Cold atoms at unitarity and inverse square interaction

Consider two identical atoms in a spherical harmonic oscillator interacting with a zero-range interaction which is tuned to produce an s-wave zero-energy bound state. The quantum spectrum of the system is known to be exactly solvable. We note that the same partial wave quantum spectrum is obtained by the one-dimensional scale-invariant inverse square potential. Long known as the Calogero-Sutherland-Moser (CSM) model, it leads to Fractional Exclusion Statistics (FES) of Haldane and Wu. The statistical parameter is deduced from the analytically calculated second virial coefficient. When FES is applied to a Fermi gas at unitarity, it gives good agreement with experimental data without the use of any free parameter.Comment: 11 pages, 3 figures, To appear in J. Phys. B. Atomic, Molecular and Optical Physic

Supersymmetry,Shape Invariance and Exactly Solvable Noncentral Potentials

Using the ideas of supersymmetry and shape invariance we show that the eigenvalues and eigenfunctions of a wide class of noncentral potentials can be obtained in a closed form by the operator method. This generalization considerably extends the list of exactly solvable potentials for which the solution can be obtained algebraically in a simple and elegant manner. As an illustration, we discuss in detail the example of the potential $$V(r,\theta,\phi)={\omega^2\over 4}r^2 + {\delta\over r^2}+{C\over r^2 sin^2\theta}+{D\over r^2 cos^2\theta} + {F\over r^2 sin^2\theta sin^2 \alpha\phi} +{G\over r^2 sin^2\theta cos^2\alpha\phi}$$ with 7 parameters.Other algebraically solvable examples are also given.Comment: 16 page

Atomic Ground-State Energies

It is demonstrated that atomic Hartree–Fock binding energies may be reproduced with great accuracy (within about four parts in a thousand) by a scaled model system in which the electrons are noninteracting, and are bound in a bare Coulomb potential. </jats:p