Threshold Behavior and Levinson's Theorem for Two-Dimensional Scattering Systems: A Surprise

The low-energy threshold behavior and Levinson's theorem are derived for general, not necessarily rotationally symmetric, two-dimensional scattering systems, with zero-energy resonances and zero-energy bound states explicity taken into account. Surprisingly, $s$-wave-type zero-energy resonances do not contribute at all to Levinson's theorem, while $p$-wave-type zero-energy resonances each contribute a term - $\pi$, exactly like (zero-energy) bound states. Some consequences of this for, e.g., the Witten index in certain supersymmetric quantum-mechanical models are briefly discussed.