Solution of Second Order Supersymmetrical Intertwining Relations in Minkowski Plane

TL;DR

This paper extends supersymmetrical intertwining relations to quantum Hamiltonians in Minkowski space, explicitly constructing potentials for second order supercharges and analyzing their properties within integrable systems.

Contribution

It generalizes SUSY intertwining relations to Minkowski space and explicitly solves for potentials with second order supercharges, including diagonalizable and nondiagonalizable cases.

Findings

Explicit potentials constructed for second order supercharges.

Solutions obtained for both diagonalizable and nondiagonalizable metric matrices.

Analysis of properties of the resulting integrable systems.

Abstract

Supersymmetrical (SUSY) intertwining relations are generalized to the case of quantum Hamiltonians in Minkowski space. For intertwining operators (supercharges) of second order in derivatives the intertwined Hamiltonians correspond to completely integrable systems with the symmetry operators of fourth order in momenta. In terms of components, the itertwining relations correspond to the system of nonlinear differential equations which are solvable with the simplest - constant - ansatzes for the "metric" matrix in second order part of the supercharges. The corresponding potentials are built explicitly both for diagonalizable and nondiagonalizable form of "metric" matrices, and their properties are discussed.