... Try thinking about this in one dimension first. Consider a pure parabolic dispersion first E(k) = hbar^2 k^2/2m and calculate a density of states. Now imagine that the dispersion is a bit slower increasing than quadratic, maybe NewE(k) = hbar^2 k^2/2m - a k^4 with very small a such that NewE(k) < E(k). Does this increase or decrease the density of states? Ie. is Density of states for NewE greater or less than Density of States for E. Can you understand why you get this result? Remember that the density DOS(E0) of states is simply a counting of all the eigenstates with energy E0 (or more precisely DOS(E0) dE0 is the number of eigenstates with energy between E0 and E0+ dE0).
Then go back to 2d. If you change the dispersion E(kx,ky) in some way such that NewE(kx,ky) < E(kx,ky), what does this do to the density of states?
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