  We propose a method for observation of the quasi-stationary states of
neutrons, localized near the curved mirror surface. The bounding effective well
is formed by the centrifugal potential and the mirror Fermi-potential. This
phenomenon is an example of an exactly solvable "quantum bouncer" problem that
could be studied experimentally. It could provide a promising tool for studying
fundamental neutron-matter interactions, as well as quantum neutron optics and
surface physics effects. We develop formalism, which describes quantitatively
the neutron motion near the mirror surface. The effects of mirror roughness are
taken into account.
