On the energy dependence of the effective interaction

The Schrödinger equation is studied in a finite subspace of the Hilbert space. The truncation of the configuration space makes the (effective) interaction energy-dependent. This dependency is investigated. An analytic expression for the matrix elements of the effective interaction is established for the schematic model, for the displaced harmonic oscillator, for a simple random-phase problem and for a pairing force model. Thereby the phase transitions from bound to unbound states and from normal to superconducting states are analysed.