Why is the $k$-space in multiples of $2\pi/L$? So when you find the solution to the Schrödinger equation you get that the wave function can have $k=n\pi/L$, $n=1, 2,3  \dots $
The problem I have is that when calculating the density of states of a fermi gas, you use the $k$-space.
In this space, you use $k = \frac{2\pi}{L}n$, $n=\dots; -2,-1, 0, +1, +2, \dots$.
Why don't these equations match as they are both Schrödinger solution and using boundary condition with sin functions?
 A: Speaking about Schrödinger's equation, without specifying the potential and which boundary conditions are present, is equivalent to say almost nothing.
The first case, $k=\pi n/L$, corresponds to the solutions of the Schrödinger's equation with zero potential, in  the interval $[0,L]$, provided hard wall boundary conditions are used, i.e. the wavefunction must vanish at the boundary points $0$ and $L$.
The second case, $k=2 \pi n/L$, corresponds to the solutions of the Schrödinger's equation with zero potential, in  the interval $[0,L]$, provided periodic boundary conditions are used, i.e. the wavefunction must satisfy the conditions $\psi(0)=\psi(L)$ and $\psi^{\prime}(0)=\psi^{\prime}(L) $.
When calculating the density of states, one exploits the fact that for large systems the density of states gets independent on the exact boundary conditions and on the precise shape of the boundary surface, so the density of states evaluated with periodic boundary conditions is used as a convenient way of doing calculations.

Added after a few comments.
First of all, I would notice that while eigenfunctions on the case of hard wall boundary conditions can be written in term of $sin$ functions, this is not the case with periodic boundary conditions where the eigenstates are of the form
$ e^{i k x}$.
The density of states in energy per unit of length is the same in the two cases as 
discussed in the first answer to a related question.
Notice however that such equality, i.e. the independence of the density of states on the boundaries for large systems is a much more general result.
