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

TL;DR

This paper rigorously analyzes the structure of non-linear supersymmetry in non-Hermitian, non-diagonalizable Hamiltonians, establishing mathematical foundations and properties of associated functions and Jordan structures.

Contribution

It provides the first rigorous mathematical characterization of SUSY for non-diagonalizable complex Hamiltonians, including potential classes and asymptotic eigenfunction behavior.

Findings

Classes of SUSY-invariant potentials identified

Asymptotic behavior of eigenfunctions derived

Index Theorem relating Jordan structures proven

Abstract

We continue our investigation of the nonlinear SUSY for complex potentials started in the Part I (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.