What is not provided in (11) is a formula for the coefficients (the formulæ for 's an 's are the same as in (6)). If is a resonant state then the assumption of compact support on initial data in (7) shows that the integrals
One possibility is to use the Lax-Phillips semigroup, - see [25],[23], and also [31] for a concise treatment in a very general setting. The semigroup not only provides the most elegant way of defining resonances, but is in fact the only purely dynamical definition: resonances are simply the eigenvalues of its generator. However, it is perhaps the beauty of this definition that makes its applicability limited (even in one dimension some modifications have to be made [22]). It is also rather cumbersome numerically.
Instead, let us go back to Green's function,
Suppose for simplicity that in the initial value problem (7). Then in the expansion (11) we can choose the resonant functions to be given by 's and we have
with the resonant state for which (14) holds. | (15) |
To find a normalization of we observe that (9) and (10) shows that we can continue analytically to
It turns out that the normalization which determines in (14) and (15) is
We cannot fully explain the origin of the normalization given in (18). It comes from the celebrated complex scaling method discovered by Aguilar-Combes and Balslev-Combes in the early 70s, and greatly developed in mathematics, computational physics, and chemistry - see [30] and [35] for a discussion and references.
Roughly speaking, we can restrict the operator to the contour . It then acts (as an unbounded operator) on rather than . The resonances with are the eigenvalues of this new non-self-adjoint operator, denoted by . Its resolvent,
David Bindel 2006-10-04