  Helium films show various quantum phases that undergo quantum phase
transitions by changing coverage n. We found anomalous elastic phenomena in
bosonic 4He and fermionic 3He films adsorbed on a glass substrate. The films
stiffen under AC strain at low temperature with an excess dissipation. The
onset temperature of the stiffening decreases to 0 K as n approaches a critical
coverage nc. The elastic anomaly is explained by thermal activation of helium
atoms from the localized to extended states with a distributed energy gap. We
determine for the first time the energy band structure of helium films from
elasticity. The ground states of 4He and 3He at n < n_c are identically gapped
and compressible, which are possibly a sort of Mott insulator or Mott glass.
