mathematical physics - Quantum mechanics on a manifold

In quantum mechanics the state of a free particle in three dimensional space is $L^2(\mathbb R^3)$, more accurately the projective space of that Hilbert space. Here I am ignoring internal degrees of freedom otherwise it would be $L^2(\mathbb R^3)\otimes S$, but let's say it is not that time of the month. The observables are operators in that space and the dynamics is describe by the Schrodinger equation or in other equivalent ways. If we have more particles then it is $L^2(\mathbb R^{3N})$. My question is: are there any examples where one considers a system with configuration space a general manifold $M$, instead of $\mathbb R^3$, say a system of particles (a particle) with some restrictions, so that the state space is $L^2(M)$. There might be physical reasons why this is of no interest and I would be interested to here them. What I am interested in is seeing is specific (or general) examples worked out in detail. For example a system with a given Hamiltonian, where one can explicitly find the spectrum. Or if that is too much to ask for an example where the system has very different properties from the usual case. Say a particle living on the upper half plane with the Lobachevsky geometry, may be some connection to number theory! I am aware that there is quantum field theory on curved spacetime, I am interested in quantum mechanics.

Edit: Just a small clarification. The examples I would like to see do not have to come from real physics, they can be toy models or completely unrealistic mathematical models. Something along the lines: take your favorite manifold $M$ and pretend this is the space we live in, what can we say about QM in it. The choice of $M$ doesn't have to do anything with general relativity. As I said the upper half plane is interesting or quotients of it by interesting discrete groups or generalizations $\Gamma\backslash G(\mathbb R)/K$ or anything at all. The answers so far are interesting. Hoping to see more.

Answer

Here is an overview of quantization methods: http://arxiv.org/abs/math-ph/0405065

Most of this article deals with QM on manifolds.