We present a unified exact analytic solution of the radial Schrödinger equation with a class of four long-range potentials, three of which are two-scale. The physics of different potentials are encapsulated in the solution variable and parameters. Based on this solution and its asymptotic behaviors, we generalize Bo Gao’s formulation of quantum defect theory for single-scale potentials to two-scale regime. As an application of this two-scale quantum defect theory, we propose a concept of harmonic oscillator quantu…