Inverse Spectral Theory for a Singular Sturm Liouville Operator with Coulomb Potential

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.


Introduction
The Sturm-Liouville equation is a second order linear ordinary differential equation of the form 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 [2] [3].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 [4] [5].Sturm-Liouville problems with a singularity at zero have various versions.The best known case is the one studied by Amirov [6] [7], in which the potential has a Coulomb-type singularity ( ) where ψ 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 λ is a parameter which corresponds to the energy [12].
Our aim here is to find asymptotic formulas for singular Sturm-Liouville operatör with Coulomb potential with domain 0,1 : , are absolutely continuous on 0,1 , 0,1 and 1 0 Also, we give the normalizing eigenfunctions and spectral functions.

Basic Properties
We consider the singular Sturm-Liouville problem ( ) where the function ( ) [ ] the solution of (2.1) satisfying the initial condition and by ( ) the solution of same equation, satisfying the initial condition The solution of problem (2.1) and (2.2) has the following form: Proof.Since ( ) , , x q ϕ λ 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 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 ϕ and ψ which will be used later.For each fixed x in [0, 1] the map where c is uniform with respect to q on bounded sets in [ ]2 0,1 L .

Lemma 3 (Counting Lemma). [13] Let
[ ] 2 0,1 q L ∈ and From this Lemma there exists an integer N such that for every n N > there is only one Thus for every n N ≥ ( ) 2 N can be chosen independent of q on bounded sets of Following theorem [13] shows that the eigenvalues ≥ are the zeroes of the map ( ) 1, , q λ ϕ λ → and these zeroes are simple.

Asymptotic Formula
We need the following lemma for proving the main result.Lemma 4. For every and Proof.Firstly, we shall prove the relation (3.1) By the Cauchy-Schwarz inequality, we get , the last two integrals are equal to ( ) Finally, we shall prove the relation (3.2) This proves the lemma.
The main result of this article is the following theorem: Proof of the Main Theorem.Since ( ) 0, , 0, q ϕ λ = it must be ( ) is a nontrivial solution of Equation (2.1) satisfying Dirichlet boundary conditions, we have From (2.7) someone gets the inequality n q t n n n n n q t t q t ct t q q t q q t q t µ µ ϕ µ µ µ µ From (3.5) integral in the equation of (3.4)  q t q t t q q t t q q q t q t q t t q q q t q t q q t O q n Thus, by using this inequality (3.4) can be written in the form From (2.8) we conclude that ( ) Since ( ) In this case, the theorem is proved.From this theorem, the map , , q q q q µ µ µ → =  from q to its sequences of Dirichlet eigenvalues sends 2  L into S. Later, we need this map to characterize spectra which is equivalent to determining the image of µ .

Inverse Spectral Theory
To each eigenvalue we associate a unique eigenfunction ( ) Let's define the normalizing eigenfunction ( ) , n g x q : ( ) Also, we need to have asymptotic estimates of the squares of the eigenfunctions and products

2 )
first introduced in an 1837 publication[1] 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 actly N roots, counted with multiplicities, in the open half plane n N > , exactly one simple root in the egg shaped region π π 2 n λ − < There are no other roots.