Schrödinger equation for a central potential I know that, for a central potential $U(r,\theta, \varphi)=U(r)$, the Schrödinger equation takes the well-known form in spherical coordinates.
I'm searching for separable solutions in the form:
\begin{equation}
\phi(r, \theta,\varphi)=a(r)b(\theta)c(\varphi)
\end{equation}
So, if I understood correctly the theory (I'm a beginner in theoretical physics so maybe I did some conceptual mistakes) my $\phi$ is the eigenfunction corresponding to some eigenvalue E.
Solving the various steps this equation appears for $c(\varphi)$:
\begin{gather}
\frac{d^2c}{d\varphi^2}+m_l^2=0
\end{gather}
And the solution should be:
\begin{equation}
c(\varphi)_{m_l}=Ae^{im_l\varphi}+Be^{-im_l\varphi}
\end{equation}
and after some considerations on the periodicity of $\varphi$, I got that $m_l$ is integer. My problem is the following:
My teacher, and also the majority of the documents I found online, states that the solution is
\begin{equation}
c(\varphi)_{m_l}=Ae^{im_l\varphi}
\end{equation}
saying "noting that $m_l$ and $-m_l$ yield the same general solution $c(\varphi)$". What does it mean? Every solution to the harmonic oscillator differential equation yield a solution of the same form, so what is the point? It seems to me that we are "throwing away" something (even if I'm sure it's not like this, but I don't understand why).
This is because I think that the equation 
$$\phi(r, \theta,\varphi)_{m_l}=a(r)b(\theta)c(\varphi)_{m_l}$$
would be different if I use $c(\varphi)_{m_l}=Ae^{im_l\varphi}+Be^{-im_l\varphi}$ from the one obtained by using $c(\varphi)_{m_l}=Ae^{im_l\varphi}$.
I feel like I'm missing something really stupid here but I'm in a loop now and I can't see a solution.
 A: The overall theme is that you're not really looking for an arbitrary solution of the time-independent Schrödinger equation: instead, you're only looking for a basis of such solutions, which you can then use to build, via linear combinations of them, any arbitrary solution. This allows you a lot of freedom, because as long as you arrive at a complete set of solutions then you're free to restrict your search area as much as you need to.
In fact, you've already done precisely this sort of narrowing-down of the search area simply by the form of wavefunctions,
$$\psi(r, \theta,\varphi)=R(r)\Theta(\theta)\Phi(\varphi), \tag1$$
that you're looking for. It's important to realize that these wavefunctions are intrinsically tied to a choice of coordinate system (and, particularly, to the choice of the direction for the $z$ axis), and that if you transform these wavefunctions to an arbitrary coordinate system then they will no longer be separable. Again, the reason you're allowed to make the Ansatz $(1)$ is that it will eventually lead you to a complete set of solutions that you can use to reconstruct any arbitrary solution.
Thus, when you're staring down at
$$
\Phi_{m_l}(\varphi)=Ae^{im_l\varphi}+Be^{-im_l\varphi}
$$
and you're trying to decide what to put in for $A$ and $B$, what you do is you realize that what you really have is simply a solution space of dimension two, and since we're looking for a basis anyways then it's perfectly OK to separate this into 
$$
\Phi_{m_l}(\varphi)=Ae^{im_l\varphi}
\quad \text{and}\quad
\Phi_{-m_l}(\varphi)=Be^{-im_l\varphi},
$$
since when taken together the two constitute a basis for the solution space. After that, it's just a bit of admin: you realize that the second solution is actually just the first solution with a different index, so you can simply keep $\Phi_{m_l}(\varphi)=Ae^{im_l\varphi}$ as your standard solution (while keeping in mind that $m_l$ might well be negative), and you will still be describing a basis for the entire space.
A: A general solution is a linear combination of such separable solutions. Since Schrödinger equation is a linear equation, superposition of solutions is also a solution. You can take $A$ and $B$ as coefficients of two separable solutions one with $e^{im\phi}$ and another with $e^{-im\phi}$ as their $\phi$ dependent part.
