  We study one-dimensional Levy processes with Levy-Khintchine exponent
psi(xi^2), where psi is a complete Bernstein function. These processes are
subordinate Brownian motions corresponding to subordinators, whose Levy measure
has completely monotone density; or, equivalently, symmetric Levy processes
whose Levy measure has completely monotone density on the positive half-line.
Examples include symmetric stable processes and relativistic processes. The
main result is a formula for the generalized eigenfunctions of transition
operators of the process killed after exiting the half-line. A generalized
eigenfunction expansion of the transition operators is derived. As an
application, a formula for the distribution of the first passage time (or the
supremum functional) is obtained.
