  Exact diagonalization expansions of Bose or Fermi gases with contact
interactions converge very slowly due to a non-analytic cusp in the wave
function. Here we develop a transcorrelated approach where the cusp is treated
exactly and folded into the many-body Hamiltonian with a similarity
transformation that removes the leading order singularity. The resulting
transcorrelated Hamiltonian is not hermitian but can be treated numerically
with a standard projection approach. The smoothness of the wave function
improves by at least one order and thus the convergence rate for the ground
state energy improves. By numerical investigation of a one-dimensional gas of
spin-$\frac{1}{2}$ fermions we find the error in the transcorrelated energy to
scale as $M^{-3}$ with a single-particle basis of $M$ plane waves compared to
$M^{-1}$ for the expansion of the original Hamiltonian and $M^{-2}$ using
conventional lattice renormalization.
