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 TMF, 1995, Volume 104, Number 2, Pages 356–367 (Mi tmf1344)

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

V. G. Bagrova, B. F. Samsonovb

a Tomsk State University
b Institute of High Current Electronics, Siberian Branch of the Russian Academy of Sciences

Abstract: 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.

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English version:
Theoretical and Mathematical Physics, 1995, 104:2, 1051–1060

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Citation: V. G. Bagrov, B. F. Samsonov, “Darboux transformation, factorization, supersymmetry in one-dimensional quantum mechanics”, TMF, 104:2 (1995), 356–367; Theoret. and Math. Phys., 104:2 (1995), 1051–1060

Citation in format AMSBIB
\Bibitem{BagSam95} \by V.~G.~Bagrov, B.~F.~Samsonov \paper Darboux transformation, factorization, supersymmetry in one-dimensional quantum mechanics \jour TMF \yr 1995 \vol 104 \issue 2 \pages 356--367 \mathnet{http://mi.mathnet.ru/tmf1344} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1488681} \zmath{https://zbmath.org/?q=an:0857.34070} \transl \jour Theoret. and Math. Phys. \yr 1995 \vol 104 \issue 2 \pages 1051--1060 \crossref{https://doi.org/10.1007/BF02065985} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1995UD33400013} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. B. F. Samsonov, “On the $N$-th order Darboux transformation”, Russian Math. (Iz. VUZ), 43:6 (1999), 62–65
2. Andrey M. Pupasov, Boris F. Samsonov, “Exact Propagators for Soliton Potentials”, SIGMA, 1 (2005), 020, 7 pp.
3. A. A. Andrianov, A. V. Sokolov, “Factorization of nonlinear supersymmetry in one-dimensional Quantum Mechanics I: general classification of reducibility and analysis of third-order algebra”, J. Math. Sci. (N. Y.), 143:1 (2007), 2707–2722
4. Andrianov, AA, “Non-linear supersymmetry for non-Hermitian, non-diagonalizable Hamiltonians: I. General properties”, Nuclear Physics B, 773:3 (2007), 107
5. Alexander A. Andrianov, Andrey V. Sokolov, “Hidden Symmetry from Supersymmetry in One-Dimensional Quantum Mechanics”, SIGMA, 5 (2009), 064, 26 pp.
6. Ryu Sasaki, Wen-Li Yang, Yao-Zhong Zhang, “Bethe Ansatz Solutions to Quasi Exactly Solvable Difference Equations”, SIGMA, 5 (2009), 104, 16 pp.
7. Mikhail V. Ioffe, “Supersymmetrical Separation of Variables in Two-Dimensional Quantum Mechanics”, SIGMA, 6 (2010), 075, 10 pp.
8. Orlando Ragnisco, Danilo Riglioni, “A Family of Exactly Solvable Radial Quantum Systems on Space of Non-Constant Curvature with Accidental Degeneracy in the Spectrum”, SIGMA, 6 (2010), 097, 10 pp.
9. Fernandez C D.J., “Supersymmetric Quantum Mechanics”, Advanced Summer School in Physics 2009: Frontiers in Contemporary Physics, 5th Edition, AIP Conference Proceedings, 1287, 2010, 3–36
10. David Bermúdez, “Complex SUSY transformations and the Painlevé IV equation”, SIGMA, 8 (2012), 069, 10 pp.
11. Christiane Quesne, “Novel enlarged shape invariance property and exactly solvable rational extensions of the Rosen–Morse II and Eckart potentials”, SIGMA, 8 (2012), 080, 19 pp.
12. Andrianov A.A. Ioffe M.V., “Nonlinear Supersymmetric Quantum Mechanics: Concepts and Realizations”, J. Phys. A-Math. Theor., 45:50 (2012), 503001
13. Marquette I. Quesne Ch., “Combined State-Adding and State-Deleting Approaches To Type III Multi-Step Rationally Extended Potentials: Applications To Ladder Operators and Superintegrability”, J. Math. Phys., 55:11 (2014), 112103
14. Gomez-Ullate D. Grandati Y. Milson R., “Extended Krein-Adler Theorem For the Translationally Shape Invariant Potentials”, J. Math. Phys., 55:4 (2014), 043510
15. Th. Th. Voronov, S. Hill, E. S. Shemyakova, “Darboux transformations for differential operators on the superline”, Russian Math. Surveys, 70:6 (2015), 1173–1175
16. Pupasov-Maksimov A.M., “Propagators of Isochronous An-Harmonic Oscillators and Mehler Formula For the Exceptional Hermite Polynomials”, Ann. Phys., 363 (2015), 122–135
17. David Hobby, Ekaterina Shemyakova, “Classification of Multidimensional Darboux Transformations: First Order and Continued Type”, SIGMA, 13 (2017), 010, 20 pp.
18. A. I. Aptekarev, M. A. Lapik, Yu. N. Orlov, “Asymptotic behavior of the spectrum of combination scattering at Stokes phonons”, Theoret. and Math. Phys., 193:1 (2017), 1480–1497
19. Li S. Shemyakova E. Voronov T., “Differential Operators on the Superline, Berezinians, and Darboux Transformations”, Lett. Math. Phys., 107:9 (2017), 1689–1714
20. Carinena J.F. Plyushchay M.S., “Abc of Ladder Operators For Rationally Extended Quantum Harmonic Oscillator Systems”, J. Phys. A-Math. Theor., 50:27 (2017), 275202
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