Phase-amplitude method for calculating resonance energies and widths for one-dimensional potentials

We present a method for evaluating resonances and bound states present in any one-dimensional potential. Semiclassical theory (WKB) provides a fast estimate of resonance positions, but it can be off in calculating resonance lifetimes. Conversely, there are a multitude of numerical techniques, solving the one-dimensional Schr\"odinger equation, that give precise resonance energies and widths, but they are much more difficult to apply. The phase-amplitude method presented here yields accurate resonance energies and widths and is as easy to apply as semiclassical theory. Long-lived vibrational states of He₂²⁺ and CO²⁺ molecular ions are computed for demonstration.