The regularized index introduced by Witten for supersymmetric theories and discussed by Callias for elliptic operators is studied. We relate this index to Levinson's theorem for potential scattering and clarify its meaning even when it takes on fractional values. The index can be (and usually is) half-integer whenever the continuum in the spectrum extends down to zero energy.

• Received 7 May 1984

DOI:https://doi.org/10.1103/PhysRevD.30.1821