Theory of resonances in many‐body systems: Spectral theory of unbounded Schrödinger operators, complex scaling, and extended virial theorem

In connection with the theory of resonance phenomena in many-body scattering systems, some aspects of current spectral theories for general operators are reviewed, including theories leading to complex eigenvalues of nonself-adjoint operators well defined over a perturbed originally self-adjoint domain. Consequently, a generalization of the virial theory is obtained. Various applications are considered, particularly, the problem of a system perturbed by an electric field. The associated Stark Hamiltonian is an unbounded operator, but still numerical results in conjunction with a simple reciprocal relation prompt the fulfillment of a virial sum rule. It is said that the extended virial theorem, based on complex scaling, seems to provide a systematic way of arriving at the resonances. 81 references. (JFP)