New class resonant states. Possible applications to astrophysics

The resonant states occur when a particle spends a long time (life time) in the region of the scattering center. To tackle the scattering problem a supplementary condition on the asymptotic behaviour of the wave function should be added in solving the Lippmann-Schwinger equation. The paper stresses the power of the methods based on the theory of functions of complex variables. By these methods the dependence of the resolvent pole position on the potential strength was studied. A global image of the set of resonant and bound states was obtained. This paper belongs to a series of papers in which a new class of poles of the resolvent (S-matrix poles) was identified. These poles do not become bound or virtual state poles even when the potential strength increases indefinitely. They remain in a finite region of the complex k-plane that lies on two points: the 'stable point' and a 'branch point'. The position of the first pole depends very weakly on the potential strength, the branch point is a point where two sheets of the Riemann surface are joint. The connection of this new class of poles with the quasi molecular states that occur in heavy ion scattering was shown. The construction of the Riemann surfaces that corresponds to the irreducible factors of the modified Fredholm determinant for various potentials can provide new information on nuclear forces