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Exactly solvable Schrödinger eigenvalue problems for new anharmonic potentials with variable bumps and depths

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Abstract

A new approach based on Darboux transformations is introduced to generate classes of solvable Schrödinger equations for new anharmonic potentials with variable bumps and depths. By introducing the concept of a transformation key, we present a method of controlling the number of bumps and their depths in these potentials. Although this method was applied to the one-dimensional generalized harmonic oscillator potential, it can be easily adapted to generate exactly solvable potentials using other known quantum potentials.

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Notes

  1. For \(n=3,\,5,\,7,\,9,\,\ldots \), the sequence that is generated \(-48a^{3} \,,\, 960 a^{4} -20160 a^{5} 483840 a^{6} -13305600 a^{7},\cdots \) deserves its own study.

References

  1. G. Darboux, Lecons sur la theorie generale des surfaces et les application geometriques du calcul infinitesimal Deuziem pattie (Gauthier-Viltars et fils, Paris, 1889)

    Google Scholar 

  2. T.E. Infeld, H. Hull, The factorization method. Rev. Mod. Phys. 53, 21 (1951)

    Article  ADS  MathSciNet  Google Scholar 

  3. S.-H. Dong, Factorization Method in Quantum Mechanics (Springer, Kluwer Academic Publisher, Berlin, 2007)

    Book  Google Scholar 

  4. W.-C. Qiang, S.-H. Dong, SUSYQM and SWKB approaches to the relativistic equations with Hyperbolic potential \(V_0 \tanh ^{2} (r/d)\). Phys. Scr. 72, 127 (2005)

    Article  ADS  Google Scholar 

  5. E. Witten, Dynamical breaking of supersymmetry. Nucl. Phys. B 188, 513 (1981)

    Article  ADS  Google Scholar 

  6. A. Anderson, R. Camporesi, Intertwining operators for solving differential equations, with applications to symmetric spaces. Commun. Math. Phys. 130, 61–82 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  7. F. Cooper, A. Khare, U. Sukhatme, Supersymmetry in Quantum Mechanics (World Scientific, Singapore, 2002)

    MATH  Google Scholar 

  8. V.G. Bagrov, B.F. Samsonov, Darboux transformation, factorization, and supersymmetry in one-dimensional quantum mechanics. Theor. Math. Phys. 104, 1051 (1995)

    Article  MathSciNet  Google Scholar 

  9. A.A. Suzko, A. Schulze-Halberg, Darboux transformations and supersymetry for the generalized Schrödinger equations in \((1+1)\) dimensions. J. Phys. A Math. Theor. (2009). https://doi.org/10.1088/1751-8113/42/29/295203

    Article  MATH  Google Scholar 

  10. A.A. Suzko, E. Velicheva, Supersymmetry and darboux transformations. J. Phys. Conf. Ser. (2012). https://doi.org/10.1088/1742-6596/343/1/012120

    Article  Google Scholar 

  11. K.R. Bryenton, Darboux-Crum Transformations, Supersymmetric Quantum Mechanics, and the Eigenvalue Problem, (B.Sc Hons) University of Prince Edward Island, Charlottetown, Canada (2016). https://doi.org/10.13140/RG.2.2.23129.98408

  12. V.B. Matveev, M.A. Salle, Darboux Transformations and Solitons (Springer, Berlin, 1991)

    Book  Google Scholar 

  13. A. Arsenault, S. Opps, N. Saad, Solvable potentials with exceptional orthogonal polynomials. Ann. Phys. (Berlin) (2015). https://doi.org/10.1002/andp.201500255

    Article  MATH  Google Scholar 

  14. I.I. Gol’dman, D.V. Krivchenkov, Problems in Quantum Mechanics (Pergamon, London, 1961)

    MATH  Google Scholar 

  15. R.L. Hall, N. Saad, A.B. von Keviczky, Matrix elements for a generalized spiked harmonic oscillator. J. Math. Phys. 39, 6345 (1998)

    Article  ADS  MathSciNet  Google Scholar 

  16. R.L. Hall, N. Saad, A.B. von Keviczky, Generalized spiked harmonic oscillator. J. Phys. A Math. Gen. 34, 1169 (2001)

    Article  ADS  MathSciNet  Google Scholar 

  17. D. Agboola, J. Links, I. Marquette, Y.-Z. Zhang, New quasi-exactly-solvable class of generalized isotonic oscillators. J. Phys. A Math. Theor. 47, 395305 (2014)

    Article  MathSciNet  Google Scholar 

  18. J.F. Cariñena, A.M. Perelomov, M.F. Rańada, M. Santander, A quantum exactly solvable nonlinear oscillator related to the isotonic oscillator. J. Phys. A Math. Theor. 41, 085301 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  19. J.F. Cariñena, M.F. Rańada, M. Santander, Two important examples of nonlinear oscillators, in Proceedings of 10th International Conference in Modern Group Analysis (2005), pp. 39–46

  20. J.M. Fellows, R.A. Smith, Factorization solution of a family of quantum nonlinear oscillators. J. Phys. A Math. Theor. 42, 335303 (2009)

    Article  MathSciNet  Google Scholar 

  21. D. Gómez-Ullate, N. Kamran, R. Milson, The Darboux transformation and algebraic deformations of shape-invariant potentials. J. Phys. A Math. Theor. 37, 1789 (2004)

    ADS  MathSciNet  MATH  Google Scholar 

  22. R.L. Hall, N. Saad, Ö. Yeşiltaş, Generalized quantum isotonic nonlinear oscillator in \(d\)-dimensions. J. Phys. A Math. Theor. 43, 465304 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  23. N. Saad, R. Hall, H. Çiftçi, Ö. Yeşiltaş, Study of the generalized quantum isotonic nonlinear oscillator potential. Adv. Math. Phys. (2011). https://doi.org/10.1155/2011/750168

    Article  MathSciNet  MATH  Google Scholar 

  24. J. Sesma, The generalized quantum isotonic oscillator. J. Phys. A Math. Theor. 43, 185303 (2010)

    Article  ADS  MathSciNet  Google Scholar 

  25. Q. Dong, G.-H. Sun, N. Saad, S.-H. Dong, Exact solutions of a nonpolynomial oscillator related to isotonic oscillator. Eur. Phys. J. Plus 134, 562 (2019)

    Article  Google Scholar 

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Acknowledgements

Partial financial support of this work under Grant No. GP249507 from the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged [NS]. We thank the anonymous reviewers for their careful reading of our manuscript and their many insightful comments and suggestions to improve it.

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Correspondence to Nasser Saad.

Appendix A: Continuation of transformations from Sect. 3

Appendix A: Continuation of transformations from Sect. 3

Here, we complete the transformations indicated by the key \({\mathcal {T}}_{m}=\{0;\,0,\,1,\,2,\,3,\,4\}\) for \(m = 4,\,5,\,6\). Since the procedure is identical to the transformations for \(m =2\), and \(m=3\) as discussed in Sects. 2.2 and 2.3, only the final results shall be reported. In all cases, \(E_{n} = a\left( 2n+1\right) \).

1.1 Fourth transformation \({\mathcal {T}}_{4}=2\)

Seed Function: Square Integrable

$$\begin{aligned} \psi _{3;{\mathcal {T}}_{4}}(x)=-\frac{16 \left( 3 a^{3} \xi +2 a^{2} \xi ^{3}\right) }{a+2 \xi ^{2}} \exp \left[ -\frac{\xi ^{2}}{2 a}\right] \end{aligned}$$
(A.1)

Potential Energy: \(x \in [-b/a,\infty )\)

$$\begin{aligned} V_{4}(x)= \xi ^{2} + 4a + \frac{6 a^{2} \left( 3 a^{2}+4 \xi ^{4}\right) }{\xi ^{2} \left( 3 a+2 \xi ^{2}\right) ^{2}} \end{aligned}$$
(A.2)

Exact Solutions: \(n=4,\,6,\,8,\,\ldots \)

$$\begin{aligned} \psi _{4;n}(x)&= \exp \left[ \frac{\xi ^{2}}{2 a} \right] \left( D_{x}^{n+4}-\frac{3 \left( a^{2}-4 a \xi ^{2}-4 \xi ^{4}\right) }{\xi \left( 3 a+2 \xi ^{2}\right) } D_{x}^{n+3} \right. \nonumber \\&\qquad \qquad \qquad +\frac{6 \left( 3 a^{2}+4 a \xi ^{2}+4 \xi ^{4}\right) }{3 a+2 \xi ^{2}}D_{x}^{n+2}\nonumber \\&\qquad \qquad \qquad \left. -\frac{2 \left( 9 a^{3}-18 a^{2} \xi ^{2}-12 a \xi ^{4}-8 \xi ^{6}\right) }{\xi \left( 3 a+2 \xi ^{2}\right) } D_{x}^{n+1} \right) \nonumber \\&\quad \exp \left[ -\frac{\xi ^{2}}{a}\right] \end{aligned}$$
(A.3)

1.2 Fifth transformation \({\mathcal {T}}_{5}=3\)

Seed Function: Nonsquare Integrable

$$\begin{aligned} \psi _{4;{\mathcal {T}}_{5}}(x)= \frac{48 \left( 3 a^{5}+12 a^{4} \xi ^{2}+4 a^{3} \xi ^{4}\right) }{\xi \left( 3 a+2 \xi ^{2}\right) } \exp \left[ -\frac{\xi ^{2}}{2 a}\right] \end{aligned}$$
(A.4)

Potential Energy: \(x \in (-\infty ,\infty )\)

$$\begin{aligned} V_{5}(x)= \xi ^{2} + 6a + \frac{16 a^{2} \left( -9 a^{3}+18 a^{2} \xi ^{2}+12 a \xi ^{4}+8 \xi ^{6}\right) }{\left( 3 a^{2}+12 a \xi ^{2}+4 \xi ^{4}\right) ^{2}} \end{aligned}$$
(A.5)

Exact Solutions: \(n=4,\,5,\,6,\,\ldots \)

$$\begin{aligned} \psi _{5;n}&= \exp \left[ \frac{\xi ^{2}}{2 a}\right] \left( D_{x}^{n+5} + \frac{16 \left( 5 a \xi ^{3}+2 \xi ^{5}\right) }{3 a^{2}+12 a \xi ^{2}+4 \xi ^{4}} D_{x}^{n+4} \right. \nonumber \\&\qquad \qquad \qquad + \frac{12 \left( 5 a^{3}+10 a^{2} \xi ^{2}+20 a \xi ^{4}+8 \xi ^{6}\right) }{3 a^{2}+12 a \xi ^{2}+4 \xi ^{4}} D_{x}^{n+3} \nonumber \\&\qquad \qquad \qquad +\frac{32 \left( 15 a^{2} \xi ^{3}+12 a \xi ^{5}+4 \xi ^{7}\right) }{3 a^{2}+12 a \xi ^{2}+4 \xi ^{4}} D_{x}^{n+2} \nonumber \\&\qquad \qquad \qquad \left. +\frac{4 \left( 45 a^{4}+120 a^{2} \xi ^{4}+64 a \xi ^{6}+16 \xi ^{8}\right) }{3 a^{2}+12 a \xi ^{2}+4 \xi ^{4}} D_{x}^{n+1} \right) \nonumber \\&\quad \exp \left[ -\frac{\xi ^{2}}{a}\right] \end{aligned}$$
(A.6)

1.3 Sixth transformation \({\mathcal {T}}_{6}=4\)

Seed Function: Square Integrable

$$\begin{aligned} \psi _{5;{\mathcal {T}}_{6}}(x)=-\frac{768 \left( 15 a^{6} \xi +20 a^{5} \xi ^{3}+4 a^{4} \xi ^{5}\right) }{3 a^{2}+12 a \xi ^{2}+4 \xi ^{4}} \exp \left[ -\frac{\xi ^{2}}{2 a}\right] \end{aligned}$$
(A.7)

Potential Energy: \(x \in [-b/a,\infty )\)

$$\begin{aligned} V_{6}(x)= \xi ^{2} + 8a + \frac{10 a^{2} \left( 45 a^{4}+120 a^{2} \xi ^{4}+64 a \xi ^{6}+16 \xi ^{8}\right) }{\xi ^{2} \left( 15 a^{2}+20 a \xi ^{2}+4 \xi ^{4}\right) ^{2}} \end{aligned}$$
(A.8)

Exact Solutions: \(n=6,\,8,\,10,\,\ldots \)

$$\begin{aligned} \psi _{6;n}&= \exp \left[ \frac{\xi ^{2}}{2 a}\right] \nonumber \\&\quad \left( D_{x}^{n+6} -\frac{5 \left( 3 a^{3}-18 a^{2} \xi ^{2}-36 a \xi ^{4}-8 \xi ^{6}\right) }{\xi \left( 15 a^{2}+20 a \xi ^{2}+4 \xi ^{4}\right) } D_{x}^{n+5} \right. \nonumber \\&\qquad +\frac{20 \left( 15 a^{3}+30 a^{2} \xi ^{2}+36 a \xi ^{4}+8 \xi ^{6}\right) }{15 a^{2}+20 a \xi ^{2}+4 \xi ^{4}} D_{x}^{n+4} \nonumber \\&\qquad -\frac{20 \left( 15 a^{4}-60 a^{3} \xi ^{2}-120 a^{2} \xi ^{4}-80 a \xi ^{6}-16 \xi ^{8}\right) }{\xi \left( 15 a^{2}+20 a \xi ^{2}+4 \xi ^{4}\right) } D_{x}^{n+3} \nonumber \\&\qquad +\frac{20 \left( 45 a^{4}+120 a^{3} \xi ^{2}+216 a^{2} \xi ^{4}+96 a \xi ^{6}+16 \xi ^{8}\right) }{15 a^{2}+20 a \xi ^{2}+4 \xi ^{4}} D_{x}^{n+2} \nonumber \\&\qquad \left. -\frac{4 \left( 225 a^{5}-450 a^{4} \xi ^{2}-600 a^{3} \xi ^{4}-720 a^{2} \xi ^{6}-240 a \xi ^{8}-32 \xi ^{10}\right) }{\xi \left( 15 a^{2}+20 a \xi ^{2}+4 \xi ^{4}\right) } D_{x}^{n+1} \right) \nonumber \\&\quad \exp \left[ -\frac{\xi ^{2}}{a}\right] \end{aligned}$$
(A.9)

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Bryenton, K.R., Saad, N. Exactly solvable Schrödinger eigenvalue problems for new anharmonic potentials with variable bumps and depths. Eur. Phys. J. Plus 135, 369 (2020). https://doi.org/10.1140/epjp/s13360-020-00378-9

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