  We analyze the limit of the spectrum of a geometric Dirac-type operator under
a collapse with bounded diameter and bounded sectional curvature. In the case
of a smooth limit space B, we show that the limit of the spectrum is given by
the spectrum of a certain first-order differential operator on B, which can be
constructed using superconnections. In the case of a general limit space X, we
express the limit operator in terms of a transversally elliptic operator on a
G-space Y, with X = Y/G. As an application, we give a characterization of
manifolds which do not admit uniform upper bounds, in terms of diameter and
sectional curvature, on the k-th eigenvalue of the square of a Dirac-type
operator. We also give a formula for the essential spectrum of a Dirac-type
operator on a finite-volume manifold with pinched negative sectional curvature.
