Representation of Hamiltonian in terms of "creation" and "destruction" operators Let's have Schrodinger equation or Dirac equation in Schrodinger form:
$$
i \partial_{0}\Psi = \hat {H}\Psi .
$$
Sometimes we can introduce some operators $\hat {A}, \hat {B}$ (the second is not always Hermitian conjugate to the first) for which
$$
\hat {H} = \hat {B}\hat {A} + \varepsilon ,
$$
which is look like rewriting $\hat {H}$ through destruction and creation operators.
Right commutation or anticommutation laws $[\hat {B}, \hat {A}] = 1, \quad [\hat {B}, \hat {A}]_{+} = 1$ are not necessarily too.
I heard that it helps to simplify the way of solution of equation. But I don't know what physical sense of $\hat {A}, \hat {B}$, so I don't understand how to use these operators fo finding energy spectra and functions for corresponding state. 
What do you know about this?
An example.
For electron in Hydrogen atom Schrodinger equation can be rewritten in form
$$
\partial^{2}_{\rho}\kappa + \frac{2m\alpha}{r}\kappa - \frac{l(l + 1)}{\rho^{2}}\kappa = -E\kappa ,
$$
where $\kappa$ is $R(\rho)\rho$ and $R(\rho )$ is the radial part of Schrodinger equation.
So $\hat {H}\kappa = E \kappa$, and we may construct some $\hat {A}, \hat {B}$ operators, for which
$$
\hat {A} = \partial_{\rho} + \omega , \quad \hat {B} = -\partial_{\rho} + \omega, \quad \omega = \gamma + \frac{\beta}{\rho },
$$
and 
$$
\hat {H} = \hat {B}\hat {A} + \epsilon = -\partial^{2}_{\rho} - \frac{2m\alpha}{r} + \frac{l(l + 1)}{\rho^{2}},
$$
from which it is possible to find $\gamma , \beta$.
I heard that this representation also can help to find the spectra and state functions, but I don't know why, because physical sense of introduced operators is incomprehensible for me.
 A: I will recommand you a marvellous little book : 
Supersymmetric Quantum Mechanics, (Asim Gangopadhyaya,Jeffry V Mallow, Constantin Raisnariu ), World Scientific 
The first idea is to factorize an hamiltonian, using generalized creation/anihilation operators $A^-$, $A^+$, coming from a superpotential $W(x)$, $A^\pm =   W(x) \mp  \frac{d}{dx}$
This could be done in two ways : $H^+ =A^- A^+$ and $H^-= A^+ A^-$, and this leads to  two potentials $V_\pm(x) = W^2(x) \pm W'(x)$. The fundamental interest of the above factorization of the hamiltonians is that they are positive definite ($A^-$ and $A^+$ are hermitian conjugate each other) 
These relations establish a link between the 2 spectra of the 2 potentials : except the ground state of $H^-$, the spectra are the same, and the eigenfunctions are related each other by the operators $A^-$, $A^+$. So, surprinsingly, two very different potentials could have the same spectra (for instance Posch-Teller potential and infinite one-dimensional square well potential)
Moreover, with a supplementary property called translational shape invariance, one may calculate the all spectra of each of the 2 potential very easily, for instance, in the case of the radial part of Coulomb potential, you may obtain, with only very basic and simple calculus (no differential equations !!), the eigenvalues and the eigenfunctions, which is quite amazing. 
The relation with the supersymmetry is writing supersymmetric charges $Q^-= \begin{pmatrix} 0&0\\ A^-&0\end{pmatrix}$, $Q^+= \begin{pmatrix} 0&A^+\\ 0&0\end{pmatrix}$
An hamiltonian is $H= \{Q^-, Q^+\} = \begin{pmatrix} H^+&0\\ 0&H^-\end{pmatrix}$. We have $(Q^-)^2 = (Q^+)^2=0$, and $[Q^\pm, H]=0$. The 2-dimensional space of the eigenfunctions may be understood as a boson+ fermion space.
A: As it stands, the question is too general to be answered. Having a general hamiltonian $H$ written in the form $$H=AB+\epsilon \tag 1$$ without imposing additional conditions on $A$ and $B$ is too general to necessarily impose useful structure. For example, given any hamiltonian $H$, you can take $A=H$, $B=1$ and $\epsilon=0$ and it will satisfy the condition (1) without really offering anything new to the solution of the problem.
This is why it is an achievement to find a useful factorization of this form: one needs to find the right pair of operators, $A$ and $B$, which will give back $H$ (of which there are very many), as well as obeying a useful commutation relation. It is precisely what commutation relations you can impose on $A$ and $B$ that will determine the spectrum of $H$, and since there are very many possible spectra there will be correspondingly many possible commutation relations.
Thus, for example,


*

*if you impose $[A,B]=1$, then $HB=B(H+1)$, and you must have equally spaced energies as in a harmonic oscillator, whereas

*if you impose $[AB,A]=0$ then you will have as much degeneracy as you can get away with by acting with powers of $A$ on a given eigenstate without getting zero, as in a rigid rotor.


Thus, the role of $A$ and $B$ is mathematically ill defined as you have stated your general question. What hope is there, then of finding physical interpretations for them?
