(adsbygoogle = window.adsbygoogle || []).push({}); Completeness of the eigenfunctions (Which vectorspace??)

Once again in need of brain power from the interwebz :)

So I get that the eigenfunctions to the hamiltonoperator forms a complete set, but I'm unsure now as to which vectorspace it is?

And we're talking the one-dimensional case!

My first guess would be that it is all the functions that are summable? But that doesn't completely satisfy me, since I know from Fourier Series and Fourier Transform know that any summable function can be written as aninfinitesum of complex exponentials, and the eigenfunctions to the hamiltonian are a finite set?

Maybe someone can give me the hints of what I need to look at, to get a full understanding?

Right now I'm working with perturbation theory, and I'm trying to reconcile myself with the fact that the the first order correction to the wavefunction can be written as a linear combination of the solutions (eigenfunctions) to the unperturbed hamiltonoperator.

Thank you.

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# Completeness of the eigenfunctions (Which vectorspace?)

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