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In this paper we discuss the $R$-matrix approach to treat the subthresholdresonances for the single-level and one channel, and for the single-level andtwo channel cases. In particular, the expression relating the ANC with theobservable reduced width, when the subthreshold bound state is the only channelor coupled with an open channel, which is a resonance, is formulated. Since theANC plays a very important role in nuclear astrophysics, these relationssignificantly enhance the power of the derived equations. We present therelationship between the resonance width and the ANC for the general case andconsider two limiting cases: wide and narrow resonances. Different equationsfor the astrophysical S-factors in the $R$-matrix approach are presented. Afterthat we discuss the Trojan Horse Method (THM) formalism. The developedequations are obtained using the surface-integral formalism and the generalized$R$-matrix approach for the three-body resonant reactions. It is shown how theTrojan Horse (TH) double differential cross section can be expressed in termsof the on-the-energy-shell astrophysical S-factor for the binary sub-reaction.Finally, we demonstrate how the THM can be used to calculate the astrophysicalS-factor for the neutron generator $^{13}{\rm C}(\alpha,\,n)^{16}{\rm O}$ inlow-mass AGB stars. At astrophysically relevant energies this astrophysicalS-factor is controlled by the threshold level $1/2^{+}, E_{x}= 6356$ keV. Here,we reanalyzed recent TH data taking into account more accurately the three-bodyeffects and using both assumptions that the threshold level is a subthresholdbound state or it is a resonance state.

Subthreshold resonances and resonances in the $R$-matrix method for binary reactions and in the Trojan Horse method