Asymptotic perturbation theory

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SUMMARY

The discussion centers on the application of asymptotic perturbation theory in solving the Schrödinger equation for various potentials beyond the harmonic oscillator. It highlights the relevance of techniques such as Padé sequence, dominant balance, and WKB (Wentzel-Kramers-Brillouin) approximation when transitioning from one-dimensional to three-dimensional spatial problems. The conversation emphasizes that while these methods remain applicable, they require modifications for partial differential equations (PDEs) and are still considered advanced research topics. For further exploration, the Journal of Approximation Theory is recommended as a resource for relevant papers.

PREREQUISITES
  • Understanding of Schrödinger equation and quantum mechanics
  • Familiarity with asymptotic analysis techniques
  • Knowledge of partial differential equations (PDEs)
  • Basic concepts of mathematical physics
NEXT STEPS
  • Research the application of Padé approximants in quantum mechanics
  • Study the WKB approximation in the context of PDEs
  • Explore dominant balance methods for multi-dimensional problems
  • Review recent articles in the Journal of Approximation Theory
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Mathematicians, physicists, and researchers interested in advanced quantum mechanics and asymptotic analysis techniques, particularly those working with multi-dimensional Schrödinger equations.

qtm912
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Having just watched Prof Carl Bender's excellent 15 lecture course in mathematical physics on YouTube, the following question arose:
The approach was to work in one space dimension and to solve the Schrödinger equation for more general potentials than the harmonic oscillator using asymptotic theory. I was wondering how radically the maths changes when we work instead in three spatial dimensions.
Are the techniques developed, such as pade sequence, dominant balance and wkb somehow still applicable, but perhaps modified to solve PDEs,
Thanks
 
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Yes, but it's not a light topic and is still at research level without a nice introductory volume. Look at papers in Journal of Approximation Theory for example.
 

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