On form-preserving transformations for the time-dependent Schrodinger equation
| dc.contributor.author | Finkel Morgenstern, Federico | |
| dc.contributor.author | González López, Artemio | |
| dc.contributor.author | Kamran, Niky | |
| dc.contributor.author | Rodríguez González, Miguel Ángel | |
| dc.date.accessioned | 2023-06-20T20:08:44Z | |
| dc.date.available | 2023-06-20T20:08:44Z | |
| dc.date.issued | 1999-07 | |
| dc.description | © 1999 American Institute of Physics. F.F., A.G.-L., and M.A.R. would like to acknowledge the partial financial support of the DGICYT under Grant No. PB95-0401. N.K. was supported in part by NSERC Grant No. 0GP0105490. | |
| dc.description.abstract | In this paper we point out a close connection between the Darboux transformation and the group of point transformations which preserve the form of the time-dependent Schroumldinger equation (TDSE). In our main result, we prove that any pair of time-dependent real potentials related by a Darboux transformation for the TDSE may be transformed by a suitable point transformation into a pair of time-independent potentials related by a usual Darboux transformation for the stationary Schroumldinger equation. Thus, any (real) potential solvable via a time-dependent Darboux transformation can alternatively be solved by applying an appropriate form-preserving point transformation of the TDSE to a time-independent potential. The pre-eminent role of the latter type of transformations in the solution of the TDSE is illustrated with a family of quasi-exactly solvable time-dependent anharmonic potentials. | |
| dc.description.department | Depto. de Física Teórica | |
| dc.description.faculty | Fac. de Ciencias Físicas | |
| dc.description.refereed | TRUE | |
| dc.description.sponsorship | DGICYT | |
| dc.description.sponsorship | NSERC | |
| dc.description.status | pub | |
| dc.eprint.id | https://eprints.ucm.es/id/eprint/31405 | |
| dc.identifier.doi | 10.1063/1.532885 | |
| dc.identifier.issn | 0022-2488 | |
| dc.identifier.officialurl | http://dx.doi.org/10.1063/1.532885 | |
| dc.identifier.relatedurl | http://scitation.aip.org | |
| dc.identifier.uri | https://hdl.handle.net/20.500.14352/59670 | |
| dc.issue.number | 7 | |
| dc.journal.title | Journal of mathematical physics | |
| dc.language.iso | eng | |
| dc.page.final | 3274 | |
| dc.page.initial | 3268 | |
| dc.publisher | American Institute of Physics | |
| dc.relation.projectID | PB95-0401 | |
| dc.relation.projectID | 0GP0105490 | |
| dc.rights.accessRights | open access | |
| dc.subject.cdu | 51-73 | |
| dc.subject.ucm | Física-Modelos matemáticos | |
| dc.subject.ucm | Física matemática | |
| dc.title | On form-preserving transformations for the time-dependent Schrodinger equation | |
| dc.type | journal article | |
| dc.volume.number | 40 | |
| dcterms.references | 1. G. Darboux, C. R. Acad. Sci. Paris 94, 1456 (1882). 2. F. Cooper, A. Khare, and U. Sukhatme, Phys. Rep. 251, 267 (1995). 3. V. B. Matveev and M. A. Salle, Darboux Transformations and Solitons (Springer, Berlin, 1991). 4. G. Bluman and V. Shtelen, J. Phys. A 29, 4473 (1996). 5. V. G. Bagrov and B. F. Samsonov, Phys. Lett. A 210, 60 (1996). 6. P. G. L. Leach, J. Math. Phys. 18, 1902 (1977). 7. L. S. Brown, Phys. Rev. Lett. 66, 527 (1991). 8. G. Bluman, SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 39, 238 (1980). 9. G. Bluman, SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 43, 1259 (1983). 10. J. R. Ray, Phys. Rev. A 26, 729 (1982). 11. R. S. Kaushal and D. Parashar, Phys. Rev. A 55, 2610 (1997). 12. J. R. Burgan, M. R. Feix, E. Fijalkow, and A. Munier, Phys. Lett. A 74, 11 (1979). 13. M. Boiti, F. Pempinelli, A. K. Pogrebkov, and M. C. Polivanov, Inverse Probl. 7, 43 (1991). 14. V. G. Bagrov and B. F. Samsonov (preprint, quant-ph/9709040). 15. V. Singh, S. N. Biswas, and K. Datta, Phys. Rev. D 18, 1901 (1978). 16. A. V. Turbiner and A. G. Ushveridze, Phys. Lett. A 126, 181 (1987). 17. A. González-López, N. Kamran, and P. J. Olver, Commun. Math. Phys. 153, 117 (1993). 18. A. G. Ushveridze, Quasi-Exactly Solvable Models in Quantum Mechanics (IOP, Bristol, 1994). 19. Note that Eq. (139 cannot in general be solved in closed form if x1 is a polynomial in x of degree k.2. 20. P. Deift and E. Trubowitz, Commun. Pure Appl. Math. 32, 121 (1979). 21. J. M. Sparenberg and D. Baye, J. Phys. A 28, 5079 (1995). 22. Our choice of Q_(n11/2) differs from the one in Ref. 14 by the irrelevant constant factor i^n. 23. If n is even Q_(n11/2) is nonzero on the whole real axis. 24. A comprehensive review of these methods is beyond the scope of this paper; see Refs. 26– 30, and references therein for a detailed treatment. 25. Some families of time-independent complex potentials which are quasi-exactly solvable have been recently proposed in the literature (Ref. 31). 26. W. Magnus, Commun. Pure Appl. Math. 7, 649 (1954). 27. J. Wei and E. Norman, J. Math. Phys. 4, 575 (1963). 28. D. R. Truax, J. Math. Phys. 22, 1959 (1981). 29. C. M. Cheng and P. C. W. Fung, J. Phys. A 21, 4115 (1988). 30. S. Zhang and F. Li, J. Phys. A 24, 6143 (1996). 31. C. M. Bender and S. Boettcher (preprint, physics/9801007). | |
| dspace.entity.type | Publication | |
| relation.isAuthorOfPublication | 207092a4-0443-4336-a037-15936f8acc25 | |
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| relation.isAuthorOfPublication | d781a665-7ef6-44e0-a0da-81f722f1b8ad | |
| relation.isAuthorOfPublication.latestForDiscovery | 207092a4-0443-4336-a037-15936f8acc25 |
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