Spectral theorem for discontinuous operators

1. Oct 14, 2008

jdstokes

Hi all,

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

$L = -\frac{d^2}{dx^2} + \mathrm{rect}(x)$

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?

2. Oct 15, 2008

ice109

why are you using the "rectangular function" which isn't a real function anyway? just break up the domain of the operator in two 3 parts - where rect(x) is zero, where rect(x) = k, and finally where rect(x) is zero again.

outside the square the two eigenfunctions will be exp(ikx) and exp(-ikx) and inside they will be exp(mx) and exp(-mx). they're not orthogonal though, they're dirac orthogonal.

Last edited: Oct 15, 2008
3. Oct 15, 2008

HallsofIvy

In what sense is a "rectangular function" not a function? What definition of function are you using? By the most common definition of "function", that certainly is a function.

4. Oct 15, 2008

ice109

yea i guess you're right, i don't know what i was thinking saying that. something along the lines that the lines up sides made the function non-single valued.

5. Oct 16, 2008

HallsofIvy

I thought that for a moment but in the picture given the value at the "break point" is specifically the average of the two constant values.

6. Oct 16, 2008

jdstokes

The eigenfunctions are defined piecewise consisting of $\exp(\pm ikx)$ and $\exp(\pm \kappa x)$, with the functions and their first derivatives matched at $\pm 1/2$.

Are you absolutely sure that these piecewise-defined eigenfunctions form a complete set (ie Dirac orthonormal)? In order for this to be the case we must have (with appropriate normalisation)

$\left(\int_{-\infty}^{-1/2} + \int_{-1/2}^{1/2} +\int_{1/2}^{\infty}\right)\psi_k \psi_l dx= \delta (k - l)$.

It is not immediately obvious to me that this will work due to the way $\psi_k$ is defined and those funny integration limits. In fact I'm starting to think that the spectral theorem is not applicable here.