  We examine the notion and properties of the non-Hermitian effective
Hamiltonian of an unstable system using as an example potential resonance
scattering with a fixed angular momentum. We present a consistent self-adjoint
formulation of the problem of scattering on a finite-range potential, which is
based on separation of the configuration space on two, internal and external,
segments. The scattering amplitude is expressed in terms of the resolvent of a
non-Hermitian operator H. The explicit form of this operator depends both on
the radius of separation and the boundary conditions at this place which can be
chosen in many different ways. We discuss this freedom and show explicitly that
the physical scattering amplitude is, nevertheless, unique though not all
choices are equally adequate from the physical point of view. The
energy-dependent operator H should not be confused with the non-Hermitian
effective Hamiltonian H_{eff} exploited usually to describe interference of
overlapping resonances. We apply the developed formalism to a chain of L
delta-barriers whose solution is also found independently in a closed form. For
a fixed band of L overlapping resonances, the smooth energy dependence of H can
be ignored so that complex eigenvalues of the LxL submatrix H_{eff} define the
energies and widths of the resonances.We construct H_{eff} for the two commonly
considered types of the boundary conditions (Neumann and Dirichlet) for the
internal motion. Formation in the outer well of a short-lived doorway state is
explicitly demonstrated together with the appearance of L-1 long-lived states
trapped in the inner part of the chain.
