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

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.KeywordsCamassa–Holm hierarchyleft-definite spectral problemsdistributional coefficientsFloquet theorysupersymmetric formalism