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Can a self adjoint operator have a continous spectrum ?

  1. Jul 21, 2007 #1
    Can a self adjoint operator have a continous spectrum ??

    If we have a self adjoint operator

    [tex] Ly_{n} = \lambda _{n} [/tex]

    can n take arbitrary real values (n >0 ) in the sense that the spectrum will be continous ?

    and in that case, what is the orthonormality condition for eigenfunctions

    [tex] <y_{n} |y_{m}>= \delta (n-m) [/tex]

    where 'd' is dirac delta, as a generalization of discrete case of Kronecker delta. could someone put an example ? (since all the cases from QM i know the spectrum is discrete) thankx
     
  2. jcsd
  3. Jul 21, 2007 #2

    matt grime

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    Just look up spectral analysis, please, Jose. C* algebras, stuff like that. (I seem to remember making this request a lot in the past - have you followed that advice?)

    The orthonormality condition is precisely what it ought to be - two things are orthogonal if their (inner) product is 0. I don't know what makes you think that Dirac deltas have anything to do with taking an inner product (which by definition is just a complex number, not a distribution).

    This is basic functional analysis, and not differential equations anyway - one of the first things you learn is that there is, I seem to think, a bijection between (arbitrary) compact subsets of C and commutative C* algebras. Perhaps I have missed some hypotheses, but that is a well known result.
     
    Last edited: Jul 21, 2007
  4. Jul 21, 2007 #3

    Gokul43201

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    If this is a physics question (rather than a math question) then yes, physicists indeed use the Dirac delta to represent the inner product between states in a space spanned by continuous eigenkets (but then, physicists do a lot of things that make mathematicians cringe). A reason for this choice is that it provides a working extension to the completeness condition. I've never seen a rigorous development for this though.

    If you want to use produce some new math, I strongly advise against starting from physics.
     
    Last edited: Jul 21, 2007
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