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We consider the inverse spectral problem for a singular Sturm-Liouville operator with Coulomb potential. In this paper, we give an asymptotic formula and some properties for this problem by using methods of Trubowitz and Poschel.

The Sturm-Liouville equation is a second order linear ordinary differential equation of the form

for some

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

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 [

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 [

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

is the radius of the nucleus, e is electronic charge. According to this, we use time-dependent Schrödinger equation

where

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

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.

We consider the singular Sturm-Liouville problem

where the function

and by

Lemma 1. The solution of problem (2.1) and (2.2) has the following form:

where

Proof. Since

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

we get

Since

we have

From (2.6) the inequality is easily checked

where c is uniform with respect to q on bounded sets in

Lemma 3 (Counting Lemma). [

and for each

There are no other roots.

From this Lemma there exists an integer N such that for every

Theorem 1. If

In particular,

Proof. The proof is similar as that of ([

We need the following lemma for proving the main result.

Lemma 4. For every f in

and

Proof. Firstly, we shall prove the relation (3.1)

By the Cauchy-Schwarz inequality, we get

Since f is in

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

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

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

To each eigenvalue we associate a unique eigenfunction

Let’s define the normalizing eigenfunction

Lemma 5. For

This estimate holds uniformly on bounded subsets of

Proof. Let

By using this estimate we have

So we get

Thus we conclude that

Dividing

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

Hence,

Let

The map

Theorem 3. Each

Its gradient is

The error terms are uniform on bounded subsets of

Proof. From [

So we calculate the integral

Finally, since

By the Cauchy-Schwarz inequality, we prove the theorem.

Let

Formula (4.3) shows that

from q to its sequences of

from

Theorem 4. [

Let

Theorem 5. [

Etibar S.Panakhov,IsmailUlusoy, (2016) Inverse Spectral Theory for a Singular Sturm Liouville Operator with Coulomb Potential. Advances in Pure Mathematics,06,41-49. doi: 10.4236/apm.2016.61005