  One-dimensional multi-component Fermi or Bose systems with strong zero-range
interactions can be described in terms of local exchange coefficients and
mapping the problem into a spin model is thus possible. For arbitrary external
confining potentials the local exchanges are given by highly non-trivial
geometric factors that depend solely on the geometry of the confinement through
the single-particle eigenstates of the external potential. To obtain accurate
effective Hamiltonians to describe such systems one needs to be able to compute
these geometric factors with high precision which is difficult due to the
computational complexity of the high-dimensional integrals involved. An
approach using the local density approximation would therefore be a most
welcome approximation due to its simplicity. Here we assess the accuracy of the
local density approximation by going beyond the simple harmonic oscillator that
has been the focus of previous studies and consider some double-wells of
current experimental interest. We find that the local density approximation
works quite well as long as the potentials resemble harmonic wells but break
down for larger barriers. In order to explore the consequences of applying the
local density approximation in a concrete setup we consider quantum state
transfer in the effective spin models that one obtains. Here we find that even
minute deviations in the local exchange coefficients between the exact and the
local density approximation can induce large deviations in the fidelity of
state transfer for four, five, and six particles.
