quantum mechanics - WHy is SUSY QM important?

why is SUSY QM important ? i mean for each one DImensional Hamiltonian , can we write 'H' as

$ H= A.A^{+}+C$ (or similar constant)

here $ A= \frac{d}{dt}+A(x)$ and $ A^{+}= -\frac{d}{dt}+A(x)$

1) can we always express 'x' and 'p' as a combination of A and its adjoint?

2) does SUSY QM makes easier to solve the Hamiltonian

3) can we apply second quantizatio formalism with operators $ A,A^{+}$ in the same way we did for the Harmonic oscillator

4) let be $ Ay(x)=\lambda _{n} y(x) $ an eigenfunction of the anhinilation operator, then is it true that $ E_{n} =C+ |\lambda _{n}|^{2} $ ¿what happens for the eigenfunctions ??

Answer

Here we will not answer all the subquestions, but just mention that there are at least two theoretical reasons why SUSY QM is important:

1. 
   

   Witten's derivation of Atiyah-Singer Index Theorem using SUSY QM.

   
   



2. 
   

   There is a big overlap between SUSY QM systems, and QM systems that can be analytically solved.