Asymptotic perturbation theory

In summary, after watching Prof Carl Bender's lecture course on mathematical physics, a question was raised about the differences in mathematical approach between one and three spatial dimensions. While techniques such as pade sequence, dominant balance, and wkb may still be applicable, they may require modification to solve PDEs in three dimensions. However, this topic remains at a research level and there is no introductory volume available. Papers in the Journal of Approximation Theory may provide further insights.
  • #1
qtm912
38
1
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 schrodinger 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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  • #2
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.
 

Related to Asymptotic perturbation theory

1. What is asymptotic perturbation theory?

Asymptotic perturbation theory is a mathematical tool used to approximate the behavior of a system as a parameter approaches a limiting value. It is commonly used in physics and engineering to study the behavior of complex systems.

2. How does asymptotic perturbation theory differ from other perturbation methods?

Unlike other perturbation methods, asymptotic perturbation theory focuses on the behavior of a system as the perturbation parameter approaches a limiting value, rather than just small perturbations. It also takes into account the overall behavior of the system instead of just the immediate effects of the perturbation.

3. What are some applications of asymptotic perturbation theory?

Asymptotic perturbation theory has many applications in physics, engineering, and other fields. It is commonly used to study the behavior of chaotic systems, as well as in the analysis of complex differential equations and in statistical mechanics.

4. How is asymptotic perturbation theory used in practical applications?

In practical applications, asymptotic perturbation theory is used to gain insight into the behavior of complex systems and to make predictions about their behavior. It can also be used to simplify complicated equations and to identify key parameters that influence the system's behavior.

5. What are the limitations of asymptotic perturbation theory?

One limitation of asymptotic perturbation theory is that it may not accurately predict the behavior of a system when the perturbation parameter is close to the limiting value. It also assumes that the system is well-behaved and does not take into account non-linear effects. Additionally, it may be difficult to apply to highly complex systems with many variables.

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