Exact solution for quantum scattering from noncentral rational scatterers for ultracold molecules to quantum dots
The double ring-shaped oscillator (DRO) was first introduced in 1989 where an exact solution was presented in a couple of articles, first using the Feynman path integral [Phys. Lett. A 134, 395 (1989)] followed by an algebraic approach [Phys. Lett. A 137, 1 (1989)]. Since then, interest has grown in the anisotropic DRO potential because of its widespread application from ultracold molecule collisions to ring-shaped molecules and double ring-shaped quantum dots, which all lack spherical symmetry [Nat. Phys. 13, 35 (2017), J. Low Temp. Phys. 190, 200 (2018)]. The quantum scattering from this potential has also been treated using the Feynman path integral formulation of the S-matrix [Phys. Lett. A 264, 45 (1999)]. In this talk, we generalize the DRO to a novel set of rational trigonometric forms of anisotropic potentials given by
Vn(r,θ) = αf(r) + A[4r2 cos2n(θ/2)]−1 + B[4r2 sin2n(θ/2)]−1
The case n = 1, is akin to the DRO introduced earlier. We solve the quantum problem for this potential using the Jacobi partial waves in contrast to the spherical harmonics normally used in partial wave analysis. An exact quantum scattering solution for this set of noncentral rational scatterers is then presented where we obtain the scattering amplitude and the total scattering cross-section.