Multivalued classical mechanics arising from singularity loops in complex time

The Journal of Chemical Physics
Werner Koch, David J Tannor

Abstract

Complex-valued classical trajectories in complex time encounter singular times at which the momentum diverges. A closed time contour around such a singular time may result in final values for q and p that differ from their initial values. In this work, we develop a calculus for determining the exponent and prefactor of the asymptotic time dependence of p from the singularities of the potential as the singularity time is approached. We identify this exponent with the number of singularity loops giving distinct solutions to Hamilton's equations of motion. The theory is illustrated for the Eckart, Coulomb, Morse, and quartic potentials. Collectively, these potentials illustrate a wide variety of situations: poles and essential singularities at finite and infinite coordinate values. We demonstrate quantitative agreement between analytical and numerical exponents and prefactors, as well as the connection between the exponent and the time circuit count. This work provides the theoretical underpinnings for the choice of time contours described in the studies of Doll et al. [J. Chem. Phys. 58(4), 1343-1351 (1973)] and Petersen and Kay [J. Chem. Phys. 141(5), 054114 (2014)]. It also has implications for wavepacket reconstruction from co...Continue Reading

References

Jan 30, 1995·Physical Review Letters·A Shudo, K S Ikeda
May 23, 2006·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Tamar Shnerb, K G Kay
Dec 28, 2006·The Journal of Chemical Physics·Yair GoldfarbDavid J Tannor
May 2, 2008·The Journal of Chemical Physics·Yair GoldfarbDavid J Tannor
Aug 12, 2014·The Journal of Chemical Physics·Jakob Petersen, Kenneth G Kay
Feb 12, 2015·The Journal of Chemical Physics·Noa Zamstein, David J Tannor
Jul 15, 2015·The Journal of Chemical Physics·Jakob Petersen, Kenneth G Kay
Feb 13, 2016·Physical Review. E·Harinder PalSteven Tomsovic

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Citations

Nov 30, 2019·Reports on Progress in Physics·C Figueira de Morisson Faria, A S Maxwell
Mar 17, 2018·The Journal of Chemical Physics·Werner Koch, David J Tannor

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