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I'm trying to extract a complete set of states, by applying the spectral theorem to the following differential operator:

[itex]L = -\frac{d^2}{dx^2} + \mathrm{rect}(x)[/itex]

where rect(x) is the (discontinuous) rectangular function:

http://en.wikipedia.org/wiki/Rectangular_function

I have a feeling that this may not be possible because of the discontinuity in rect(x).

On the one hand, tt should be possible to approximate rect(x) by a sequence of functions for which the spectral theorem applies. But on the other hand, I don't think eigenspectra of this sequence is guaranteed to converge to that of L.

Can anyone more familiar with functional analysis confirm my suspicion?

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# Spectral theorem for discontinuous operators

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