  The perfectly matched layers (PML) and exterior complex scaling (ECS) methods
for absorbing boundary conditions are analyzed using spectral decomposition.
Both methods are derived through analytical continuations from unitary to
contractive transformations. We find that the methods are mathematically and
numerically distinct: ECS is complex stretching that rotates the operator's
spectrum into the complex plane, whereas PML is a complex gauge transform which
shifts the spectrum. Consequently, the schemes differ in their time-stability.
Numerical examples are given.
