Spectral Analysis for Fractional Hydrogen Atom Equation


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.

Share and Cite:

Bas, E. and Metin, F. (2015) Spectral Analysis for Fractional Hydrogen Atom Equation. Advances in Pure Mathematics, 5, 767-773. doi: 10.4236/apm.2015.513070.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Miller, K.S. and Ross, B. (1993) An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley and Sons, New York.
[2] Podlubny, I. (1999) Fractional Differential Equations. Academic Press, San Diego.
[3] Kilbas, A.A., Srivastava, H.M. and Trujillo, J.J. (2006) Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam.
[4] Klimek, M. (2009) On Solutions of Linear Fractional Differential Equations of a Variational Type. The Publishing Office of Czestochowa, University of Technology, Czestochowa.
[5] Al-Mdallal, Q.M. (2009) An Efficient Method for Solving Fractional Sturm-Liouville Problems. Chaos Solitons and Fractals, 40, 183-189.
[6] Erturk, V.S. (2011) Computing Eigenelements of Sturm-Liouville Problems of Fractional Order via Fractional Differential Transform Method. Mathematical and Computational Applications, 16, 712-720.
[7] Klimek, M. and Argawal, O.P. (2012) On a Regular Fractional Sturm-Liouville Problem with Derivatives of Order in (0,1). Proceedings of the 13th International Carpathian Control Conference, Vysoke Tatry (Podbanske), Slovakia, 28-31 May 2012.
[8] Baleanu, D., Octavian, M.G. and Agarwal, P.R. (2011) Asymptotic Integration of (1+Alpha)-Order Fractional Differential Equations. Computers & Mathematics with Applications, 62-63, 1492-1500.
[9] Yilmazer, R. (2010) N-Fractional Calculus Operator N^{μ} Method to a Modified Hydrogen Atom Equation. Mathematical Communications, 15, 489-501.
[10] Bas, E. (2013) Fundamental Spectral Theory of Fractional Singular Sturm-Liouville Operator. Journal of Function Spaces and Applications, Article ID: 915830.
[11] Samko, S.G., Kilbass, A.A. and Marichev, O.I. (1993) Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publishers, Philadelphia.
[12] Hilfer, R. (2000) Applications of Fractional Calculus in Physics. World Scientific, Singapore.
[13] Carpinteri, A. and Mainardi, F., Eds. (1998) Fractals and Fractional Calculus in Continum Mechanics. Springer-Verlag, Telos.
[14] Johnson, R.S. (2006) An Introduction to Sturm-Liouville Theory. University of Newcastle, Upon Tyne Publishing, Newcastle upon Tyne.
[15] Zettl, A. (2005) Sturm-Liouville Theory, Mathematical Surveys and Monographs. Volume 121, American Mathematical Society, Providence.
[16] Amrein, W.O., Hinz, A.M. and Pearson, D.B. (2005) Sturm-Liouville Theory: Past and Present. Birkhauser, Basel, Switzerland.
[17] Panakhov, E.S. and Yilmazer, R. (2012) A Hochstadt-Lieberman Theorem for the Hydrogen Atom Equation. Computational & Applied Mathematics, 11, 74-80.
[18] Levitan, B.M. and Sargsjan, I.S. (1975) Introduction to Spectral Theory: Self Adjoint Ordinary Differential Operators. American Mathematical Society, Providence.
[19] Qi, J. and Chen, S. (2011) Eigenvalue Problems of the Model from Nonlocal Continuum Mechanics. Journal of Mathematical Physics, 52, Article ID: 073516.
[20] Bas, E., Panakhov, E. and Yilmazer, R. (2013) The Uniqueness Theorem for Hydrogen Atom Equation. TWMS Journal of Pure and Applied Mathematics, 4, 20-28.
[21] West, B.J., Bologna, M. and Grigolini, P. (2003) Physics of Fractal Operators. Springer Verlag, New York.
[22] Atanackovic, T.M. and Stankovic, B. (2009) Generalized Wave Equation in Nonlocal Elasticity. Acta Mechanica, 208, 1-10.
[23] Granas, A. and Dugundji, J. (2003) Fixed Point Theory. Springer-Verlag, New York.

Copyright © 2022 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.