It is pretty cool (in my opinion) that one can solve Schrödinger's equation for the harmonic oscillator by using ladder operators, rather than just integrating it.
In particular, it is possible to extract the spectrum of the hamiltonian without almost any calculus at all ( => this method is called: algebraic method).
My question is now mathematical: When is it possible to find such ladder operators for a given Hamiltonian and how do I find them? Cause so far, the operators where always given in advance and just used for the harmonic oscillator(or very similar potentials). Probably one needs a positive and somehow well-behaving spectrum of the Hamiltonian, but what else?
Any hint is interesting!
This is an interesting question. Actually this algebraic method is known as "factorization", at least this is what Schrodinger called it. This is how he solved the Hydrogen atom initially, he did not use his wave equation (which is very strange), he used factorization methods. This algebraic/ factorization approach is now called "supersymmetric" quantum mechanics. Of course Erwin didn't call it SUSY than.
However if you want literature on it, here is one 1/2 great Schrodinger's papers on exactly what you are asking. http://www.jstor.org/stable/20490756
The only restriction on the operators is that they are hermitian since the time evolution operator is unitary. If you have bound states and have a negative energy spectrum, this is allowed. Thus positive definite matrices is not needed as a postulate for factorizing the Hamiltonian.
If you are looking for more after Erwin's paper, let me know. Cheers