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Complex classical trajectories in complex time can diverge at singular times. This study develops a calculus to predict trajectory behavior near these singularities, crucial for wavepacket reconstruction.

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Area of Science:

  • Quantum mechanics
  • Theoretical chemistry
  • Mathematical physics

Background:

  • Complex-valued classical trajectories in complex time exhibit singularities where momentum diverges.
  • Closed time contours around these singularities can lead to differing initial and final values of position (q) and momentum (p).

Purpose of the Study:

  • To develop a calculus for determining the asymptotic time dependence of momentum (p) near potential singularities.
  • To connect the exponent of this dependence to the number of singularity loops and distinct solutions of Hamilton's equations.

Main Methods:

  • Developing a calculus to analyze asymptotic time dependence of momentum (p) near potential singularities.
  • Investigating various potentials including Eckart, Coulomb, Morse, and quartic potentials.
  • Comparing analytical and numerical calculations of exponents and prefactors.

Main Results:

  • The exponent of asymptotic time dependence is identified with the number of singularity loops yielding distinct solutions.
  • Quantitative agreement was demonstrated between analytical and numerical results for exponents and prefactors.
  • A connection was established between the exponent and the time circuit count.

Conclusions:

  • The developed theory provides theoretical underpinnings for time contour choices in studies of complex classical trajectories.
  • The findings have implications for wavepacket reconstruction involving multiple trajectory branches.
  • This work clarifies the behavior of complex classical trajectories near singularities.