Fourier transform approach to potential harmonics

Hyperspherical lowest-order constrained-variational approximation to resonant Bose-Einstein condensates

“…The set {λ} represents a set of quantum numbers, here unspecified, that serve to distinguish the various basis states. In the case of non-interacting particles, a = 0, these are the usual hyperspherical harmonics and are independent of ρ [27,28]. In the present circumstance, however, eigenfunctions of Λ 2 N −1 are crafted subject to the Bethe-Peierls boundary conditions, in which case these functions depend also parametrically on ρ, as we will see in the next section.…”

“…The angular coordinates on this hypersphere, collectively denoted by Ω, may be chosen in a great many different ways [27,28]. A main point, however, is that all such angular coordinates are bounded and therefore eigenstates of kinetic energy operators expressed in these coordinates have discrete spectra and are characterized by a collection of as many as 3N − 4 discrete quantum numbers.…”

Hyperspherical approach to dipolar Bose-Einstein condensates beyond the mean-field limit

“…Where in the first sum, x i is understood to be an element of the 2N dimensional vector (x 1 , y 1 , x 2 , y 2 , • • • , x N , y N ). These operators obey the eigenvalue equations [37,38] Λ…”

“…We work in momentum space because it will later allow us to deal with the 1/r 3 singularity in the dipole-dipole potential. We define the following unit vectors [38]…”

“…Notice that û depends only on position coordinates and ŵ depends only on momentum coordinates. Using these unit vectors, we can embed the plane wave in the larger hyperspheres [38]…”