  This paper is concerned with reconstruction issue of inverse obstacle
problems governed by partial differential equations and consists of two parts.
(i) The first part considers the foundation of the probe and enclosure methods
for an impenetrable obstacle embedded in a medium governed by the stationary
Schr\"odinger equation. Under a general framework, some natural estimates for a
quantity computed from a pair of the Dirichlet and Neumann data on the outer
surface of the body occupied by the medium are given. The estimates enables us
to derive almost immediately the necessary asymptotic behaviour of indicator
functions for both methods. (ii) The second one considers the realization of
the enclosure method for a penetrable obstacle embedded in an absorbing medium
governed by the stationary Schr\"odinger equation. The unknown obstacle
considered here is modeled by a perturbation term added to the background
complex-valued $L^{\infty}$-potential. Under a jump condition on the term
across the boundary of the obstacle and some kind of regularity for the
obstacle surface including Lipschitz one as a special case, the enclosure
method using infinitely many pairs of the Dirichlet and Neumann data is
established.
