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

TL;DR

This paper explores the structure and properties of non-Hermitian, non-diagonalizable Hamiltonians using nonlinear supersymmetry, introducing new theorems, classifying invariant potentials, and analyzing Jordan cell transformations.

Contribution

It presents a comprehensive analysis of nonlinear SUSY for complex, non-diagonalizable Hamiltonians, including theorems, classification of invariant potentials, and the role of SUSY transformations in Jordan cell dynamics.

Findings

Classified potentials invariant under SUSY transformations.

Established theorems characterizing the structure of nonlinear SUSY.

Analyzed the impact of SUSY transformations on Jordan cells and provided illustrative examples.

Abstract

We study complex potentials and related non-diagonalizable Hamiltonians with special emphasis on formal definitions of associated functions and Jordan cells. The nonlinear SUSY for complex potentials is considered and the theorems characterizing its structure are presented. We present the class of 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 Jordancells in SUSY partner Hamiltonians is thoroughly analyzed and summarized in the Index Theorem. The properties of non-diagonalizable 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.