Is the concept of the non-Hermitian effective Hamiltonian relevant in the case of potential scattering?

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 the separation of the configuration space into two segments, internal and external. The scattering amplitude is expressed in terms of the resolvent of a non-Hermitian operator H. The explicit form of this operator depends on both 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, although 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) which is usually exploited to describe interference of overlapping resonances. We note that the simple Breit-Wigner approximation is as a rule valid for any individual resonance in the case of few-channel scattering on a flat billiardlike cavity, leaving no room for nontrivial H(eff) to appear. The physics is appreciably richer in the case of an open chain of L connected similar cavities whose spectrum has a band structure. For a fixed band of L overlapping resonances, the smooth energy dependence of H can be ignored so that the constant LxL submatrix H(eff) approximately describes the time evolution of the chain in the energy domain of the band and the complex eigenvalues of H(eff) define the energies and widths of the resonances. We apply the developed formalism to the problem of a chain of L delta barriers, whose solution is also found independently in a closed form. We construct H(eff) for the two commonly considered types of boundary conditions (Neumann and Dirichlet) for the internal motion. Although the final results are in perfect coincidence, somewhat different physical patterns arise of the trend of the system with growing openness. Formation in the outer well of a short-lived doorway state shifted in energy is explicitly demonstrated together with the appearance of L-1 long-lived states trapped in the inner part of the chain.