Quantisation is a important step to properly define a quantum system from a classical one. In a nutshell : On a symplectic manifold $(M,\omega)$ and an algebra of function $f$ on $M$, one defines an algebra of self adjoint operator $\hat{f}$ on a Hilbert space with the main rule (defined by Dirac in 1925) that the Poisson brackets corresponds to the commutator $$\widehat{\{f,g\}}=[\hat{f},\hat{g}].$$
The evolution given by the Hamilton equation in the classical system is then given by the Schrodinger equation in the quantum system. The usual way of quantisation is to set $x\rightarrow X$ (multiplicative operator), and $p\rightarrow i\nabla_x$ (derivative operator). and formally extend this to any analytic function $f(x,p)$
However this raises questions:
1- the quantisation is not unique. Example $px^2$ can be quantised as $X^2\nabla_x$ or $\nabla_x X^2$.
2- the quantisation strongly depend on the choice of coordinate.
These are very important issues both for physics and mathematics. Ideally one should be able to construct a quantisation only from a geometrical properties of $(M,\omega)$. This is the aim of geometric quantisation https://en.wikipedia.org/wiki/Geometric_quantization. Unfortunattely I am not yet very familiar with this field. So here are my questions.
Let $(M,\omega,H)$ with $H$ an hamiltonian on $M$,
1-Can we define conditions such that the quantisation of $H$ is unique up to an unitary transform (ie whatever the choice of coordinate on $M$)?
2-If the quantisation is not unique, do we still have properties that can be deduced only from the classical system?
3-Can we get some estimate of the spectrum of $\hat{H}$ whatever the quantisation is ?
A good practical example I am interesting in for these questions is the simple harmonic oscilator.
Any partial answer would be great. I would be happy as well with a good reference to geometric quantisation. Thanks a lot
