Some remarks on the spectral problem underlying the camassa-holm hierarchy

Academic Article

Abstract

  • © 2014 Springer International Publishing Switzerland. We study particular cases of left-definite eigenvalue problems Aψ = λBψ, with A ≥ εI for some ε > 0 and B 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 L2(R; dx) associated with a differential expression of the form τ = −(d/dx)p(x)(d/dx) + q(x), x ∈ R, and B represents an operator of multiplication by r(x) in L2 (R; dx), which, in general, is not a weight, that is, it is not nonnegative (or nonpositive) a.e. on 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 L2(R; dx). Our approach relies on rewriting the eigenvalue problem Aψ = λBψ in the form A−1/2BA−1/2χ = λ−1 χ, χ=A1/2ψ, and a careful study of (appropriate realizations of) the operator A−1/2BA−1/2 in L2 (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 T0 = −d2/dx2 defined on H2(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.
  • Authors

    Digital Object Identifier (doi)

    Author List

  • Gesztesy F; Weikard R
  • Start Page

  • 137
  • End Page

  • 188
  • Volume

  • 240