Padmanabhan, T. Demystifying the constancy of the Ermakov-Lewis invariant for a time-dependent oscillator. (English) Zbl 1383.34055 Mod. Phys. Lett. A 33, No. 7-8, Article ID 1830005, 5 p. (2018). Summary: It is well known that the time-dependent harmonic oscillator (TDHO) possesses a conserved quantity, usually called Ermakov-Lewis invariant. I provide a simple physical interpretation of this invariant as well as a whole family of related invariants. This interpretation does not seem to have been noticed in the literature before. The procedure also allows one to tackle some key conceptual issues which arise in the study of quantum fields in the external, time-dependent backgrounds like in the case of particle production in an expanding universe and Schwinger effect. Cited in 2 ReviewsCited in 8 Documents MSC: 34C14 Symmetries, invariants of ordinary differential equations 34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations 81T99 Quantum field theory; related classical field theories Keywords:Ermakov-Lewis invariant; time-dependent harmonic oscillator; Schwinger effect; Bogoliubov coefficients PDFBibTeX XMLCite \textit{T. Padmanabhan}, Mod. Phys. Lett. A 33, No. 7--8, Article ID 1830005, 5 p. (2018; Zbl 1383.34055) Full Text: DOI arXiv References: [1] V. Ermakov, Univ. Izv. Kiev, Series III9, 1 (1880); W. E. Milne, Phys. Rev.35, 863 (1930); E. Pinney, Proc. Am. Math. Soc.1, 681 (1950); M. Kruskal, J. Math. Phys.3, 806 (1962); H. R. Lewis, Jr., Phys. Rev. Lett.18, 510 (1967) [Erratum-ibid.636 (1967)]; H. R. Lewis, Jr., J. Math. Phys.9, 1976 (1968); H. R. Lewis, Jr. and W. B. Riesenfeld, J. Math. Phys.10, 1458 (1969); D. C. Khandekar and S. V. Lawande, J. Math. Phys. (N. Y.)16, 384 (1975); J. G. Hartley and J. R. Ray, Phys. Rev. A24, 2873 (1981); I. A. Pedrosa, J. Math. Phys. (N. Y.)28, 2662 (1987); C. M. A. Dantas, I. A. Pedrosa and B. Baseia, Phys. Rev. A45, 1320 (1992); Braz. J. Phys.22, 33 (1992); K. H. Yeon, J. H. Kim, C. J. Um, T. F. George and L. N. Pandey, Phys. Rev. A50, 1035 (1994); P. G. L. Leach, Austral. Math. Soc.20, 97 (1977); C. Athorne et al., Phys. Lett. A143, 207 (1990); J. Y. Ji, J. H. Kim and S. P. Kim, Phys. Rev. A51, 4268 (1995); J. Y. Ji, J. H. Kim, S. P. Kim and K. S. Son, Phys. Rev. A52, 3352 (1995); S. P. Kim, J. Phys. A27, 3927 (1994). [2] P. G. L. Leach and K. Andriopoulos, Appl. Anal. Discr. Math.2, 146 (2008); P. B. E. Padilla, arXiv:math-ph/0002005v3. [3] Ray, J. R. and Reid, J. L., Phys. Lett. A71, 317 (1979). [4] V. Aldaya et al., J. Phys. A: Math. Theor.44, 065302 (2011); J. Guerrero, F. F. López-Ruiz, Phys. Scripta90, 074046 (2015). This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.