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Spectral theorem for discontinuous operators

  1. Oct 14, 2008 #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?
     
  2. jcsd
  3. Oct 15, 2008 #2
    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
  4. Oct 15, 2008 #3

    HallsofIvy

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    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.
     
  5. Oct 15, 2008 #4
    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.
     
  6. Oct 16, 2008 #5

    HallsofIvy

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    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.
     
  7. Oct 16, 2008 #6
    The eigenfunctions are defined piecewise consisting of [itex]\exp(\pm ikx)[/itex] and [itex]\exp(\pm \kappa x)[/itex], with the functions and their first derivatives matched at [itex]\pm 1/2[/itex].

    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)

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

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