Analytical solution of the Schrödinger equation with $1/r^3$ and attractive $1/r^2$ potentials: Universal three-body parameter of mixed-dimensional Efimov states

TL;DR

The paper addresses Efimov physics in mixed-dimensional settings by solving the Schrödinger equation with long-range $1/r^3$ dipolar and attractive $1/r^2$ potentials using an extended quantum defect theory with complex angular momentum. The authors derive analytical expressions for binding energies and wave functions, distinguishing the repulsive and attractive $1/r^3$ cases: the former yields a universal three-body parameter fixed by the dipole length $\\beta_3$, while the latter exhibits explicit short-range dependence through the quantum defect $K^c$. Numerical benchmarks show excellent agreement with the analytic results and reveal how finite transverse confinement influences the spectrum, preserving discrete scaling in the appropriate limits. The findings provide a universal framework for describing mixed-dimensional Efimov states, offer practical guidance for realizing polar-molecule/heavy-atom mixtures in quasi-1D traps, and clarify the role of short-range physics in these long-range interacting systems.

Abstract

We study the Schrödinger equation with $1/r^3$ and attractive $1/r^2$ potentials. Using the quantum defect theory, we obtain analytical solutions for both repulsive and attractive $1/r^3$ 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 $1/r^3$ and $1/r^2$ 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.

Figures (7)

• Figure 1: Schematic illustration of the three-body system studied in Eq. (\ref{['eq:SchEqVBO']}) : we consider a system two heavy particles (mass $M$) with dipole moments $\bm{d}$ interacting with a light particle (mass $m$) with an $s$-wave scattering length $a^\mathrm{(HL)}$ (black dotted line). The heavy particles are assumed to be confined in an axially symmetric harmonic trap (trapping length $l$), while the light particle move freely in a three dimensional space. The direction of the dipole moment $\theta_\mathrm{rd}$ can be controlled by an external field.
• Figure 2: (a) Energy spectra of the Efimov bound states under a repulsive $1/r^3$ potential at the unitary limit $1/a^{(\mathrm{HL})}=0$ calculated with $R_\mathrm{max}/R_\mathrm{min} = 2000$. Blue squares (orange circles) show the energies obtained by numerically solving Eq. (\ref{['eq:onedimSchEq']}) for mass ratios corresponding to $^{167}$Er-$^{6}$Li atoms $M/m = 27.7520...$ ($^{6}$Li atom and $^{23}$Na$^{40}$K molecule $M/m = 10.4659...$), respectively. Solid curves denote the analytical result in Eq. (\ref{['eq:EnergyRepDipole']}) with $\mu=M/2$. (b) Energy spectra for finite heavy-light $s$-wave scattering length $a^{(\mathrm{HL})}$, calculated with $\beta_3 / R_\mathrm{min} = 16.0$. We only show three and one states among the whole Efimov series for $^{167}$Er-$^{6}$Li (blue) and $^{23}$Na$^{40}$K-$^{6}$Li (orange), respectively. Dashed lines denote the heavy-light dimer states, into which the Efimov trimers dissociate for $a^{(\mathrm{HL})}\rightarrow +0$.
• Figure 3: Efimov states' wave function for a repulsive $1/r^3$ potential, obtained by solving Eq. (\ref{['eq:onedimSchEq']}) with $M/m = 27.7520...$, $R_\mathrm{min} / \beta_3 = 0.05$, and $R_\mathrm{max} / \beta_3 = 20.0$. Blue (red) solid curves are the ground and first-excited Efimov states, respectively. Dashed curves denote analytical solutions for $r \ll 1/\sqrt{M |E|}$ in Eqs. (\ref{['eq:fcgc_shortI_rep']}) and (\ref{['eq:QDTWF_fcKcgc']}) with $K^c= f^c (R_{\mathrm{min}})/g^c (R_{\mathrm{min}})$.
• Figure 4: (a) Energy spectra of the Efimov bound states under an attractive $1/r^3$ potential at the unitary limit $1/a^{(\mathrm{HL})}=0$. Solid curves denote the analytical result in Eqs. (\ref{['eq:EnergyAttDipole']}) and (\ref{['eq:kc']}) with $\mu=M/2$. (b) Energy spectra for finite heavy-light $s$-wave scattering length $a^{(\mathrm{HL})}$, obtained with $\beta_3 / R_\mathrm{min} = 10.0$. We only show three and two states among the whole Efimov series for $^{167}$Er-$^{6}$Li (blue) and $^{23}$Na$^{40}$K-$^{6}$Li (orange), respectively.
• Figure 5: Efimov states' wave function for an attractive $1/r^3$ potential, obtained by solving Eq. (\ref{['eq:onedimSchEq']}) with $M/m = 27.7520...$, $R_\mathrm{min} / \beta_3 = 0.05$, and $R_\mathrm{max} / \beta_3 = 20.0$. Blue (red) solid curves are the ground and first-excited Efimov states, respectively. Dashed curves denote analytical solutions for $r \ll 1/\sqrt{M |E|}$ in Eqs. (\ref{['eq:zeroenergy_wavefunction_attractive']}) and (\ref{['eq:QDTWF_fcKcgc']}) with $K^c= f^c (R_{\mathrm{min}})/g^c (R_{\mathrm{min}})$.