, 93:76 | Cite as

Higher-dimensional fractional time-independent Schrödinger equation via fractional derivative with generalised pseudoharmonic potential

  • Tapas Das
  • Uttam GhoshEmail author
  • Susmita Sarkar
  • Shantanu Das


In this paper, we obtain approximate bound-state solutions of N-dimensional time-independent fractional Schrödinger equation for the generalised pseudoharmonic potential which has the form \(V(r^{\alpha })=a_1r^{2\alpha } + ({a_2}/{r^{2\alpha }})+a_3\). Here \(\alpha \;(0<\alpha <1)\) acts like a fractional parameter for the space variable r. The entire study consists of the Jumarie-type fractional derivative and the elegance of Laplace transform. As a result, we can successfully express the approximate bound-state solution in terms of Mittag–Leffler function and fractionally defined confluent hypergeometric function. Our study may be treated as a generalisation of all previous works carried out on this topic when \(\alpha =1\) and N arbitrary. We provide numerical result of energy eigenvalues and eigenfunctions for a typical diatomic molecule for different \(\alpha \) close to unity. Finally, we try to correlate our work with a Cornell potential model which corresponds to \(\alpha = {1}\) \(/\) \({2}\) with \(a_3=0\) and predicts the approximate mass spectra of quarkonia.


Fractional radial Schrödinger equation generalised pseudoharmonic potential bound-state solutions Mittag–Leffler function 


02.30.−f 03.65.Db 03.65.Ge 02.30.Rz 



The authors would like to thank the anonymous reviewers for their careful reading, useful comments and constructive suggestions for the improvement of the manuscript of the present research work.


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Copyright information

© Indian Academy of Sciences 2019

Authors and Affiliations

  1. 1.Kodalia Prasanna Banga High School (HS)KolkataIndia
  2. 2.Department of Applied MathematicsUniversity of CalcuttaKolkataIndia
  3. 3.Reactor Control System Design Section (E & I Group)Bhabha Atomic Research CentreMumbaiIndia

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