The situation changes dramatically when we consider with
satisfying (1), on all of the real line or
on a half line. The wave equation looks the same except for
the lack of boundary conditions. We will assume that the
initial values are zero outside some bounded set:
To define resonant states, we make the obvious observation that any solution of
To describe the solutions of (7) in terms of resonances
and resonant states let us make the simplifying assumption that
the resonances are simple. That means that the poles of, say,
are simple in the sense of meromorphic
functions. That is the case for a generic potential [17].
The operator then has a finite number of negative eigenvalues,
We then have the following analogue of
(5) for the solutions of (7):
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The expansion (11)
shows that in bounded sets, such as , and for large positive times,
solutions to the wave equation are desribed by modes specific to
and independent of initial conditions. That is very much like the
more standard (5). The waves ``resonate'' with frequencies
given by
and decay rates given by
.
This interpretation of resonances is quite common in popular culture, especially in the context of ``bells sounding its last dying notes''1. An amusing example comes from Vikram Seth's beautiful translations of Li Bai's poems [28]:
A thousand valleys' rustling pines resound.The word
My heart was cleansed, as if in flowing water.
In bells of frost I heard the resonance die.
David Bindel 2006-10-04