Efficiently Solving the Eigenvalue Problem: Sturm-Liouville Equation [URGENT]"

In summary, the conversation discusses the problem of finding solutions to the given eigenvalue equation and using the Laplacian in spherical coordinates to expand the left-hand side. The concept of g(x) as a solution is also mentioned, with the suggestion of applying the same approach. The question of where the term with k comes from is also addressed.
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[URGENT] Eigenvalue problem

Homework Statement


[PLAIN]http://img228.imageshack.us/img228/4990/111em.png


Homework Equations


Sturm-Liouville equation?


The Attempt at a Solution


I guess I'm just totally lost here. I've no idea how to start. It seems to me that maybe solving for solutions directly is ok, but that's near impossible in this case. I think there's some clever way around.
 
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Plug f(x) into the eigenvalue equation and use the Laplacian in spherical coordinates in n dimensions to expand the lefthand side.
 
  • #3


Thanks!

How about g(x)? How do I prove g(x) is also a solution?
 
  • #4


Same way, I'd imagine.
 
  • #5


Well, I tried to do the same thing, but I just can't reach the same conclusion. Where does k come from? How do I get that term with k?

Thanks,
 
  • #6


It should come from the Laplacian acting on the spherical harmonic.
 

Related to Efficiently Solving the Eigenvalue Problem: Sturm-Liouville Equation [URGENT]"

1. What is the Sturm-Liouville equation?

The Sturm-Liouville equation is a second-order linear differential equation that is commonly used in mathematics and physics. It is used to solve problems involving eigenvalues and eigenfunctions, which are important concepts in areas such as quantum mechanics and vibration analysis.

2. What is the eigenvalue problem?

The eigenvalue problem is a mathematical problem that involves finding the values (known as eigenvalues) for which a given linear operation has a non-zero solution. In the context of the Sturm-Liouville equation, the eigenvalue problem involves finding the values for which the equation has non-trivial solutions.

3. Why is efficiently solving the eigenvalue problem important?

Efficiently solving the eigenvalue problem is important because it allows us to find the eigenvalues and corresponding eigenfunctions for a given linear operation in a timely manner. This is crucial in many areas of mathematics and science, as it allows us to make accurate predictions and solve complex problems.

4. What are some methods for efficiently solving the eigenvalue problem?

Some common methods for efficiently solving the eigenvalue problem include the power method, the inverse power method, and the Rayleigh quotient iteration method. These methods involve iteratively computing approximations of the eigenvalues and eigenfunctions until a desired level of accuracy is reached.

5. How does the Sturm-Liouville equation relate to other mathematical concepts?

The Sturm-Liouville equation is closely related to other mathematical concepts such as orthogonal polynomials, spectral theory, and differential operators. It is also used in various applications such as quantum mechanics, signal processing, and fluid mechanics. Understanding the Sturm-Liouville equation can help in developing a deeper understanding of these related concepts and their applications.

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