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 onedimensional Schrödinger equation with turning point on the half line.
MSC: 58C40, 34L25.
Keywords:
initial value problem; the eigenvalues; normalizing numbers; scattering function; asymptotic formula1 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 SturmLiouville 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 [710]. Notice that the paper [11] presented an approximate construction of the Jost function for some SturmLiouville
boundary value problem in the case
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
and μ is a complex spectral parameter. To study the eigenvalues of (1.1)(1.2), we first
consider the case when
For
From now on we consider
It is easy to see that
where
For
as the equation of the eigenvalues
From this we have
Together with the solution
With the aid of (1.7), we have
where the coefficients
It should be noted, here, that the equation of the eigenvalues can be obtained, also,
from the condition that the solution
Now for
For
where
For
where
where the coefficients
Further, (1.1), for
Now we find the characteristic equation of the eigenvalues of (1.1)(1.2). Since
the solution (1.15) belongs to
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
Proof Let
We prove that
multiplying both sides of this by
Integrating the first integral by parts and using (1.22), (1.15) we obtain
where
Integrating the difference
We prove the reality of
For
To prove that, for
Taking the conjugate of (1.26) we have
It is clear, from (1.26) and (1.27), that
From (1.28) we see that
Remark 1 For
Lemma 1.2For all
Proof Since the function
so that
To prove that
Lemma 1.3For all
where
satisfies the properties
It should be noted here that the function
Proof As mentioned before (1.30) for all
where
Substituting (1.36) into (1.35) we arrive at the required formula (1.32). Further,
since
from which we have
and
□
2 The asymptotic formulas of eigenvalues and normalizing numbers
The eigenvalues
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
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
To make (2.12) more accurate, we must refine (2.11). With the aid of Lemma 1.1,
substituting (2.13) into
and from this and by virtue of the inequality
From (2.12), it is easy to see that
The estimation of
Therefore
Finally
Definition (The normalizing numbers)
The numbers
are called the normalizing numbers of problem (1.1)(1.2) (notice that
To evaluate the asymptotic formula of
where dots and dashes denote the differentiation with respect to λ and x, respectively,
from which it follows that
From (1.18), using integration by parts and then putting
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
We substitute (2.23), (2.24), (2.25), and (2.26) into (2.27),
where
By substituting from (2.29) into (2.28) we obtain the required asymptotic formula
for
where
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The two authors typed read and approved the final manuscript also they contributed to each part of this work equally.
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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