Know a simple, linear, complex, eigenvalue BVP?

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Discussion Overview

The discussion revolves around finding a one-dimensional, linear, complex, second-order eigenvalue boundary value problem (BVP) that admits analytic solutions. Participants explore potential examples and clarify the nature of the functions involved.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant requests examples of a 1D, linear, complex, second-order eigenvalue BVP with analytic solutions.
  • Another participant suggests the square well as a potential example but questions whether it qualifies as complex.
  • A participant interprets the request as involving complex-valued functions and notes that a linear differential equation would typically reduce to solving two real eigenvalue problems for the real and imaginary parts of the function.
  • Further clarification is provided regarding the nature of the eigenvalues and eigenvectors, indicating a desire for them to have both real and complex components.
  • One participant expresses uncertainty about the square well being complex and shares their own example of a differential equation, ##x''+ix'+x=0##, while admitting they are unsure of its suitability.
  • Participants clarify that a real-world example is not necessary, emphasizing the search for a "toy problem" to work with instead.

Areas of Agreement / Disagreement

Participants do not reach a consensus on a specific example of a suitable BVP. Multiple competing views and interpretations of what constitutes a complex eigenvalue problem remain present.

Contextual Notes

There are unresolved assumptions regarding the definitions of "complex" in the context of the problem, as well as the nature of the eigenvalues and eigenvectors being sought.

member 428835
Hi PF!

I'm trying to find a 1D, linear, complex, 2nd order, eigenvalue BVP: know any that admit analytic solutions? Can't think of any off the top of my head.

Thanks!
 
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Would the square well be such a problem?
 
I take you to mean the function is complex valued over the real line, so it could be written ##f=g(x)+ih(x)##. If the differential equation is linear, then wouldn't the problem always reduce to the solving a pair of eigenvalue problems for real functions, ##g(x)## and ##h(x)##?

Edit: I also meant to ask if you are looking for a real-world example.
 
jedishrfu said:
Would the square well be such a problem?
Is this complex? Everything I've checked (here and here) appears to be real.

Haborix said:
I take you to mean the function is complex valued over the real line, so it could be written ##f=g(x)+ih(x)##.
Sorry, perhaps I did not specify: I mean't so that the eigenvalues/vectors have a real and complex part. Perhaps something like ##x''+ix'+x=0##, but I'm at a loss.

Haborix said:
Edit: I also meant to ask if you are looking for a real-world example.
Does not need to be a real-world example. I am looking for a good toy problem to work with.
 

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