  In the present paper, we develop the Floer-style elliptic Morse theory for
the Hamiltonian-perturbed contact action functional attached to the Legendrian
links. Motivated by the present authors' construction [OY2] of the a perturbed
action functional defined on the Carnot path space introduced in [OY2] as the
canonical generating function, we apply a Floer-type theory to the
aforementioned functional and associate the Legendrian contact instanton
cohomology, denote by $HI^*(J^1B,H;R)$, to each Legendrian submanifold contact
isotopic to the zero section of one-jet bundle. Then we give a Floer theoretic
construction of Legendrian spectral invariants and establish their basic
properties. This theory subsumes the Lagrangian intersection theory and
spectral invariants on the cotangent bundle previously developed by the
first-named author in [Oh1,Oh2]. The main ingredient for the study is the
interplay between the geometric analysis of the Hamiltonian-perturbed contact
instantons and the calculus of contact Hamiltonian geometry.
