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Bounded Solutions of KdV and Non-Periodic One-Gap Potentials in Quantum Mechanics

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Abstract

We describe a broad new class of exact solutions of the KdV hierarchy. In general, these solutions do not vanish at infinity, and are neither periodic nor quasi-periodic. This class includes algebro-geometric finite-gap solutions as a particular case. The spectra of the corresponding Schrödinger operators have the same structure as those of N-gap periodic potentials, except that the reflectionless property holds only in the infinite band. These potentials are given, in a non-unique way, by 2N real positive functions defined on the allowed bands. In this letter we restrict ourselves to potentials with one allowed band on the negative semi-axis; however, our results apply in general. We support our results with numerical calculations.

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Authors and Affiliations

  1. Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY, 10012, USA

    Dmitry V. Zakharov

  2. Department of Mathematics, University of Illinois at Urbana–Champaign, Urbana, IL, 61801, USA

    Sergey A. Dyachenko

  3. Department of Mathematics, University of Arizona, Tucson, AZ, 85791, USA

    Vladimir E. Zakharov

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  1. Dmitry V. Zakharov
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  2. Sergey A. Dyachenko
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  3. Vladimir E. Zakharov
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Correspondence to Dmitry V. Zakharov.

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Zakharov, D.V., Dyachenko, S.A. & Zakharov, V.E. Bounded Solutions of KdV and Non-Periodic One-Gap Potentials in Quantum Mechanics. Lett Math Phys 106, 731–740 (2016). https://doi.org/10.1007/s11005-016-0838-6

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  • DOI: https://doi.org/10.1007/s11005-016-0838-6

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