The Levinson theorem for the Schr\"odinger equation with a spherically symmetric potential in D dimensions is uniformly established by the Sturm-Liouville theorem. It is shown that the Levinson theorem for the cases without a half bound state does not depend on the spatial dimension D, namely, the phase-shift δₗ(0) of the scattering state with angular momentum l at zero momentum is equal to the total number nₗ of bound states multiplied by π. When a half bound state occurs the Levinson theorem may be modified.
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Dong et al. (2002) studied this question.
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