Jost function for some Sturm-Liouville boundary value problem in the case \rho (x)=1 by means of the collocation method. In the present paper we introduce and investigate the eigenvalues and the normalizing numbers as well as the scattering function for some version of the one-dimensional Schrödinger equation with turning point on the half line as in (1.1), (1.2). In [12, 13], and [14] the weight functions introduced are considered as applications of the discontinuous wave speed problem on a non-homogeneous medium as in our case, while the introduction of the weight function \rho (x) which is given by (1.3) as ± signs causes an excess of analytical difficulties. In [15] the author studied the spectral property in a finite interval, while in the present work we consider the half line which gives rise both to a continuous and a discrete spectrum; the latter is treated by the scattering function. In [16] the author considered the weight function of the form

and the spectra were both continuous and discrete as in our problem. We must notice that the result of this paper is a starting point in calculating the regularized trace formula and solving the inverse scattering problem, which will be investigated later on.

Consider the initial value problem

where

q(x) is a finite real valued function which satisfies

and *μ* is a complex spectral parameter. To study the eigenvalues of (1.1)-(1.2), we first consider the case when q(x)\equiv 0 and h=0.

For q(x)\equiv 0 and h=0 problem (1.1)-(1.2) takes the form

From now on we consider Im\lambda \ge 0 because according to (1.6) *μ* covers all the complex plane. Denote by {\phi}_{o}(x,\lambda ) the solution of (1.4) with the initial conditions {\phi}_{o}(0,\lambda )=1, {\phi}_{o}^{\prime}(0,\lambda )=0. According to (1.3), (1.4) is equivalent to the two equations

It is easy to see that

where {a}_{o}(\lambda ), {b}_{o}(\lambda ) are calculated from the requirements {\phi}_{o}(1-0,\lambda )={\phi}_{o}(1+0,\lambda ) and {\phi}_{o}^{\prime}(1-0,\lambda )={\phi}_{o}^{\prime}(1+0,\lambda ), so that (1.8) takes the form

For Im\lambda =0, the function {\phi}_{o}(x,\lambda ) does not belong to {L}_{2}(0,\mathrm{\infty}) also, for Im\lambda >0, {e}^{i\lambda x}\to 0 as x\to \mathrm{\infty} whereas {e}^{-i\lambda x}\to \mathrm{\infty} as x\to \mathrm{\infty}, so that it is convenient to consider

as the equation of the eigenvalues {\mu}_{o}={\lambda}_{o}^{2}.

From this we have {\lambda}_{o}=(n+\frac{1}{4})\pi i, n=0,1,\pm 1,\pm 2,\dots or

Together with the solution {\phi}_{o}(x,\lambda ) of (1.4) we introduce the second solution {f}_{o}(x,\lambda ), which is known as the Jost solution. This solution is defined by the condition

With the aid of (1.7), we have

where the coefficients {c}_{o}(\lambda ), {d}_{o}(\lambda ) are calculated from the requirements {f}_{o}(1-0,\lambda )={f}_{o}(1+0,\lambda ) and {f}_{o}^{\prime}(1-0,\lambda )={f}_{o}^{\prime}(1+0,\lambda ), and the solution becomes

It should be noted, here, that the equation of the eigenvalues can be obtained, also, from the condition that the solution {f}_{o}(x,\lambda )\in {L}_{2}(0,\mathrm{\infty};\rho ); this condition implies that {f}_{o}^{\prime}(0,\lambda )=0, which is the same as (1.10).

Now for q(x)\ne 0, h\ne 0 we denote by f(x,\lambda ) the solution of (1.1) which satisfies the condition

For x>1, (1.1) takes the form -{y}^{\u2033}+q(x)y={\lambda}^{2}y, and in the following, we study its solution and the related spectrum. From [4] this solution has the following representation:

where Im\lambda \ge 0, K(x,x)=\frac{1}{2}{\int}_{0}^{x}q(t)\phantom{\rule{0.2em}{0ex}}dt, 1<x<\mathrm{\infty}.

For 0\le x\le 1, the solution f(x,\lambda ) has the form

where \phi (x,\lambda ), \theta (x,\lambda ) is the fundamental system of solutions of (1.1) subject to the initial conditions

where the coefficients a(\lambda ), b(\lambda ) are calculated from the requirements f(1-0,\lambda )=f(1+0,\lambda ), {f}^{\prime}(1-0,\lambda )={f}^{\prime}(1+0,\lambda ), from which

Further, (1.1), for 0\le x\le 1, takes the form -{y}^{\u2033}+q(x)y=-{\lambda}^{2}y, and the fundamental system of solution of this follows from [[4], p.18] by the representation

Now we find the characteristic equation of the eigenvalues of (1.1)-(1.2). Since the solution (1.15) belongs to {L}_{2}(0,\mathrm{\infty}), Im\lambda >0 it follows that, for \mu ={\lambda}^{2} to be an eigenvalue, it must satisfy the initial condition (1.2), namely

From (1.15) and (1.16) we have

In the following lemmas we study some properties of the eigenvalues of problem (1.1)-(1.2).

**Lemma 1.1** *Under the conditions* q(x)>0 (0<x<\mathrm{\infty}), *the roots of* (1.20), *for* Im\lambda >0, *are simple and lie only on the imaginary axis*.

*Proof* Let {\lambda}_{o}, where Im{\lambda}_{o}>0, be a zero of the function {f}^{\prime}(0,\lambda )-hf(0,\lambda ), so that

We prove that {\lambda}_{o}=i{\tau}_{o}, {\tau}_{o}>0. Since f(x,{\lambda}_{o}) is a solution of (1.1) we have

multiplying both sides of this by \overline{f(x,{\lambda}_{o})} and integrating both sides from 0 to ∞, we have

Integrating the first integral by parts and using (1.22), (1.15) we obtain

where {\int}_{0}^{\mathrm{\infty}}\rho (x){|f(x,{\lambda}_{o})|}^{2}\phantom{\rule{0.2em}{0ex}}dx\ne 0, from which we deduce that {\lambda}_{o}^{2} is real and hence {\lambda}_{o} is pure imaginary. We turn now to the proof that the roots are simple from (1.22), this is carried out by proving that {f}^{\prime}(0,\lambda )-hf(0,\lambda )=0 implies [{\dot{f}}^{\prime}(0,\lambda )-h\dot{f}(0,\lambda )]\ne 0, where ‘dot’ denotes differentiation with respect to *λ*.

Integrating the difference [\dot{f}(x,\lambda )\times \text{(1.23)}]-[f(x,\lambda )\times \frac{d}{d\lambda}\text{(1.23)}] with respect to *x* from 0 to ∞ and using (1.20) we get after some calculation that

We prove the reality of f(x,\lambda ).

For x>1, \lambda =i\tau the function f(x,\lambda )={e}^{-\tau x}+{\int}_{x}^{\mathrm{\infty}}K(x,t){e}^{-\tau t}\phantom{\rule{0.2em}{0ex}}dt is real because reality of K(x,t) comes from the reality of q(x).

To prove that, for 0\le x<1, we observe that *φ* and *θ* are real. Let \lambda =i\tau; since \phi (x,\lambda ) is a solution of (1.1)-(1.2), we have

Taking the conjugate of (1.26) we have

It is clear, from (1.26) and (1.27), that \phi (x,\lambda )=\overline{\phi (x,\lambda )}. In a similar way we can prove that \theta (x,\lambda ) is also real so that the solution f(x,\lambda ) for 0\le x<1 is real from which we have {f}^{2}(x,\lambda )={|f(x,\lambda )|}^{2} and (1.25) takes the form

From (1.28) we see that \frac{d}{d\lambda}[{f}^{\prime}(0,\lambda )-hf(0,\lambda )]\ne 0, which completes the proof. □

**Remark 1** For Im{\lambda}_{n}>0 and {f}^{\prime}(0,{\lambda}_{n})-hf(0,{\lambda}_{n})=0, the function f(0,{\lambda}_{n}) is the eigenfunction of problem (1.1)-(1.2) that corresponds to the negative eigenvalues {\mu}_{n}={\lambda}_{n}^{2}=-{\chi}_{n}^{2}.

**Lemma 1.2** *For all* Re\lambda \ne 0 *the function* {f}^{\prime}(0,\lambda )-hf(0,\lambda ) *does not tend to zero*, *i*.*e*.

*Proof* Since the function f(x,\lambda ) is the solution of (1.1), f(x,-\lambda ) is also a solution, and it can be shown that these two solutions are linearly independent and their Wronskian is

so that W[f(x,\lambda ),f(x,-\lambda )]\ne 0, for Re\lambda \ne 0, so that f(x,\lambda ) and f(x,-\lambda ) is a fundamental system of solutions of (1.1). In particular, putting x=0 into (1.30) we have

To prove that {f}^{\prime}(0,\lambda )-hf(0,\lambda )\ne 0, Re\lambda \ne 0, -\mathrm{\infty}<\lambda <\mathrm{\infty}, assume to the contrary *i.e.* {f}^{\prime}(0,\lambda )-hf(0,\lambda )=0, Re\lambda \ne 0, -\mathrm{\infty}<\lambda <\mathrm{\infty}. From (1.31) and (1.20) we reach to contradiction to the assumption, and, consequently, we deduce that {f}^{\prime}(0,\lambda )-hf(0,\lambda )\ne 0, Re\lambda \ne 0, -\mathrm{\infty}<\lambda <\mathrm{\infty}. Notice that \overline{f(x,\lambda )}=f(x,-\lambda ). □

**Lemma 1.3** *For all* Re\lambda \ne 0 *the following equality holds*:

*where* \phi (x,\lambda ) *is the solution of problem* (1.1)-(1.2) *and the function*

*satisfies the properties*

It should be noted here that the function S(\lambda ) defined by (1.33) is called the scattering function of problem (1.1)-(1.2) and the function {f}^{\prime}(0,\lambda )-hf(0,\lambda ) is called the denominator of S(\lambda ).

*Proof* As mentioned before (1.30) for all Re\lambda \ne 0, f(x,\lambda ) and f(x,-\lambda ) is a fundamental system of solutions of (1.1)-(1.2), so that any linear combination of them is again a solution of (1.1)-(1.2):

where A(\lambda ), B(\lambda ) are calculated from the initial conditions \phi (0,\lambda )=1, {\phi}^{\prime}(0,\lambda )=h in the form

Substituting (1.36) into (1.35) we arrive at the required formula (1.32). Further, since \overline{f(x,\lambda )}=f(x,-\lambda ), it follows from (1.33) that

from which we have

and

□

## 2 The asymptotic formulas of eigenvalues and normalizing numbers

The eigenvalues \mu ={\lambda}^{2} of problem (1.1)-(1.2) are the roots of the equation

In the following we prove that (2.1) has an infinite number of roots and find their asymptotic formula. From (1.15), (1.17), (1.18), and (1.19) we have

Now, we calculate the asymptotic formula of f(1,\lambda ), {f}^{\prime}(1,\lambda ), \phi (1,\lambda ) and {\phi}^{\prime}(1,\lambda ). Integrating (1.15) by parts we have, for x\ge 1, Im\lambda >0,

Similarly from (1.18) we have

The following group of inequalities follows from (2.3)-(2.6):

Substituting (2.7)-(2.10) into (2.2), we obtain

comparing (1.10) and (2.11) we see that {f}^{\prime}(0,\lambda )-hf(0,\lambda ) and {f}_{o}(0,\lambda )={e}^{i\lambda}[cosh\lambda +isinh\lambda ] have the same number of zeros inside the quadratic contour {\mathrm{\Gamma}}_{n} where \{{\mathrm{\Gamma}}_{n}:|Re\lambda |\le \pi (n-\frac{1}{4}),0<Im\lambda \le \pi (n-\frac{1}{4})\}, but since {f}_{o}(0,\lambda ) has exactly *n* zeros, namely {\lambda}_{k}^{o}=i\pi (k-\frac{1}{4}), k=1,2,\dots ,n, {f}^{\prime}(0,\lambda )-hf(0,\lambda ) has an infinite number of zeros, as n\to \mathrm{\infty}, with limiting point at infinity. Denote by {\lambda}_{n} the zeros of {f}^{\prime}(0,\lambda )-hf(0,\lambda )=0, so that, by the Rouche theorem, we have

To make (2.12) more accurate, we must refine (2.11). With the aid of Lemma 1.1, {\lambda}_{n} lies on the imaginary axis, so that it is sufficient to know the asymptotic of {f}^{\prime}(0,\lambda )-hf(0,\lambda ) for small *λ*. Let \lambda =i\tau, \tau >0, we find the asymptotic formula of {f}^{\prime}(0,i\tau )-hf(0,i\tau ) for \tau \to \mathrm{\infty}. From (2.3), (2.4), (2.5), and (2.6), we have

substituting (2.13) into {f}^{\prime}(0,i\tau )-hf(0,i\tau )=0, and putting {\lambda}_{n}=i{\tau}_{n} we have

and from this and by virtue of the inequality |cos{\tau}_{n}|\ge \delta >0 ∀*n*, we have

From (2.12), it is easy to see that

The estimation of {\epsilon}_{n} follows from (2.15) and (2.16) in the form

Therefore

Finally

**Definition** (The normalizing numbers)

The numbers

are called the normalizing numbers of problem (1.1)-(1.2) (notice that f(x,{\lambda}_{n}) are the eigenfunctions of problem (1.1)-(1.2) corresponding to the eigenvalues {\lambda}_{n}). From (1.28) and the reality of f(x,{\lambda}_{n}), we have

To evaluate the asymptotic formula of {a}_{n} we evaluate the asymptotic formula of the right hand side of (2.21). From (1.15), (1.17) we have

where dots and dashes denote the differentiation with respect to *λ* and x, respectively, a(\lambda ) and b(\lambda ) are given by (1.17)

from which it follows that

From (1.18), using integration by parts and then putting x=1, \lambda =i\tau, we obtain

From (1.19), carrying out a similar calculation with respect to *θ*, we obtain

With the aid of (1.15), similar expressions can be calculated with respect to f(1,i\tau ):

From (2.21) and (2.22), the normalizing numbers {a}_{n} can be written in the form

We substitute (2.23), (2.24), (2.25), and (2.26) into (2.27), {\lambda}_{n}=i{\tau}_{n}, and we find

where \alpha =1+2K(1,1)+2A(1,1), \beta =1+2A(1,1), {\alpha}_{1}=B(1,1)+K(1,1), and {\beta}_{1}=K(1,1)-B(1,1). Further, from (2.16) and (2.17) we have

By substituting from (2.29) into (2.28) we obtain the required asymptotic formula for {a}_{n}:

where

## References

Everitt WN: Charles Sturm and the development of Sturm-Liouville theory in the years 1900 to 1950. In

*Sturm-Liouville Theory*. Birkhäuser, Basel; 2005:271-331.Levitan BM:

*Inverse Sturm-Liouville Problems*. Nauka, Moscow; 1984. English transl., VNU Sci. Press, Utrecht (1987)Alimov SA: The work of A.N. Tikhonov on inverse problems for the Sturm-Liouville equation.

*Usp. Mat. Nauk*1976, 31(6(192)):84-88. English transl., Russ. Math. Surv. 31(6), 87-92 (1976)Marchenko VA:

*Sturm-Liouville Operators and Their Applications*. Naukova Dumka, Kiev; 1977. English transl., Birkhäuser (1986)Tikhonov AN: On the uniqueness of the solution of electric investigation problem.

*Doklady of the USSR*1949, 69(6):797-800. (in Russian)Gasymov MG: Direct and inverse problems of spectral analysis for a class of equations with discontinuous coefficients.

*Proceedings of the International Conference on Non-Classical Methods in Geophysics*1977, 37-44. (in Russian), NovosibirskAliev AR, Mirzoev SS: On boundary value problem solvability theory for a class of high-order operator-differential equations.

*Funkc. Anal. Prilozh.*2010, 44(3):63-65. English transl., Funct. Anal. Appl. 44(3), 209-211 (2010) 10.4213/faa2996Mirzoev SS, Aliev AR, Rustamova RA: Solvability conditions for boundary-value problems for elliptic operator-differential equations with discontinuous coefficient.

*Mat. Zametki*2012, 92(5):789-793. English transl., Math. Notes 92(5), 722-726 (2012) 10.4213/mzm10132Mirzoev SS, Aliev AR, Rustamova RA: On the boundary value problem with the operator in boundary conditions for the operator-differential equation of second order with discontinuous coefficients.

*J. Mat. Fiz. Anal. Geom.*2013, 9(2):207-226.Aliev AR: On the generalized solution of the boundary-value problem for the operator-differential equations of the second order with variable coefficients.

*J. Mat. Fiz. Anal. Geom.*2006, 2(1):87-93.Aliev AR, Gasymova SG, Gasymova DG, Ahmadzadeh ND: Approximate construction of the Jost function by the collocation method for Sturm-Liouville boundary value problem.

*Azerb. J. Math.*2013, 3(2):45-61.Freiling G, Yurko VA:

*Inverse Sturm-Liouville Problems and Their Applications*. Nova Science Publishers, New York; 2001.Efendiev RF, Orudzhev HD: Inverse wave spectral problem with discontinuous wave speed.

*J. Mat. Fiz. Anal. Geom.*2010, 6(3):255-265.Efendiev RF: Spectral analysis for one class of second-order indefinite non-self-adjoint differential operator pencil.

*Appl. Anal.*2011, 90(12):1837-1849. 10.1080/00036811.2010.532491El-Raheem ZFA, Nasser AH: On the spectral property of a Dirichlet problem with explosive factor.

*Appl. Math. Comput.*2003, 138(2-3):355-374. 10.1016/S0096-3003(02)00134-0Gasymov MG, Kakhramanov AS, Petrosyan SK: On the spectral theory of linear differential operators with discontinuous coefficients.

*Akad. Nauk Azerbaĭdzhan. SSR Dokl.*1987, 43(3):13-16. (in Russian)

## Acknowledgements

We are indebted to an anonymous referee for a detailed reading of the manuscript and useful comments and suggestions, which helped us improve this work. This work is supported by the Research Support Unit of Alexandria University.

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El-Raheem, Z.F., Nasser, A. On the spectral investigation of the scattering problem for some version of one-dimensional Schrödinger equation with turning point.
*Bound Value Probl* **2014**, 97 (2014). https://doi.org/10.1186/1687-2770-2014-97

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DOI: https://doi.org/10.1186/1687-2770-2014-97