  Given a countable metric space, we can consider its end. Then a basis of a
Hilbert space indexed by the metric space defines an end of the Hilbert space,
which is a new notion and different from an end as a metric space.
  Such an indexed basis also defines unitary operators of finite propagation,
and these operators preserve an end of a Hilbert space. Then, we can define a
Hilbert bundle with end, which lightens up new structures of Hilbert bundles.
In a special case, we can define characteristic classes of Hilbert bundles with
ends, which are new invariants of Hilbert bundles. We show Hilbert bundles with
ends appear in natural contexts. First, we generalize the pushforward of a
vector bundle along a finite covering to an infinite covering, which is a
Hilbert bundle with end under a mild condition. Then we compute characteristic
classes of some pushforwards along infinite coverings. Next, we will show the
spectral decompositions of nice differential operators give rise to Hilbert
bundles with ends, which elucidate new features of spectral decompositions. The
spectral decompositions we will consider are the Fourier transform and the
harmonic oscillators.
