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Feb26-12, 05:26 PM
P: 29
1. The problem statement, all variables and given/known data
Let A be a linear transformation on the space of square summable sequences [itex]\ell[/itex]2 such that (A[itex]\ell[/itex])n = [itex]\ell[/itex]n+1 + [itex]\ell[/itex]n-1 - 2[itex]\ell[/itex]n. Find the spectrum of A.

2. The attempt at a solution
I see that A is self-adjoint, so its spectrum must be a subset of the real line. We also showed in class that it is bounded, and it's clearly the discrete analog of the second derivative, but I'm not sure how to use these facts. It seems to me that for any real λ one can construct a sequence [itex]\ell[/itex] such that (A - λ)[itex]\ell[/itex] = 0, since we then have the condition [itex]\ell[/itex]n+1 = -[itex]\ell[/itex]n-1 + (2 + λ)[itex]\ell[/itex]n, which can be used to recursively construct a sequence setting [itex]\ell[/itex]0 = a and [itex]\ell[/itex]1 = b for arbitrary a and b. I'm guessing that the problem with this is that such a sequence would only converge for certain values of λ, but I don't really know how I could show this. Any help would be greatly appreciated!
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