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

Abstract

We consider left-definite eigenvalue problems $A \psi = \lambda B \psi$, with $A \geq \varepsilon I$ for some $\varepsilon > 0$ 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 Schr\"odinger or Sturm-Liouville operator $T$ in $L^2(\bbR; dx)$ associated with a differential expression of the form $\tau = - (d/dx) p(x) (d/dx) + q(x)$, $x \in \bbR$, and $B$ represents an operator of multiplication by $r(x)$ in $L^2(\bbR; dx)$, which, in general, is not a weight, that is, it is not nonnegative a.e.\ on $\bbR$. 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 $L^2(\bbR; dx)$. Our approach relies on rewriting the eigenvalue problem $A \psi = \lambda B \psi$ in the form $A^{-1/2} B A^{-1/2} \chi = \lambda^{-1} \chi$, $\chi = A^{1/2} \psi$, and a careful study of (appropriate realizations of) the operator $A^{-1/2} B A^{-1/2}$ in $L^2(\bbR; dx)$. In the course of our treatment we employ a supersymmetric formalism which permits us to factor the second-order operator $T$ into a product of two first-order 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 $r$.