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
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
With the aid of (1.7), we have
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:
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).
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.
Taking the conjugate of (1.26) we have
satisfies the properties
from which we have
2 The asymptotic formulas of eigenvalues and normalizing numbers
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
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
From (2.12), it is easy to see that
Definition (The normalizing numbers)
from which it follows that
From (1.19), carrying out a similar calculation with respect to θ, we obtain
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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