Received 21 September 2015; accepted 18 January 2016; published 21 January 2016
The Sturm-Liouville equation is a second order linear ordinary differential equation of the form
for some and. It was first introduced in an 1837 publication  by the eminent French mathematicians Joseph Liouville and Jacques Charles François Sturm. The Sturm-Liouville Equation (1.1) can easily be reduced to form
If we assume that p(x) has a continuous first derivative, and p(x), r(x) have a continuous second derivative, then by means of the substitutions
where c is given by
Equation (1.1) assumes the form (1.2) replaced by; where
The transformation of the general second order equation to canonical form and the asymptotic formulas for the eigenvalues and eigenfunctions was given by Liouville. A deep study of the distribution of the zeros of eigenfunctions was done by Sturm. Firstly, the formula for the distribution of the eigenvalues of the single dimensional Sturm operator defined in the whole of the straight-line axis with increasing potential in the infinity was given by Titchmarsh in 1946   . Titchmarsh also showed the distribution formula for the Schrödinger Operator. In later years, Levitan improved the Titchmarsh’s method and found important asymptotic formula for the eigenvalues of different differential operators   . Sturm-Liouville problems with a singularity at zero have various versions. The best known case is the one studied by Amirov   , in which the potential has a Coulomb-type singularity
at the origin. In these works, properties of spectral characteristic were studied for Sturm-Liouville operators with Coulomb potential, which have discontinuity conditions inside a finite interval. Panakhov and Sat estimated nodal points and nodal lengths for the Sturm-Liouville operators with Coulomb potential  - . Basand Metin defined a fractional singular Sturm-Liouville operator having Coulomb potential of type A/x  .
Let’s give some fundamental physical properties of the Sturm-Liouville operator with Coulomb potential. Learning about the motion of electrons moving under the Coulomb potential is of significance in quantum theory. Solving these types of problems provides us with finding energy levels of not only hydrogen atom but
also single valance electron atoms such as sodium. For the Coulomb potential is given by, where r
is the radius of the nucleus, e is electronic charge. According to this, we use time-dependent Schrödinger equation
where is the wave function, h is Planck’s constant and m is the mass of electron.
In this equation, if the Fourier transform is applied
it will convert to energy equation dependent on the situation as follows:
Therefore, energy equation in the field with Coulomb potential becomes
If this hydrogen atom is substituted to other potential area, then energy equation becomes
If we make the necessary transformation, then we can get a Sturm-Liouville equation with Coulomb potential
where is a parameter which corresponds to the energy  .
Our aim here is to find asymptotic formulas for singular Sturm-Liouville operatör with Coulomb potential with domain
Also, we give the normalizing eigenfunctions and spectral functions.
2. Basic Properties
We consider the singular Sturm-Liouville problem
where the function. Let us denote by the solution of (2.1) satisfying the initial condition
and by the solution of same equation, satisfying the initial condition
Lemma 1. The solution of problem (2.1) and (2.2) has the following form:
Proof. Since satisfies Equation (2.1), we have
Integrating the first integral on the right side by parts twice and taking the conditions (2.2) into account, we find that
which is (2.4).
Lemma 2. The solution of problem (2.1) and (2.3) has the following form:
Proof. The proof is the same as that of Lemma 1.
Now we give some estimates of and which will be used later. For each fixed x in [0, 1] the map is an entire function on which is real-valued on  . Using the estimate
From (2.6) the inequality is easily checked
where c is uniform with respect to q on bounded sets in.
Lemma 3 (Counting Lemma).  Let and be an integer. Then has exactly N roots, counted with multiplicities, in the open half plane
and for each, exactly one simple root in the egg shaped region
There are no other roots.
From this Lemma there exists an integer N such that for every there is only one eigenvalue in Thus for every
can be chosen independent of q on bounded sets of. Following theorem  shows that the eigenvalues are the zeroes of the map and these zeroes are simple.
Theorem 1. If is Dirichlet eigenvalue of q in, then
In particular,. Thus, all roots of are simple.
Proof. The proof is similar as that of ( , Pöschel and Trubowitz).
3. Asymptotic Formula
We need the following lemma for proving the main result.
Lemma 4. For every f in,
Proof. Firstly, we shall prove the relation (3.1)
By the Cauchy-Schwarz inequality, we get
Since f is in, the last two integrals are equal to
So (3.3) is equivalent to
Finally, we shall prove the relation (3.2)
This proves the lemma.
The main result of this article is the following theorem:
Theorem 2. For,
Proof of the Main Theorem. Since it must be. Because is a nontrivial solution of Equation (2.1) satisfying Dirichlet boundary conditions, we have
From (2.7) someone gets the inequality
From (3.5) integral in the equation of (3.4) takes the form
By using difference formulas for sine we have
From Lemma 4 we get
Thus, by using this inequality (3.4) can be written in the form
From (2.8) we conclude that
Since and, (3.7) is equivalent to
So we get
From (2.8) we have
In this case, the theorem is proved.
From this theorem, the map
from q to its sequences of Dirichlet eigenvalues sends into S. Later, we need this map to characterize spectra which is equivalent to determining the image of.
4. Inverse Spectral Theory
To each eigenvalue we associate a unique eigenfunction normalized by
Let’s define the normalizing eigenfunction:
Lemma 5. For,
This estimate holds uniformly on bounded subsets of.
Proof. Let and. By the basic estimate for,
By using this estimate we have
So we get
Thus we conclude that
Dividing by we get
Also, we need to have asymptotic estimates of the squares of the eigenfunctions and products
Lemma 6. For,
This estimate holds uniformly on bounded subsets of.
Proof. We know that
By the basic estimate for, we have
The map is real analytic on. Now we give asymptotic behavior for.
Theorem 3. Each is a compact, real analytic function on with
Its gradient is
The error terms are uniform on bounded subsets of.
Proof. From  we have
So we calculate the integral
Finally, since, we get
By the Cauchy-Schwarz inequality, we prove the theorem.
Formula (4.3) shows that belongs to. By Theorem 3, the map
from q to its sequences of -values maps into the. So we obtain a map
from into the.
Theorem 4.  is one-to-one on.
Let be the Frechet derivative of the map at q.
Theorem 5.  is an isomorphism from onto.
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 Birkhoff, G.D. (1908) Boundary Value and Expansion Problems of Ordinary Linear Differential Equations. Transactions of the American Mathematical Society, 9, 219-231.
 Levitan, B.M. (1978) On the Determination of the Sturm-Liouville Operator from One and Two Spectra. Mathematics of the USSR Izvestija, 12, 179-193.
 Topsakal, N. and Amirov, R. (2010) Inverse Problem for Sturm-Liouville Operators with Coulomb Potential Which Have Discontinuity Conditions inside an Interval. Mathematical Physics, Analysis and Geometry, 13, 29-46.
 Sat, M. (2014) Half Inverse Problem for the Sturm-Liouville Operator with Coulomb Potential. Applied Mathematics and Information Sciences, 8, 501-504.
 Bas, E. and Metin, F. (2013) Fractional Singular Sturm-Liouville Operator for Coulomb Potential. Advances in Difference Equations, Article ID: 300.
 Guillot, J.-C. and Ralston, J.V. (1988) Inverse Spectral Theory for a Singular Sturm-Liouville Operatör on [0,1]. Journal of Differential Equations, 76, 353-373.