Non-linear Supersymmetry for non-Hermitian, non-diagonalizable Hamiltonians: I. General properties

We study complex potentials and related non-diagonalizable Hamiltonians with special emphasis on formal definitions of associated functions and Jordan cells. The non-linear SUSY for complex potentials is considered and the theorems characterizing its structure are presented. We define the class of complex potentials invariant under SUSY transformations for (non-)diagonalizable Hamiltonians and formulate several results concerning the properties of associated functions . We comment on the applicability of these results for softly non-Hermitian PT-symmetric Hamiltonians. The role of SUSY (Darboux) transformations in increasing/decreasing of Jordan cells in SUSY partner Hamiltonians is thoroughly analyzed and summarized in the Index Theorem. The properties of nondiagonalizable Hamiltonians as well as the Index Theorem are illustrated in the solvable examples of non-Hermitian reflectionless Hamiltonians . The rigorous proofs are relegated to the Part II of this paper. At last, some peculiarities in resolution of identity for discrete and continuous spectra with a zero-energy bound state at threshold are discussed.