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SCATTERING RESONANCES AS VISCOSITY LIMITS: A
NUMERICAL EXAMPLE

We present a simple numerical companion to Scattering resonances as viscosity
limits.

The code viscap.m computes resonances of

where V is a step potential and the eigenvalues of

The input is given ε,V,r where

are values of the potential in r(K) < x < r(K + 1) where

In the output [z,w,v] , z is the vector of approximate eigenvalues of D_{x}^{2} + V - iεx^{2} and w is
the vector of resonances. Both z and w are pruned to include only the eigenvalues near 0 and
to exclude spurious einvalues with arg z near -π∕4. The last output consists of all the
computed eigenvalues of the discretization of D_{x}^{2} + V -iεx^{2}: most of them are spurious or due
to pseudospectral effects.

The resonances are computed using squarepot.m written by
David Bindel.

Here are some examples:

viscap(0.5, [ 50 30 50 -30], [ -2 -1 0 1 2]); outputs the figure

[ z, w, v ] = viscap(0.2, [ 40 20 80 ], [-2 -1 1 2]); outputs z,w,v and the figure

The majority of the eigenvalues of the discretization of D_{x}^{2} + V - iεx^{2} are
either near arg z = -π∕4 or are due to pseudospectral effects.

plot(v,’r*’,’MarkerSize’,10)
outputs the figure

The smaller the ε the more pronounced are the pseudospectral effects: most of the computed
eigenvalues are not close to the eigenvalues of the operator.