  We consider complex eigenstates of unstable Hamiltonian and its physically
meaningful regions. Starting from a simple model of a discrete state
interacting with a continuum via a general potential, we show that its
Lippmann-Schwinger solution set can be decomposed into a free-field set, a set
containing lower half plane pole of Green's function and a set containing upper
half pole of Green's function. From here distinctive complex eigenstates
corresponding to each pole are constructed. We note that on the real line
square integrable functions can be decomposed into Hardy class above and below
functions which behave well in their respective complex half planes. Test
function restriction formulas which remove unphysical growth are given. As a
specific example we consider Friedrichs model which solutions and complex
eigenstates are known, and compare numerically calculated total time evolution
with test function restricted complex eigenstates for various cases. The
results shows that test function restricted complex eigenstates capture the
essence of decay phenomena quite well.
