General form of the quantum-defect theory for $-1/r^{\alpha }$ type of potentials with $\alpha >2$

We present a general formulation of the quantum-defect theory for systems with an attractive $1/r^{\alpha }$ type of asymptotic potentials with $\alpha >2$. Its structural differences from the standard theory for $\alpha \le 2$, or from the semiclassical theory, is fully explained through a systematic understanding of quantum reflection above the threshold, and quantum connection formulas below the threshold.