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We study the well-known Friedrichs model, in which a discrete state iscoupled to a continuum state. By examining the pole behaviors of the Friedrichsmodel in a specific form factor thoroughly, we find that, in general, when thebare discrete state is below the threshold of the continuum state, there shouldalso be a virtual-state pole accompanying the bound-state pole originating fromthe bare discrete state as the coupling is turned on. There are also othersecond-sheet poles originating from the singularities of the form factor. Wegive a general argument for the existence of these two kinds of states. As thecoupling is increased to a certain value, the second-sheet poles may merge andbecome higher-order poles. We then discuss the completeness relationsincorporating bound states, virtual states, and resonant states correspondingto higher-order poles.

Virtual states and generalized completeness relation in the Friedrichs Model