Time-dependent Darboux (supersymmetric) transformations for non-Hermitian quantum systems
Abstract
We propose time-dependent Darboux (supersymmetric) transformations that provide a scheme for the calculation of explicitly time-dependent solvable non-Hermitian partner Hamiltonians. Together with two Hermitian Hamilitonians the latter form a quadruple of Hamiltonians that are related by two time-dependent Dyson equations and two intertwining relations in form of a commutative diagram. Our construction is extended to the entire hierarchy of Hamiltonians obtained from time dependent Darboux-Crum transformations. As an alternative approach we also discuss the intertwining relations for Lewis-Riesenfeld invariants for Hermitian as well as non-Hermitian Hamiltonians that reduce the time-dependent equations to auxiliary eigenvalue equations. The working of our proposals is discussed for a hierarchy of explicitly time-dependent rational, hyperbolic, Airy function and nonlocal potentials.
Keywords
Cite
@article{arxiv.1811.00149,
title = {Time-dependent Darboux (supersymmetric) transformations for non-Hermitian quantum systems},
author = {Julia Cen and Andreas Fring and Thomas Frith},
journal= {arXiv preprint arXiv:1811.00149},
year = {2019}
}
Comments
22 pages, 4 figures