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Darboux transformation, factorization, and supersymmetry in one-dimensional quantum mechanics. (English. Russian original) Zbl 0857.34070
Theor. Math. Phys. 104, No. 2, 1051-1060 (1995); translation from Teor. Mat. Fiz. 104, No. 2, 356-367 (1995).
Summary: We introduce an \(N\)-th 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 a 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.

MSC:
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
81U40 Inverse scattering problems in quantum theory
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
17A70 Superalgebras
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