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TheN-body problem with spin-orbit or Coulomb interactions

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Abstract

A study is made of the resolventR(λ) for a system ofn particles with spin-orbit coupling, an interaction which necessarily has a long range in momentum space. For short-range interactions, it has been known for several years thatR(λ) satisfies a Fredholm equation whose kernel is in the Schmidt-class. The corresponding spin-orbit kernel is not in the Schmidt-class, but it is shown that it does belong to a certain class of compact operators which is larger than the Schmidt-class. A modified Fredholm theory is presented which applies to all operators in this larger class. This enablesR(λ) to be found for all values of λ in the complex plane cut along the continuous spectrum of the Hamiltonian. It is shown that the modified Fredholm theory also holds for the Coulomb interaction.

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van Winter, C., Brascamp, H.J. TheN-body problem with spin-orbit or Coulomb interactions. Commun.Math. Phys. 11, 19–55 (1968). https://doi.org/10.1007/BF01654300

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Keywords

  • Neural Network
  • Statistical Physic
  • Complex System
  • Nonlinear Dynamics
  • Long Range