  We reduce the problem of many-channel hadron scattering at nonrelativistic
energies to calculations on the scale of a few fermis. Having thus disentangled
kinematics from interior quark dynamics, we study their interplay when a quark
state occurs near a hadronic threshold. Characteristic parameters, such as the
observed peak width, the decay width, and the shape of a cross-section itself
are highly affected by the threshold. A general pole-form expression for the
S-matrix in an arbitrary background is given, and the pole structure of S is
examined. We show that at a hadronic threshold two poles in S are generally
important. We also classify the S-matrix pole structure considering an example
where nonsingular coupled channels are closed at the threshold. The framework
of our paper is the P-matrix formalism, which is reviewed and extended for use
together with conventional methods of computing quark-gluon dynamics. Results
and applications are illustrated for the doubly strange two-baryon system, the
detailed analysis of which we postpone till our forthcoming paper.
