  We analyze the $1+1$ dimensional Nambu-Jona-Lasinio model non-perturbatively.
In addition to its simple ground state saddle points, the effective action of
this model has a rich collection of non-trivial saddle points in which the
composite fields $\sigx=\lag\bar\psi\psi\rag$ and $\pix=\lag\bar\psi
i\gam_5\psi\rag$ form static space dependent configurations because of
non-trivial dynamics. These configurations may be viewed as one dimensional
chiral bags that trap the original fermions (``quarks") into stable extended
entities (``hadrons"). We provide explicit expressions for the profiles of
these objects and calculate their masses. Our analysis of these saddle points
is based on an explicit representation we find for the diagonal resolvent of
the Dirac operator in a $\{\sigx, \pix\}$ background which produces a
prescribed number of bound states. We analyse in detail the cases of a single
as well as two bound states. We find that bags that trap $N$ fermions are the
most stable ones, because they release all the fermion rest mass as binding
energy and become massless. Our explicit construction of the diagonal resolvent
is based on elementary Sturm-Liouville theory and simple dimensional analysis
and does not depend on the large $N$ approximation. These facts make it, in our
view, simpler and more direct than the calculations previously done by Shei,
using the inverse scattering method following Dashen, Hasslacher, and Neveu.
Our method of finding such non-trivial static configurations may be applied to
other $1+1$ dimensional field theories.
