  We present an analytical study of three-state Feshbach resonances induced by
second-order couplings. Such resonances arise when the scattering amplitude is
modified by the interaction with a bound state that is not directly coupled to
the scattering state containing incoming flux. Coupling occurs indirectly
through an intermediate state. We consider two problems: (i) the intermediate
state is a scattering state in a distinct open channel; (ii) the intermediate
state is an off-resonant bound state in a distinct closed channel. The first
problem is a model of electric-field-induced resonances in ultracold collisions
of alkali metal atoms [Phys. Rev. A 75, 032709 (2007)] and the second problem
is relevant for ultracold collisions of complex polyatomic molecules, chemical
reaction dynamics, photoassociation of ultracold atoms, and electron - molecule
scattering. Our analysis yields general expressions for the energy dependence
of the T-matrix elements modified by three-state resonances and the dependence
of the resonance positions and widths on coupling amplitudes for the
weak-coupling limit. We show that the second problem can be generalized to
describe resonances induced by indirect coupling through an arbitrary number of
sequentially coupled off-resonant bound states and analyze the dependence of
the resonance width on the number of the intermediate states.
