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Supersymmetry of the Sturm—Liouville and Korteveg—de Vries Operators

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Book cover Operator Methods in Ordinary and Partial Differential Equations

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 132))

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Abstract

In 70’s A.A. Kirillov identified the (stationary) Sturm-Liouville operator \( {L_2} = \frac{{{d^2}}}{{d{x^2}}} + F \) with an element of the dual space ĝ* of the nontrivial central extension ĝ = viτ, called the Virasoro algebra, of the Witt algebra witt =veτ c[x-1, x]. He interpreted the (stationary) KdV operator \({{L}_{3}} = \frac{{{{d}^{3}}}}{{d{{x}^{3}}}} + \frac{d}{{dx}}F + \frac{d}{{dx}} \) in terms of the stabilizer of L 2. He also found a supersymmetry that reduces solutions of L 3 (f) = 0 to solutions of L 2 (g) = 0 by studying the nontrivial central extension of a simplest super analog of the Virasoro algebra, the Neveu-Schwarz superalgebra. Kirillov also wrote the first superversion of KdV equation.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Stockholm, Roslagsv. 101, Kraftriket hus 6, Stockholm, SE-106 91, Sweden

    Dimitry Leites

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  1. Dimitry Leites
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Editor information

Editors and Affiliations

  1. Institute of Applied Mathematics, University of Bonn, Bonn, 53115, Germany

    Sergio Albeverio

  2. Department of Physics, Stockholm University, Stockholm, 10691, Sweden

    Nils Elander

  3. School of Mathematics and Statistics, University of Birmingham, Edgbaston, Birmingham, England, B15 2TT, UK

    W. Norrie Everitt

  4. Department of Mathematics, Stockholm University, Stockholm, 10691, Sweden

    Pavel Kurasov

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© 2002 Springer Basel AG

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Leites, D. (2002). Supersymmetry of the Sturm—Liouville and Korteveg—de Vries Operators. In: Albeverio, S., Elander, N., Everitt, W.N., Kurasov, P. (eds) Operator Methods in Ordinary and Partial Differential Equations. Operator Theory: Advances and Applications, vol 132. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8219-4_21

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  • DOI: https://doi.org/10.1007/978-3-0348-8219-4_21

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9479-1

  • Online ISBN: 978-3-0348-8219-4

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