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.
Keywords:initial value problem; the eigenvalues; normalizing numbers; scattering function; asymptotic formula
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 . He gave a new approach in the spectral theory of singular differential operator of the second order by using contour integration. Also Levitan  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 , Gasymov . 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  presented an approximate construction of the Jost function for some Sturm-Liouville boundary value problem in the case 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  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 which is given by (1.3) as ± signs causes an excess of analytical difficulties. In  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  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
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 and .
For and problem (1.1)-(1.2) takes the form
From now on we consider because according to (1.6) μ covers all the complex plane. Denote by the solution of (1.4) with the initial conditions , . According to (1.3), (1.4) is equivalent to the two equations
It is easy to see that
where , are calculated from the requirements and , so that (1.8) takes the form
For , the function does not belong to also, for , as whereas as , so that it is convenient to consider
as the equation of the eigenvalues .
From this we have , or
Together with the solution of (1.4) we introduce the second solution , 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 , are calculated from the requirements and , 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 ; this condition implies that , which is the same as (1.10).
Now for , we denote by the solution of (1.1) which satisfies the condition
For , (1.1) takes the form , and in the following, we study its solution and the related spectrum. From  this solution has the following representation:
where , , .
For , the solution has the form
where , is the fundamental system of solutions of (1.1) subject to the initial conditions
where the coefficients , are calculated from the requirements , , from which
Further, (1.1), for , takes the form , and the fundamental system of solution of this follows from [, 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 , it follows that, for 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 1.1Under the conditions ( ), the roots of (1.20), for , are simple and lie only on the imaginary axis.
Proof Let , where , be a zero of the function , so that
We prove that , . Since is a solution of (1.1) we have
multiplying both sides of this by and integrating both sides from 0 to ∞, we have
Integrating the first integral by parts and using (1.22), (1.15) we obtain
where , from which we deduce that is real and hence is pure imaginary. We turn now to the proof that the roots are simple from (1.22), this is carried out by proving that implies , where ‘dot’ denotes differentiation with respect to λ.
Integrating the difference with respect to x from 0 to ∞ and using (1.20) we get after some calculation that
We prove the reality of .
For , the function is real because reality of comes from the reality of .
To prove that, for , we observe that φ and θ are real. Let ; since 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 . In a similar way we can prove that is also real so that the solution for is real from which we have and (1.25) takes the form
From (1.28) we see that , which completes the proof. □
Remark 1 For and , the function is the eigenfunction of problem (1.1)-(1.2) that corresponds to the negative eigenvalues .
Lemma 1.2For all the function does not tend to zero, i.e.
Proof Since the function is the solution of (1.1), is also a solution, and it can be shown that these two solutions are linearly independent and their Wronskian is
so that , for , so that and is a fundamental system of solutions of (1.1). In particular, putting into (1.30) we have
To prove that , , , assume to the contrary i.e. , , . From (1.31) and (1.20) we reach to contradiction to the assumption, and, consequently, we deduce that , , . Notice that . □
Lemma 1.3For all the following equality holds:
where is the solution of problem (1.1)-(1.2) and the function
satisfies the properties
It should be noted here that the function defined by (1.33) is called the scattering function of problem (1.1)-(1.2) and the function is called the denominator of .
Proof As mentioned before (1.30) for all , and 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 , are calculated from the initial conditions , in the form
Substituting (1.36) into (1.35) we arrive at the required formula (1.32). Further, since , it follows from (1.33) that
from which we have
2 The asymptotic formulas of eigenvalues and normalizing numbers
The eigenvalues 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 , , and . Integrating (1.15) by parts we have, for , ,
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 and have the same number of zeros inside the quadratic contour where , but since has exactly n zeros, namely , , has an infinite number of zeros, as , with limiting point at infinity. Denote by the zeros of , 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, lies on the imaginary axis, so that it is sufficient to know the asymptotic of for small λ. Let , , we find the asymptotic formula of for . From (2.3), (2.4), (2.5), and (2.6), we have
substituting (2.13) into , and putting we have
and from this and by virtue of the inequality ∀n, we have
From (2.12), it is easy to see that
The estimation of follows from (2.15) and (2.16) in the form
Definition (The normalizing numbers)
are called the normalizing numbers of problem (1.1)-(1.2) (notice that are the eigenfunctions of problem (1.1)-(1.2) corresponding to the eigenvalues ). From (1.28) and the reality of , we have
To evaluate the asymptotic formula of 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, and are given by (1.17)
from which it follows that
From (1.18), using integration by parts and then putting , , 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 :
From (2.21) and (2.22), the normalizing numbers can be written in the form
We substitute (2.23), (2.24), (2.25), and (2.26) into (2.27), , and we find
where , , , and . Further, from (2.16) and (2.17) we have
By substituting from (2.29) into (2.28) we obtain the required asymptotic formula for :
The authors declare that they have no competing interests.
The two authors typed read and approved the final manuscript also they contributed to each part of this work equally.
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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