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Some Remarks on the Spectral Problem Underlying the Camassa–Holm Hierarchy

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Operator Theory in Harmonic and Non-commutative Analysis

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 240))

Abstract

We study particular cases of left-definite eigenvalue problems \(A\psi\;=\;B\psi\), with \(A\;\geq\;\varepsilon\it{I}\)for some \(\varepsilon\;>\;0\;\mathrm{and}\;\it{B}\; \mathrm{self-adjoint}\), but B not necessarily positive or negative definite, applicable, in particular, to the eigenvalue problem underlying the Camassa–Holm hierarchy. In fact, we will treat a more general version where A represents a positive definite Schrödinger or Sturm–Liouville operator T in \(L^2(\mathbb{R};dx)\) associated with a differential expression of the form \(\tau\;=\;-(d/dx)p(x)(d/dx)+q(x),x\in\mathbb{R}\), and B represents an operator of multiplication by \(r(x)\;\mathrm{in}\;L^2(\mathbb{R};dx)\), which, in general, is not a weight, that is, it is not nonnegative (or nonpositive) a.e. on \(\mathbb{R}\). In fact, our methods naturally permit us to treat certain classes of distributions (resp., measures) for the coefficients q and r and hence considerably extend the scope of this (generalized) eigenvalue problem, without having to change the underlying Hilbert space \(L^2(\mathbb{R};dx)\). Our approach relies on rewriting the eigenvalue problem \(A\psi\;=\;B\psi\) in the form \(A^{-1/2}BA^{-1/2}\chi\;=\lambda^{-1}\chi,\; \chi\;=\;A^{1/2}\psi\), and a careful study of (appropriate realizations of) the operator \(A^{-1/2}BA^{-1/2}\;\mathrm{in}\;L^2(\mathbb{R};dx)\).

In the course of our treatment, we review and employ various necessary and sufficient conditions for q to be relatively bounded (resp., compact) and relatively form bounded (resp., form compact) with respect to \(T_{0}\;=\;-d^2/dx^{2}\; \mathrm{defined\; on}\;H^2(\mathbb{R})\). In addition, we employ a supersymmetric formalism which permits us to factor the second-order operator T into a product of two firstorder operators familiar from (and inspired by) Miura’s transformation linking the KdV and mKdV hierarchy of nonlinear evolution equations. We also treat the case of periodic coefficients q and r, where q may be a distribution and r generates a measure and hence no smoothness is assumed for q and r.

Mathematics Subject Classification (2010). Primary 34B24, 34C25, 34K13, 34L05, 34L40, 35Q58, 47A10, 47A75; Secondary 34B20, 34C10, 34L25, 37K10, 47A63, 47E05.

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Authors and Affiliations

  1. Department of Mathematics, University of Missouri, Columbia, MO, 65211, USA

    Fritz Gesztesy

  2. Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL, 35294, USA

    Rudi Weikard

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  1. Fritz Gesztesy
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  2. Rudi Weikard
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Correspondence to Fritz Gesztesy .

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Editors and Affiliations

  1. Department of Mathematics, Virginia Polytechnic Institute, Blacksburg, Virginia, USA

    Joseph A. Ball

  2. Department of Mathematics, University of Newcastle upon Tyne, Newcastle upon Tyne, United Kingdom

    Michael A. Dritschel

  3. Department of Mathematics, University of Auckland, Auckland, New Zealand

    A.F.M. ter Elst

  4. Mathematical Sciences Institute, Université Lille 1Villeneuve d'Ascq, France and The Australian National University, Canberra, Aust Capital Terr, Australia

    Pierre Portal

  5. School of Mathematics and Statistics, University of New South Wales, Sydney, New South Wales, Australia

    Denis Potapov

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Gesztesy, F., Weikard, R. (2014). Some Remarks on the Spectral Problem Underlying the Camassa–Holm Hierarchy. In: Ball, J., Dritschel, M., ter Elst, A., Portal, P., Potapov, D. (eds) Operator Theory in Harmonic and Non-commutative Analysis. Operator Theory: Advances and Applications, vol 240. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-06266-2_7

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