According to this this the Darboux transformation preserves the discrete spectrum of the Haniltonian in quantum mechanics. Is there a proof for this? My best guess is that it has to do with the fact that $$Q^{\pm}$$ are ladder operators but I'm not sure.
The spectrum is preserved except for one eigenvalue. The two Hamitonians are related by
H1=LL^*+const, H2=L^*L+const with different constants. Having found L reduces the proof to a simple algebraic manipulation.