Publication: The Darboux transformation and algebraic deformations of shape-invariant potentials
Loading...
Official URL
Full text at PDC
Publication Date
2004-02-06
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
Iop science
Citations
Exportar
Abstract
We investigate the backward Darboux transformations (addition of the lowest bound state) of shape-invariant potentials on the line, and classify the subclass of algebraic deformations, those for which the potential and the bound states are simple elementary functions. A countable family, m = 0, 1,.2,..., of deformations exists for each family of shape-invariant potentials. We prove that the m_th deformation is exactly solvable by polynomials, meaning that it leaves invariant an infinite flag of polynomial modules P_(m)^(m) Ϲ P_(-m+1)^(m) Ϲ (...) , where P_n^(m) is a codimension m subspace of <1, z,..., z_(n)>. In particular, we prove that the first (m = 1) algebraic deformation of the shape-invariant class is precisely the class of operators preserving the infinite flag of exceptional monomial modules P_n^(1) = <1, z_(2),..., z_(n)>. By construction, these algebraically deformed Hamiltonians do not have an sl(2) hidden symmetry algebra structure.
Description
©Iop science.
The research of DGU is supported in part by a CRM-ISM Postdoctoral Fellowship and the Spanish Ministry of Education under grant EX2002-0176. The research of NK and RM is supported by the National Science and Engineering Research Council of Canada. The authors would like to thank Prof. González-López and Prof. Gesztesy for interesting discussions, as well as the referees, who made very interesting remarks on the first version of the paper
UCM subjects
Unesco subjects
Keywords
Citation
[1] Darboux G, Théorie Générale des Surfaces, vol. II, Gauthier-Villars, 1888.
[2] Jacobi CG, 1837 J. Reine Angew. Math. 17, 68.
[3] Schrödinger E 1941 Proc. Roy. Irish Acad. 47 A, 53 (Preprint physics/9910003).
[4] Infeld L and Hull T E 1951 Rev. Mod. Phys. 23 21.
[5] Cooper F, Khare A and Sukhatme U 1995 Phys. Rep. 251 267.
[6] Deift P and Trubowitz E 1979 Duke Math J. 45, 267.
[7] Gesztesy F, Simon B and Teschl G 1996 J. d’Analyse Math. 70, 267
[8] Calogero F and Degasperis A 1982 Spectral transform and solitons I, Studies in Mathematics and its Applications (New York:Elsevier).
[9] Sukumar CV 1985 J. Phys. A 18 2917.
[10] Sparenberg J-M and Baye D 1995 J. Phys. A 28 5079.
[11] Bagrov V G and Samsonov B F 1995 Theoret. and Math. Phys. 104 1051.
[12] Gendenshtein L 1983 JETP Lett 38 356.
[13] Mielnik B 1984 J. Math. Phys. 25 3387.
[14] Lévai G, Baye D and Sparenberg J-M 1997 J. Phys. A 30 8257
[15] Turbiner A V 1988 Commun. Math. Phys. 118 467.
[16] Kamran N and Olver P J 1990 J. Math. Anal. Appl. 145 342.
[17] González-Lopez A, Kamran N and Olver P J 1993 Commun. Math. Phys. 153 117.
[18] Morse P M 1929 Phys. Rev. 57 57.
[19] Pöschl G and Teller E 1933 Z. Physik 83 143.
[20] Milson R 1998 Internat. J. Theoret. Phys. 37 1735.
[21] Gómez-Ullate D, González-López A and Rodríguez M A 2000 J. Phys. A 33 7305.
[22] Post G and Turbiner A V 1995 Russian J. Math. Phys. 3 113.
[23] Finkel F and Kamran N 1998 Adv. in Applied Math. 20 300.
[24] Baye D, Sparenberg J-M and Lévai G 1997 Inverse and Algebraic Quantum Scattering Theory (Lecture notes in Physics 488) ed B Apagyi, G Endrédi and P Lévay (Berlin: Springer) p 295
[25] Gómez-Ullate D, Kamran N and Milson R, in preparation.
[26] González-López A and Tanaka T, hep-th/0307094.
[27] Shifman M 1989 Int. J. Modern Phys. A 4 3311.
[28] Schminke U W 1978, Proc. Roy. Soc. Edinburgh Sec. A 80, 67.
[29] Erdélyi A et al. 1953 Higher Transcendental Functions, Vol. I, (New York:McGraw-Hill).
[30] Dubov S Y, Eleonskii V M and Kulagin N E 1992, Sov. Phys. JETP 75 446.
[31] Bagrov V G and Samsonov B F 1997 Pramana J. Phys. 49 563.
[32] Matveev V and Salle M A 1991 Darboux transformations and solitons, Springer Series in Nonlinear Dynamics (Berlin:Springer)