Painlevé IV Transcendents Generated from the Complex Oscillator
55
7 Conclusions
In this article we have analyzed the link that exists between systems ruled by
second-degree PHA (which have third-order ladder operators) and the PIV equation.
Starting from the SUSY partners of the complex oscillator, an algorithm for
generating PIV transcendents has been implemented (see also [17]). The nonHermitian Hamiltonians giving place to these transcendents have two infinite energy
ladders: an infinite one starting from
ω
2 and going outside the origin along θ -
direction, plus a finite ladder beginning from an arbitrary complex number ε k and
pointing in the same direction. A deeper study about the classification of the PIV
transcendents generated from the SUSY partners of the complex oscillator seems to
be required.
References
1. D.J. Fernández, in Integrability, Supersymmetry and Coherent States, ed. by S. Kuru, et al.
CRM Series in Mathematical Physics, (Springer, Cham, 2019), pp. 37–68
2. V.B. Matveev, M.A. Salle, Darboux Transformations and Solitons (Springer, Berlin, 1991)
3. B.K. Bagchi, Supersymmetry in Quantum and Classical Mechanics (Chapman & Hall, Boca
Raton, 2001)
4. C.V. Sukumar, in AIP Conf. Proc., vol. 744, 166 (2004)
5. A.P. Veselov, A.B. Shabat, Funct. Anal. Appl. 27, 81 (1993)
6. S.Y. Dubov, V.M. Eleonsky, N.E. Kulagin, Chaos 4, 47 (1994)
7. V.E. Adler, Physica D 73, 335 (1994)
8. A. Andrianov, F. Cannata, M. Ioffe, D. Nishnianidze, Phys. Lett. A 266, 341 (2000)
9. J.M. Carballo, D.J. Fernández, J. Negro, L.M. Nieto, J. Phys. A 37, 10349 (2004)
10. J. Mateo, J. Negro, J. Phys. A Math. Theor. 41, 045204 (2008)
11. D. Bermudez, D.J. Fernández, SIGMA 7, 025 (2011)
12. C. Rogers, A.P. Bassom, P.A. Clarkson, J. Math. Ann. App. 462, 1225 (2018)
13. D. Levi, P. Winternitz, Painlevé Transcendents, NATO ASI B278 (Plenum Press, New York,
1992)
14. A.P. Bassom, P.A. Clarkson, A.C. Hicks, Stud. Appl. Math. 95, 1 (1995)
15. D.J. Fernández, V. Hussin, J. Phys. A 32, 3603 (1999)
16. I. Marquette, J. Math. Phys. 50, 095202 (2009)
17. D.J. Fernández, J.C. González, Ann. Phys. 359, 213 (2015)
55
7 Conclusions
In this article we have analyzed the link that exists between systems ruled by
second-degree PHA (which have third-order ladder operators) and the PIV equation.
Starting from the SUSY partners of the complex oscillator, an algorithm for
generating PIV transcendents has been implemented (see also [17]). The nonHermitian Hamiltonians giving place to these transcendents have two infinite energy
ladders: an infinite one starting from
ω
2 and going outside the origin along θ -
direction, plus a finite ladder beginning from an arbitrary complex number ε k and
pointing in the same direction. A deeper study about the classification of the PIV
transcendents generated from the SUSY partners of the complex oscillator seems to
be required.
References
1. D.J. Fernández, in Integrability, Supersymmetry and Coherent States, ed. by S. Kuru, et al.
CRM Series in Mathematical Physics, (Springer, Cham, 2019), pp. 37–68
2. V.B. Matveev, M.A. Salle, Darboux Transformations and Solitons (Springer, Berlin, 1991)
3. B.K. Bagchi, Supersymmetry in Quantum and Classical Mechanics (Chapman & Hall, Boca
Raton, 2001)
4. C.V. Sukumar, in AIP Conf. Proc., vol. 744, 166 (2004)
5. A.P. Veselov, A.B. Shabat, Funct. Anal. Appl. 27, 81 (1993)
6. S.Y. Dubov, V.M. Eleonsky, N.E. Kulagin, Chaos 4, 47 (1994)
7. V.E. Adler, Physica D 73, 335 (1994)
8. A. Andrianov, F. Cannata, M. Ioffe, D. Nishnianidze, Phys. Lett. A 266, 341 (2000)
9. J.M. Carballo, D.J. Fernández, J. Negro, L.M. Nieto, J. Phys. A 37, 10349 (2004)
10. J. Mateo, J. Negro, J. Phys. A Math. Theor. 41, 045204 (2008)
11. D. Bermudez, D.J. Fernández, SIGMA 7, 025 (2011)
12. C. Rogers, A.P. Bassom, P.A. Clarkson, J. Math. Ann. App. 462, 1225 (2018)
13. D. Levi, P. Winternitz, Painlevé Transcendents, NATO ASI B278 (Plenum Press, New York,
1992)
14. A.P. Bassom, P.A. Clarkson, A.C. Hicks, Stud. Appl. Math. 95, 1 (1995)
15. D.J. Fernández, V. Hussin, J. Phys. A 32, 3603 (1999)
16. I. Marquette, J. Math. Phys. 50, 095202 (2009)
17. D.J. Fernández, J.C. González, Ann. Phys. 359, 213 (2015)