Proof of the Levinson theorem by the Sturm–Liouville theorem

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Abstract
The Levinson theorem is proved by the Sturm–Liouville theorem in this paper. For the potential ∫10r‖V(r)‖dr <∞,V(r)→b/r2 when r→∞, the modified Levinson theorem is derived as nl=(1/π)δl(0) +(a−l)/2− 1/2 sin2{δl(0)+[(a−l)/2]π}, if a(a+1)≡b+l(l+1)> 3/4 or a=0. Two examples which violate the Levinson theorem and satisfy the modified Levinson theorem are discussed.