Let the density of states be given by

$$ g(\epsilon) = \int \frac{\mathop{d^3 q}}{4\pi^3} \delta(\epsilon - \epsilon(\vec{q})), $$

where $\epsilon(\vec{q}) = \frac{\hbar^2}{2m}q_\perp^2 + h_\pm(q_\parallel)$ and where $h_\pm(q_\parallel)$ is a function which depends on $q_\parallel$ alone. $q_\perp$ and $q_\parallel$ denote the components of the vector $\vec{q}$ which are perpendicular/parallel to a vector $\vec{K}$ of the reciprocal lattice. I want to show that $$ g(\epsilon) =\frac{1}{4\pi^2}\left(\frac{2m}{\hbar^2}\right) (q_\parallel^\text{max}-q_\parallel^\text{min})$$ with $q_\parallel^\text{max}$ and $q_\parallel^\text{min}$ the two solutions to the equation $\epsilon = h_\pm(q_\parallel)$.

Unfortunately, I have no idea how to derive the above equation. Hope someone will help me with this.


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