  An imaginary-time nonuniform mesh method is presented and used to find the
first 50 eigenstates and energies of up to five strongly interacting spinless
quantum Lennard-Jones particles trapped in a one-dimensional harmonic
potential. We show that the use of tailored grids reduces drastically the
computational effort needed to diagonalize the Hamiltonian and results in a
favorable scaling with dimensionality. Solutions to both bosonic and fermionic
counterparts of this strongly interacting system are obtained, the bosonic case
clustering as a Tonks-Girardeau crystal exhibiting the phenomenon of
fermionization. The numerically exact excited states are used to describe the
melting of this crystal at finite temperature.
