How far can you get (in quantum mechanics) with just commutation relations? Clearly it is possible to derive a set of commutation relations from some Hamiltonian, and certainly they give useful and interesting invariants when investigating the behavior of quantum systems.  But my question is can we go backwards?  
In other words, can one set up a collection of (anti)commutation relations which determine a particular C* algebra, all of whose representations are solutions to some Schrodinger equation?
And if not, how much extra data do we need to reconstruct the exact Schrodinger equation?
 A: I believe this question is suffering from a confusion regarding the C* algebra of operators in the quantum mechanics. To verify that I am thinking the same as the OP, I will quickly recapitulate the question in my own words:
When you have a linear differential equation, like 
$$ (\partial_y^2 - \partial_x^2 - x\partial_x)\psi = 0 $$
You can think of the operator algebra containing $p_x=-i\partial_x$,$p_y = -i\partial_y$ as well as functions of $x,y$ as having an algebraic relation imposed on it, namely;
$$ p_x^2 - p_y^2 + i x p_x = 0 $$
This algebraic relation implies other relations, together with $[x,p]=i$, and whatever else follows. The collection of all things in the old C* algebra that are made equal to zero by the relation of the equation is called the ideal corresponding to the differential equation. The question is what is this ideal for the Schrodinger equation, and can it be expressed using commutators?
The problem is that the Schrodinger equation is not naturally living in the space of functions on x,t and operators on them. The space of operators is for functions only of x. So the ideal corresponding to the Schrodinger equation is all due to the relation defining the Hamiltonian
$$ i\partial_t = {p^2\over 2m} + V(x)$$
And this relation doesn't alter the algebraic relations between p and x, it just serves to define H. So there is no alteration in the C* algebra of operators by defining the Hamiltonian, and the question doesn't make sense.
Perhaps you mean to define the space of all operators on functions of space and time together, and then quotient that by a nontrivial ideal. But I couldn't figure out how to to do this. A reference would be nice, because the question is very terse.
