  A general expression for the Green's function of a system of $N$ particles
(bosons/fermions) interacting by contact potentials, including impurities with
Dirac-delta type potentials is derived. In one dimension for $N>2$ bosons from
our {\it spectral determinant method} the numerically calculated energy levels
agree very well with those obtained from the exact Bethe ansatz solutions while
they are an order of magnitude more accurate than those found by direct
diagonalization. For N=2 bosons the agreement is shown analytically. In the
case of N=2 interacting bosons and one impurity, the energy levels are
calculated numerically from the spectral determinant of the system. The
spectral determinant method is applied to an interacting fermion system
including an impurity to calculate the persistent current at the presence of
magnetic field.

Mesoscale and Nanoscale Physics;Strongly Correlated Electrons