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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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##### References:
 [1] G. Darboux,Lecons sur la theorie generale des surfaces et les application geometriques du calcul infinitesimal. Deuxiem partie, Gauthier-Villars et fils, Paris (1889). · JFM 21.0744.02 [2] T.E. Infeld and H. Hull,Rev. Mod. Phys.,53, 21 (1951). · Zbl 0043.38602 [3] H. Hull and T.E. Infeld,Phys. Rev.,74, 905 (1948). · Zbl 0033.04105 [4] B. Melnik,J. Math. Phys.,25, No. 12, 3387–3389 (1984). · Zbl 0558.35064 [5] N.A. Alves and E.D. Filho,J. Phys. A: Math. Gen.,21, No. 15, 3215–3225 (1988). [6] L.D. Faddeev,Usp. Mat. Nauk,105, No. 4(88), 57–119 (1959). [7] B.M. Levitan,Inverse Sturm-Liouville Problems [In Russian] Nauka, Moscow (1984). · Zbl 0575.34001 [8] Z.S. Agranovich and V.A. Marchenko,Inverse Scattering Problem [In Russian], Kharkov Univ., Kharkov (1969). · Zbl 0117.06003 [9] P.B. Abraham and H.E. Moses,Phys. Rev. A.,22, No. 4, 1333–1340 (1980). [10] D.L. Pursey,Phys. Rev. D.,33, No. 4, 1048–1055 (1986). [11] M. Luban and D.L. Pursey,Phys. Rev. D.,33, No. 2, 431–436 (1986). [12] V.G. Bagrov, A.V. Shapovalov, and I.V. Shirokov,Teor. Mat. Fiz.,87, No. 3, 426–433 (1992). [13] V.G. Bagrov, A.V. Shapovalov, and I.V. Shirokov,Phys. Lett. A.,147, No. 7, 348–350 (1990). [14] E. Witten,Nucl. Phys.,B185, 513 (1981). · Zbl 1258.81046 [15] V.G. Bagrov and A.S. Vshivtsev,Izv. Vyssh. Uchebn. Zaved., Fiz., No. 7, 19 (1988). [16] J. Delsart,J. Math. Pures Appl.,17, 213–230 (1938). [17] A.P. Veselov and A.B. Shabat,Funkts. Anal. Prilozh.,27, No. 2, 1–21 (1993). · Zbl 0994.33501 [18] E.K. Sklyanin,Funkts. Anal. Prilozh.,16, 27 (1982);17, 34 (1983). [19] Ya.I. Granovskii, A.S. Zhedanov, and I.M. Lutsenko,Zh. Exp. Teor. Fiz.,99, 369 (1991). [20] M.G. Krein,Dokl. Akad. Nauk SSSR,113, No. 5, 970–973 (1957). [21] V.P. Berezovskii and A.I. Pashnev,Teor. Mat. Fiz.,70, 146 (1987). [22] Yu.S. Dubov, V.M. Eleonskii, and N.E. Kulagin,Zh. Exp. Teor. Fiz.,102, 814 (1992). [23] G.A. Natanzon,Vestn. Leningr. Gos. Univ., No. 10, 22 (1971);Teor. Mat. Fiz.,38, 219 (1979).
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