Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Sum over Eigenvalues

  1. Aug 28, 2006 #1
    "sum" over Eigenvalues....

    Is there any mathematical meaning or it's used in Calculus or other 2branch" of mathematics de expression:

    [tex] \sum_{n} e^{-u\lambda (n) } [/tex]

    where every "lambda" is just an Eigenvalue of a linear operator:

    [tex] L[y]=-\lambda _{n} y [/tex]

    We Physicist know it as the "partition function" and is used in Statistical Mechanics and Quantum physics...in first approximmation:

    [tex] \sum_{n} e^{-u\lambda (n) } = \iint dxdpe^{-uH} [/tex]

    Where all the "eigenvalues" are positive....here i would like to hear if this "approach" using integrals would be rigorous from a math point of view...:tongue2:
  2. jcsd
  3. Aug 29, 2006 #2


    User Avatar
    Science Advisor
    Gold Member

    I think it is rigorous as far as [tex]u[/tex] is small enough. That's what happens with the constant that multiplies the square of the principal quantum number when summing for obtaining the partition function for the translational motion, isn't it?.

    We Engineers call it the "partition function" too. :)
  4. Aug 29, 2006 #3
    Uh..sorry then "Clausius"... perhaps you as an engineer have heard about "Semiclassical approach" in Physics so you approximate the series by means of an integral, to calculate "Thermodinamic" entities...:rolleyes:
  5. Aug 29, 2006 #4

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    It's spectral theory as done by analysts, say, or measure theorists. (Lots of things are called spectral and are unrelated, so be careful).
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Sum over Eigenvalues
  1. Lost on eigenvalues (Replies: 3)

  2. Eigenvalue problem (Replies: 14)

  3. Eigenvalue pde (Replies: 2)

  4. Eigenvalues of D.E (Replies: 1)