Density of states in deferent dimension

What is density of states in general case? It is the ratio of the number of states $Delta g$, which are lie in the energy interval $Delta E$, $$g(E)=frac{Delta g}{Delta E}.$$ Consider now the case of infinitesimal quantities and RHS of this expression becomes the derivative, $$g(E)=frac{d g}{d E}.$$ To be more concrete, let us consider 3D case with a quantum particle in infinite square well with width $L$. The particle wave-vector is the discrete quantity, $$k_i=frac{pi n}{L}, ninmathbb{N}.$$ Now consider case of large $L$, so $k$ seems continuous. What is the number of states in this case? All the states lie in the sphere with radius $k$. It is straightforward to understand that different states correspond to 1/8 of this sphere. In real space, for the single value of $k$ we have the volume $(pi/ L)^3$. So, $$g(k)=frac{1}{8}frac{4}{3}pi k^3frac{1}{(pi L)^3}$$ Taking into account that $E(k)=k^2/(2m)$ and using the relation $$g(E)=frac{partial g(k)}{partial k}frac{partial k}{partial E},$$ you can easily obtain that $g(E)sim E^{1/2}$. Then, in similar way you should consider case of 2D & 1D. There are two key poitns. First, in 2D instead of sphere you will consider a circle (in 1D you'll consider a segment). Second, the dispersion law is important: in case of free particles you have $E(k)=k^2/(2m)$ but the sitatuation can be more complicated (hopefully, enough weak interaction just renormalizes particle mass, roughly speaking).

Physically, you should understand how to calculate the volume in momentum space of a system with fixed energy $E$, know which volume in real space has one state and then just calculate the density.