We first have to define the scattering matrix. For that
we will take a quantum mechanical point of view and
consider the time dependent Schrödinger equation,
For
the terms including
``move''
to the left and the ones including
``move'' to the
right. Hence the terms with coefficients and are
incoming (they move towards the perturbation) and the terms with
and are outgoing.
The scattering matrix is defined to be the operator which maps
the incoming coefficients to the outgoing coefficients, that is
(21)
The matrix
is meromorphic for
where poles with
correspond to the square root
of the eigenvalues of . Its poles in the lower half plane
coincide with resonances.
The coefficients in
(22)
are the transmission,
, and reflection,
,
coefficients.
We have the following general properties of
(23)
Since on the real axis
,
the scattering phase is defined (up to an integer) as
and
We know (see [34] and references given there) that
(24)
where for simplicity we assumed that there is no resonance at
.
Here
closure of the convex hull of
.
The convergence is guaranteed by the following consequence of Carleman's
formula (see [34]):
We then see that
and
If is isolated from other resonances
we expect from this formula that
which is a form of the Breit-Wigner formula. Note that
the integral of of the right hand side in is equal to
and that is the phase shift at the crossing of a resonance is .
But one has to note that this approximation is not really rigorous
without further parameters (such as the semiclassical parameter)
since we do not know the size of the contribution of other terms.
In particular, high energy behaviour of
is
given by