In this section we discuss other methods for computing resonances. They are essential for effective codes for higher dimensional problems for which analogues of (30) are anavailable or become more complicated.
Often, resonances are computed by changing the equation so that it is no longer posed on all of , but instead is posed on some interval with homogeneous Dirichlet or Neumann boundary conditions. For example, if the support of lies strictly within the interval , we might add a complex absorbing potential outside of , or we might scale the coordinate system into the complex plane by the method of perfectly matched layers2. The change to the equation should be designed so that the modified equation mimics the behavior of the original problem in the range .
To be more concrete, suppose that we modify the equation on the interval so that we still have a nonsingular, second-order, ordinary differential equation in whose coefficients depend on . Now we specify two linearly independent solutions and on which satisfy the modified domain equation together with the initial conditions
In summary, by changing the Schrödinger equation outside the interval , imposing homogeneous Dirichlet boundary conditions at , and then transporting the conditions at to conditions at , we arrive at the equations
The relation between outgoing wave boundary conditions and wave behavior at the boundary of a bounded absorber is useful for applications and experiments as well as for calculations. Experiments to observe acoustic (or electromagnetic) resonances and scattering are generally conducted in anechoic chambers, which are lined with baffles of sound-absorbing material. These baffles prevent incoming reflected waves from interfering with the experiment. Just as one can mimic the ``radiation-only'' property of an infinite domain with a finite absorber, models set in infinite domains are often approximations of models over a large finite domain in which the medium through which waves propogate is slightly dissipative.
David Bindel 2006-10-04