Non-linear supersymmetry for non-Hermitian, non-diagonalizable Hamiltonians: II. Rigorous results

We continue our investigation of the non-linear SUSY for complex potentials started in http://dx.doi.org/10.1016/j.nuclpysb.2007.03.016 [A.A. Andrianov, F. Cannata, A.V. Sokolov, math-ph/0610024] and prove the theorems characterizing its structure in the case of non-diagonalizable Hamiltonians. This part provides the mathematical basis of previous studies. The classes of potentials invariant under SUSY transformations for non-diagonalizable Hamiltonians are specified and the asymptotics of formal eigenfunctions and associated functions are derived. Several results on the normalizability of associated functions at infinities are rigorously proved. Finally the Index Theorem on relation between Jordan structures of intertwined Hamiltonians depending of the behavior of elements of canonical basis of supercharge kernel at infinity is proven.