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Feb26-12, 10:06 PM
P: 29
Quote Quote by Dick View Post
I don't know, I haven't followed the whole argument out. I'm not really an expert, that was just supposed to be a hint. But I do think e^(kn) is an eigensequence if k<0 and it is square summable. What's the eigenvalue? You probably know more about this than I do.
Oh, I don't think I mentioned that these are two-sided sequences, so e^(kn) for k<0 would diverge at -∞. I guess the problem is that I don't yet have a clear understanding of what a spectrum contains beyond eigenvalues. I seem to remember from class an example where an operator "seemed" to have an eigenvector, but the vector wasn't an element of the Hilbert space, so the value it corresponded to wasn't an eigenvalue, but was still an element of the spectrum. I believe the example given was the hydrogen atom potential, where the bound states are eigenvectors and the scattering states are part of the continuous spectrum since they aren't normalizable. It seems to me that this operator is similar, with the e^(kn)-like sequences acting like scattering states for all k (including k=0, I don't know what I was saying before). Can anyone clarify this?