We study the resonances of Schrödinger operators on the infinite product
, where
is odd,
is the unit circle,
and the potential
lies in
.
This paper shows that at high energy, resonances of the Schrödinger operator
on
which
are near the continuous spectrum are approximated by the resonances of
on
, where the
potential
is given
by averaging over
the unit circle. These resonances are, in turn, given in terms of the resonances of a Schrödinger
operator on
which lie in a bounded set. If the potential is smooth, we obtain improved
localization of the resonances, particularly in the case of simple, rank
poles of the corresponding
scattering resolvent on .
In that case, we obtain the leading order correction for the location of the corresponding
high-energy resonances. In addition to direct results about the location of resonances, we
show that at high energies away from the resonances, the resolvent of the model operator
on
approximates
that of
on
. If
,
in certain cases this implies the existence of an asymptotic expansion
of solutions of the wave equation. Again for the special case of
, we
obtain a resonant rigidity type result for the zero potential among all real-valued
smooth potentials.
Keywords
Schrödinger operator, resonance, infinite cylindrical end,
scattering theory
