- #1
jdstokes
- 523
- 1
Hi all,
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?
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?