  We consider classes of simply connected planar domains which are isophasal,
ie, have the same scattering phase $s(\l)$ for all $\l > 0$. This is a
scattering-theoretic analogue of isospectral domains. Using the heat invariants
and the determinant of the Laplacian, Osgood, Phillips and Sarnak showed that
each isospectral class is sequentially compact in a natural $C$-infinity
topology. In this paper, we show sequential compactness of each isophasal class
of domains. To do this we define the determinant of the exterior Laplacian and
use it together with the heat invariants (the heat invariants and the
determinant being isophasal invariants). We show that the determinant of the
interior and exterior Laplacians satisfy a Burghelea-Friedlander-Kappeler type
surgery formula. This allows a reduction to a problem on bounded domains for
which the methods of Osgood, Phillips and Sarnak can be adapted.
