  We formulate the Born approximation for finding resonance poles in the
complex plane for potential scattering problems. Using the method, we study the
distribution of resonance poles for several scattering potentials. In
particular, we find for an exponential potential with a cutoff that the cutoff
generates an infinite series of extra resonance poles below and along the real
axis, which would not exist without the cutoff. We also find for a Gaussian
potential that the series of resonance poles approach the imaginary axis of the
complex energy plane from left. In other words, the real parts of the resonant
eigenenergis are all negative.
