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Exponentially accurate semiclassical tunneling wave functions in one dimension

by V. Gradinaru and G. A. Hagedorn and A. Joye

(Report number 2010-10)

Abstract
We study the time behavior of wave functions involved in tunneling through a smooth potential barrier in one dimension in the semiclassical limit. We determine the leading order component of the wave function that tunnels. It is exponentially small in 1/h. For a wide variety of incoming wave packets, the leading order tunneling component is Gaussian for sufficiently small h. We prove this for both the large time asymptotics and for moderately large values of the time variable.

Keywords:

@Techreport{GHJ10_425,
  author = {V. Gradinaru and G. A. Hagedorn and A. Joye},
  title = {Exponentially accurate semiclassical tunneling wave functions in one dimension},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2010-10},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2010/2010-10.pdf },
  year = {2010}
}

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