Fractional singular Sturm-Liouville operator for Coulomb potential
© Bas and Metin; licensee Springer. 2013
Received: 26 April 2013
Accepted: 9 September 2013
Published: 8 November 2013
In this paper, we define a fractional singular Sturm-Liouville operator having Coulomb potential of type . Our main issue is to investigate the spectral properties for the operator. Furthermore, we prove new results according to the fractional singular Sturm-Liouville problem.
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Many important results of the existence of solutions of various classes of fractional differential equations were given by Oldham, Spainer, Kilbas, Marichev, Gorenflo, Miller, Podlubny, Baleanu, Agarwal, Ross, Srivastava etc. Most of the mathematical theory applicable to the study of fractional calculus was developed prior to the turn of the twentieth century [4–19]. For last centuries, the theory of fractional derivatives developed mainly as a pure theoretical field of mathematics useful only for mathematicians.
Furthermore, Sturm-Liouville problems have been known since 1836. The importance of mathematics arises from the study of problems in the real world. The spectral characteristics are spectra, spectral functions, scattering data, norming constants for using Sturm-Liouville problems. The concept of Sturm-Liouville problems plays an important role in mathematics and physics. The progress in applied mathematics was obtained by the extension and development of many important analytical approaches and methods. There have been numerous studies that focus on this problem since then [19–27]. Nowadays, new approaches to fractional Sturm-Liouville problem are produced [17–20].
where λ is a parameter which corresponds to the energy and C is a constant .
We should note that Klimek and Agrawal  defined a fractional Sturm-Liouville operator, introduced a regular fractional Sturm-Liouville problem and investigated the properties of eigenfunctions and eigenvalues of the operator. In this paper, our aim is to introduce a singular fractional Sturm-Liouville problem with Coulomb potential and prove spectral properties of spectral data for the operator. We also show that the fractional Sturm-Liouville operator with Coulomb potential is self adjoint, in addition to .
Let us give some important necessary data that will be used in the main results.
Definition 1 
where Γ denotes the gamma function.
Definition 2 
Definition 3 
If these conditions are satisfied, the series in (7) is convergent for any .
Theorem 4 
If , then the series in (7) is absolutely convergent for all .
If , then the series in (7) is absolutely convergent for and for and .
Theorem 5 
Any such value of q is called a Lipschitz constant for T, and the smallest one is sometimes called ‘the best Lipschitz constant’ of T.
Now, let us take up a fractional singular Sturm-Liouville problem for Coulomb potential.
3 Main results
Let us denote a fractional Sturm-Liouville problem for Coulomb potential with the differential part containing the left- and right-sided derivatives. Let us use the form of the integration by parts formulas (8), (9) 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 first-order derivatives. In the corresponding fractional version, we note that both left and right derivatives 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.
where . Fractional boundary value problem (12)-(14) is a fractional Sturm-Liouville problem for Coulomb potential.
Theorem 8 Fractional Sturm-Liouville operator for Coulomb potential is self-adjoint .
The proof is completed. □
Theorem 9 The eigenvalues of a fractional singular Sturm-Liouville operator with Coulomb potential (12)-(14) are real.
and because y is a non-trivial solution and , it easily seen that . This proves the theorem. □
where . □
and the obtained Fox-Wright function (29) is continuous in the interval for any positive order α.
and the function ψ is defined in (28).
Substituting the above solution into (31), we recover equivalent integral equation (30).
where denotes the supremum norm on the space . □
exists and such an eigenvalue is simple.
Hence, a unique fixed point enounced as exists that solves equation (12), (30) and satisfies boundary conditions (13), (14) provided (37) is applied. In that case, such eigenvalues are simple. The proof is completed. □
In the study, we focus on the spectral properties of the singular fractional Sturm-Liouville problem via Coulomb potential. We pointed that its eigenvalues related to the Coulomb potential with the certain boundary conditions are real and its eigenfunctions corresponding to distinct eigenvalues are orthogonal. We also prove that the fractional Sturm Liouville operator having Coulomb potential is self-adjoint. We give that some results of Sturm Liouville theory for fractional theory.
The authors sincerely thank the editor and reviewer for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript.
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