Conditions for Efimov physics for finite-range potentials

We consider a system of three identical bosons near a Feshbach resonance in the universal regime with large scattering length usually described by model-independent zero-range potentials. We employ the adiabatic hyperspherical approximation and derive the rigorous large-distance equation for the adiabatic potential for finite-range interactions. The effective range correction to the zero-range approximation must be supplemented by a new term of the same order. The nonadiabatic term can be decisive. Efimov physics is always confined to the range between effective range and scattering length. The analytical results agree with numerical calculations for realistic potentials.

Figure 1

(Color online) Adiabatic eigenvalues $\lambda \left(\rho \right)$ from Eq. (16) as function of hyper-radius $\rho$ for large scattering length $a$ and negative $R_e$. Different values of the model-dependent length $R_0$ are used, showing that the universal solution ${\lambda }_0$ exists in the region $\left|R_0\right|⪡\rho /\sqrt{\mu }⪡\left|a\right|$. Lengths are in units of $\left|R_e\right|$.

Figure 2

(Color online) Exact numerical adiabatic eigenvalues $\lambda \left(\rho \right)$ for a potential with barrier, Eq. (18), (solid line) compared to solutions of the eigenvalue equation, Eq. (16). The zero-range model (dotted line) includes only $a$. Crosses include $a,R_e$ terms and circles include $a,R_e,R_V$ terms. The inset shows details around ${\lambda }_0$. The arrow indicates the effect of the correction ${\lambda }_0-2R_0/R_e$.