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For some partial differential equations in physics, people may separate the variables and get some eigenfunctions. And then for any solutions for that equation, people often suppose them to be a Fourier series of those eigenfunctions. But why is it right? I think it is a little too wayward ... I know that there is some relations between this procedure and the theory of compact self-adjoint operators, but don't know the details ...

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closed as not a real question by David Roberts, Igor Rivin, Bill Johnson, Mark Sapir, Ryan Budney Dec 27 '11 at 20:21

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

What is your background, and what do you already know about the problem? You will be more likely to attract answers if you read and implement some of the strategies listed there. As the question stands, it is likely to be closed (but don't panic, if the question is improved, it can be reopened). – David Roberts Dec 26 '11 at 16:57
I believe that separation of variables is usually completely rigorous if done right. On the other hand, I am voting to close this question. – Igor Rivin Dec 26 '11 at 17:05
Thank you for your comments. I will improve it to meet the stands. – braill Dec 26 '11 at 17:26
That's much better... – Igor Rivin Dec 26 '11 at 17:35
The rigorous analysis of the "eigenfunction expansion" for self-adjoint operators on a Hilbert space is provided by the Spectral Theorem. You might look at Akhiezer and Glazman, Theory of Linear Operators in Hilbert Space, Appendix II: Differential Operators, for a thorough treatment of these questions. – Robert Israel Dec 26 '11 at 19:00