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In this paper, spectral analysis of fractional Sturm Liouville problem defined on (0, 1], having the singularity of type at zero and researched the fundamental properties of the eigenfunctions and eigenvalues for the operator. We show that the eigenvalues and eigenfunctions of the problem are real and orthogonal, respectively.

Fractional calculus have been available and applicable to various fields of science, the investigation of the theory of fractional differential equations has only been started recently. At the same time, fractional differential equations have great interest due to their numerous applications in many fields of science, such as physics, mechanics, chemistry, finance, electromagnetics, acoustics, viscoelasticity, electrochemistry, economy, etc. So far, there have been several fundamental works on the fractional derivative and fractional differential equations, written by Miller and Ross [

In recent years, fractional Sturm-Liouville problems have been studied and these studies, which the eigenvalues and eigenfunctions associated to these operators and also theirs properties, have been published by M. Klimek and O. P. Argawal [

Sturm-Liouville problems have been studied for over two hundred years. The great progress has been made related to spectral theory. And this topic has an increasing interest for years from different points of view [

Firstly, we consider singular Sturm-Liouville equation

In quantum mechanics the study of the energy levels of the hydrogen atom leads to this equation [

Substitution

In most cases we took the reference potential to be

When

where the fractional order

where

Definition 1. [

where

Definition 2. [

Analogous formulas yield the left and right-sided Caputo derivatives of order

Property 3. [

Property 4. [

hold for any

Now, let’s take up a fractional Sturm-Liouville problem for hydrogen atom equation.

Let’s denote a singular fractional Sturm-Liouville problem for hydrogen atom equation with the differential part containing the left and right-sided derivatives. Let’s use the form of the integration by parts formulas (7) and (8) for this new approximation. Main properties of eigenfunctions and eigenvalues in the theory of classical Sturm- Liouville problems are related to the integration by parts formula for the appear and the essential pairs are the left Riemann-Liouville derivative with the right Caputo derivative and the right Riemann-Liouville derivative with the left Caputo one.

Definition 5. For

consider the fractional hydrogen atom equation

where

where

Theorem 6. Fractional hydrogen operator is self-adjoint on (0, 1].

Proof. Let us consider the following equation

Considering property 3 to the first integral in the last equation, we obtain the identity

Similarly, we obtain

The right hand sides of the Equations (14) and (15) are equal hence we may see that the left sides are equal that is

Therefore,

Theorem 7. The eigenvalues of fractional hydrogen atom operator (11)-(13) are real.

Proof. Let us observe that following relation results from property (3)

Suppose that

where

Now, we integrate over interval (0, 1] and applying relation (16) we note that the right-hand side of the integrated equality contains only boundary terms:

by virtue of the boundary conditions (18), (19), (21), (22), we find

and because y is a non-trivial solution and

Theorem 8. The eigenfunctions corresponding to distinct eigenvalues of fractional hydrogen atom operator (11)-(13) are orthogonal weight function

Proof. We have by assumptions fractional Sturm-Liouville for hydrogen atom operator fulfilled by two different eigenvalues

we multiply Equation (23) by function

Integrating over interval (0, 1] and applying relation (16) we note that the right-hand side of the integrated equality contains only boundary terms:

Using the boundary conditions (24), (25), (27), (28), we find that

where

ErdalBas,FundaMetin, (2015) Spectral Analysis for Fractional Hydrogen Atom Equation. Advances in Pure Mathematics,05,767-773. doi: 10.4236/apm.2015.513070