Analytical solution of the Schrödinger equation with 1/r³ and attractive 1/r² potentials: Universal three-body parameter of mixed-dimensional Efimov states

We study the Schrödinger equation with 1/r³ and attractive 1/r² potentials. Using the quantum defect theory, we obtain analytical solutions for both repulsive and attractive 1/r³ interactions. The obtained discrete-scale-invariant energies and wave functions, validated by excellent agreement with numerical results, provide a natural framework for describing the universality of Efimov states in mixed dimension. Specifically, we consider a three-body system consisting of two heavy particles with large dipole moments confined to a quasi-one-dimensional geometry and resonantly interacting with an unconfined light particle. With the Born-Oppenheimer approximation, this system is effectively reduced to the Schrödinger equation with 1/r³ and 1/r² potentials, and manifests the Efimov effect. Our analytical solution suggests that, for repulsive dipole interactions, the three-body parameter of the mixed-dimensional Efimov states is universally set by the dipolar length scale, whereas for attractive interactions it explicitly depends on the short-range phase. We also investigate the effects of finite transverse confinement and find that our analytical results are useful for describing the Efimov states composed of two polar molecules and a light atom.