  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 $r$.
