  We introduce a class of L\'{e}vy processes subject to specific regularity
conditions, and consider their Feynman-Kac semigroups given under a Kato-class
potential. Using new techniques, first we analyze the rate of decay of
eigenfunctions at infinity. We prove bounds on $\lambda$-subaveraging
functions, from which we derive two-sided sharp pointwise estimates on the
ground state, and obtain upper bounds on all other eigenfunctions. Next, by
using these results, we analyze intrinsic ultracontractivity and related
properties of the semigroup refining them by the concept of ground state
domination and asymptotic versions. We establish the relationships of these
properties, derive sharp necessary and sufficient conditions for their validity
in terms of the behavior of the L\'{e}vy density and the potential at infinity,
define the concept of borderline potential for the asymptotic properties and
give probabilistic and variational characterizations. These results are amply
illustrated by key examples.
