# Band structure and dispersion relations

-Let's suppose we have 2 gases ..one is a "Fermi" gas under an Harmonic potential and the other is a "Bose" gas under another Harmonic potential... in both cases (as an approximation) the particles (bosons and electrons are Non-interacting) then we could write the partition functions.

$$\prod _{k=1}^{\infty}(1+be^{\beta \omega (k)})^{-1}$$

Where "b" is a constant equal to 1 (electrons) or -1 (bosons)..my question is HOw could we calculate the "band structure" for the Fermion gas... ¿are the band structure and dispersion relations the "same" concept but one is valid for Bosons and other for Fermions?.. I've read "Ashcroft: Solid State..." where you can find lot's of method to calculate band structure..but what's the best?.., Is there a differential equation or other type of equation satisfied for the $$\omega (k)$$ in the "Fermionic" case?..thanks.

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