Darboux transformation, factorization, supersymmetry in one-dimensional quantum mechanics

We introduce an $N$-order Darboux transformation operator as a particular case of general transformation operators. It is shown that this operator can always be represented as a product of $N$ first-order Darboux transformation operators. The relationship between this transformation and the factorization method is investigated. Supercharge operators are introduced. They are differential operators of order $N$. It is shown that these operators and super-Hamiltonian form a superalgebra of order $N$. For $N=2$, we have a quadratic superalgebra analogous to the Sklyanin quadratic algebras. The relationship between the transformation introduced and the inverse scattering problem in quantum mechanics is established. An elementary $N$-parametric potential that has exactly $N$ predetermined discrete spectrum levels is constructed. The paper concludes with some examples of new exactly soluble potentials.