Point Transformations: Exact Solutions of the Quantum Time-Dependent Mass. . .
303
4 Concluding Remarks
It was shown that a set of orthonormal solutions for the time-dependent mass nonstationary oscillator can be found by constructing the appropriate point transformation
and deforming the solutions of the stationary oscillator. The latter is possible since
the point transformation preserves the structure of the inner product. Although the
Hamiltonian depends explicitly on time, a spectral problem can be identified for
the appropriate constant of motion that emerges from the transformed Hamiltonian
of the stationary oscillator. The procedure has been developed in general for any
time-dependent mass, frequency and an external driving force. Among the examples
that could be addressed we have the Caldirola–Kanai oscillator, which leads to the
quantum Arnold transformation [17], and the Hermite oscillator [18]. A detailed
discussion will be provided elsewhere.
Acknowledgments K.Z. acknowledges the support from the Mathematical Physics Laboratory,
Centre de Recherches Mathématiques, through a postdoctoral fellowship. He also acknowledges
the support by Conacyt (Mexico), grant number A1-S-24569. V.H. acknowledges the support of
research grants from NSERC of Canada.
References
1. D.E. Pritchard, Phys. Rev. Lett. 51, 1336 (1983)
2. M. Combescure, Ann. Inst. Henri Poincaré A 44, 293 (1986)
3. B.M. Mihalcea, Phys. Scr. 2009, 014006 (2009)
4. R.J. Glauber, in Proceedings of the International Enrico Fermi School, Course 118, Varenna,
Italy, July 1–19, 1992, ed. by E. Arimondo, W.D. Philips, F. Sttrumia (North Holland,
Amsterdam, 1992), p. 643
5. H.R. Lewis, J. Math. Phys. 9, 1976 (1968)
6. H.R. Lewis, Jr., W.B. Riesenfeld, J. Math. Phys. 10, 1458 (1969)
7. V.V. Dodonov, V.I. Man’ko, L. Rosa, Phys. Rev. A 57, 2851 (1998)
8. J.R. Ray, Phys. Rev. A. 26, 729 (1982)
9. F. Finkel, A. González-López, N. Kamran, M.A. Rodríguez, J. Math. Phys. 40, 3268 (1999)
10. K. Zelaya, O. Rosas-Ortiz, Phys. Scr. 95, 064004 (2020)
11. F.W.J. Olver, et al. (eds.), NIST Handbook of Mathematical Functions (Cambridge University
Press, New York, 2010)
12. W.H. Steeb, Invertible Point Transformations and Nonlinear Differential Equations (World
Scientific Publishing, Singapore, 1993)
13. V. Ermakov, Kiev University Izvestia, Series III 9, 1 (1880) (in Russian). English translation
by A.O. Harin, Appl. Anal. Discrete Math. 2, 123 (2008)
14. O. Rosas-Ortiz, O. Castaños, D. Schuch, J. Phys. A: Math. Theor. 48, 445302 (2015)
15. Z. Blanco-Garcia, O. Rosas-Ortiz, K. Zelaya, Math. Meth. Appl. Sci. 42, 4925 (2019)
16. A. Mostafazadeh, Phys. Rev. D 98, 046022 (2018)
17. J. Guerrero, V. Aldaya, F.F. López-Ruiz, F. Cossio, Int. J. Geom. Meth. Mod. 9, 126011 (2012)
18. N. Ünal, J. Math. Phys. 59, 062104 (2018)
303
4 Concluding Remarks
It was shown that a set of orthonormal solutions for the time-dependent mass nonstationary oscillator can be found by constructing the appropriate point transformation
and deforming the solutions of the stationary oscillator. The latter is possible since
the point transformation preserves the structure of the inner product. Although the
Hamiltonian depends explicitly on time, a spectral problem can be identified for
the appropriate constant of motion that emerges from the transformed Hamiltonian
of the stationary oscillator. The procedure has been developed in general for any
time-dependent mass, frequency and an external driving force. Among the examples
that could be addressed we have the Caldirola–Kanai oscillator, which leads to the
quantum Arnold transformation [17], and the Hermite oscillator [18]. A detailed
discussion will be provided elsewhere.
Acknowledgments K.Z. acknowledges the support from the Mathematical Physics Laboratory,
Centre de Recherches Mathématiques, through a postdoctoral fellowship. He also acknowledges
the support by Conacyt (Mexico), grant number A1-S-24569. V.H. acknowledges the support of
research grants from NSERC of Canada.
References
1. D.E. Pritchard, Phys. Rev. Lett. 51, 1336 (1983)
2. M. Combescure, Ann. Inst. Henri Poincaré A 44, 293 (1986)
3. B.M. Mihalcea, Phys. Scr. 2009, 014006 (2009)
4. R.J. Glauber, in Proceedings of the International Enrico Fermi School, Course 118, Varenna,
Italy, July 1–19, 1992, ed. by E. Arimondo, W.D. Philips, F. Sttrumia (North Holland,
Amsterdam, 1992), p. 643
5. H.R. Lewis, J. Math. Phys. 9, 1976 (1968)
6. H.R. Lewis, Jr., W.B. Riesenfeld, J. Math. Phys. 10, 1458 (1969)
7. V.V. Dodonov, V.I. Man’ko, L. Rosa, Phys. Rev. A 57, 2851 (1998)
8. J.R. Ray, Phys. Rev. A. 26, 729 (1982)
9. F. Finkel, A. González-López, N. Kamran, M.A. Rodríguez, J. Math. Phys. 40, 3268 (1999)
10. K. Zelaya, O. Rosas-Ortiz, Phys. Scr. 95, 064004 (2020)
11. F.W.J. Olver, et al. (eds.), NIST Handbook of Mathematical Functions (Cambridge University
Press, New York, 2010)
12. W.H. Steeb, Invertible Point Transformations and Nonlinear Differential Equations (World
Scientific Publishing, Singapore, 1993)
13. V. Ermakov, Kiev University Izvestia, Series III 9, 1 (1880) (in Russian). English translation
by A.O. Harin, Appl. Anal. Discrete Math. 2, 123 (2008)
14. O. Rosas-Ortiz, O. Castaños, D. Schuch, J. Phys. A: Math. Theor. 48, 445302 (2015)
15. Z. Blanco-Garcia, O. Rosas-Ortiz, K. Zelaya, Math. Meth. Appl. Sci. 42, 4925 (2019)
16. A. Mostafazadeh, Phys. Rev. D 98, 046022 (2018)
17. J. Guerrero, V. Aldaya, F.F. López-Ruiz, F. Cossio, Int. J. Geom. Meth. Mod. 9, 126011 (2012)
18. N. Ünal, J. Math. Phys. 59, 062104 (2018)