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Number of bound states and index theorems in quantum mechanics?

  1. Oct 9, 2012 #1

    tom.stoer

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    Just an idea: is there an index theorem for an n-dimensional Hamiltonian

    [tex] H = -\triangle^{(n)} + V(x)[/tex]

    which "counts" the bound states

    [tex] (H - E) \,u_E(x) = 0[/tex]

    i.e. eigenfunctions and eigenvalues in the discrete spectrum of H?
     
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  3. Oct 9, 2012 #2

    dextercioby

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    Due to the huge dosis of arbitrary in your setting (the potential in n variables needn't be seprarable), I'm quite sure the answer is <no>.
     
  4. Oct 9, 2012 #3

    tom.stoer

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    and what would be the conditions for an index theorem?
     
  5. Oct 9, 2012 #4

    dextercioby

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  6. Oct 9, 2012 #5

    tom.stoer

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    yes, something like that
     
  7. Oct 9, 2012 #6

    tom.stoer

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    I probably should not ask for an index theorem but simply for an analytical index; the idea is to have a formula to calculate the number of bound states instead of solving the SE and counting them
     
    Last edited: Oct 9, 2012
  8. Oct 10, 2012 #7

    A. Neumaier

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    This was studies a lot about 40 years ago. Volume 3 of Thirring's ''Course in mathematical physics'' has some results. scholar.google for >>bounds on the number of "bound states"<<
    or variations turns up useful references. For related recent results see, e.g.,
    http://arxiv.org/pdf/1108.1002 or
    http://www2.imperial.ac.uk/~alaptev/Papers/Lapt_Beijing.pdf
     
  9. Oct 10, 2012 #8

    tom.stoer

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