An open $U_q(sl_2)$ -invariant spin chain of spin $S$ and length $N$ with inhomogeneous coupling is investigated as an example of a non-Hermitian (quasi-Hermitian) model. For several particular cases of such a chain, the ranges of the deformation parameter $\gamma$ are determined for which the spectrum of the model is real. For a certain range of $\gamma$ , a universal metric operator is constructed, and thus, the quasi-Hermitian nature of the model is established. This universal metric operator is nondynamical, its structure is determined only by the symmetry of the model. The results apply, in particular, to all known homogeneous $U_q(sl_2)$ -invariant integrable spin chains with nearest-neighbor interaction. In addition, the most general form of a metric operator for a quasi-Hermitian operator in finite-dimensional spaces is discussed.An open $U_q(sl_2)$-invariant spin chain of spin $S$ and length $N$ with inhomogeneous coupling is investigated as an example of a non-Hermitian (quasi-Hermitian) model. For several particular cases of such a chain, the ranges of the deformation parameter $\gamma$ are determined for which the spectrum of the model is real. For a certain range of $\gamma$, a universal metric operator is constructed, and thus, the quasi-Hermitian nature of the model is established. This universal metric operator is nondynamical, its structure is determined only by the symmetry of the model. The results apply, in particular, to all known homogeneous $U_q(sl_2)$-invariant integrable spin chains with nearest-neighbor interaction. In addition, the most general form of a metric operator for a quasi-Hermitian operator in finite-dimensional spaces is discussed.

arXiv:0911.4476St Petersburg Mathematical JournalMar 17, 20117 citations

Metric structures characterized by norm properties in non-Hermitian systems

Xin Wang, Haodi Liu, Libin Fu

Physical Review AOct 15, 20260 citations

Metric versus observable operator representation, higher spin models

Andreas Fring, Thomas Frith

We elaborate further on the metric representation that is obtained by transferring the time-dependence from a Hermitian Hamiltonian to the metric operator in a related non-Hermitian system. We provide further insight into the procedure on how to employ the time-dependent Dyson relation and the quasi-Hermiticity relation to solve time-dependent Hermitian Hamiltonian systems. By solving both equations separately we argue here that it is in general easier to solve the former. We solve the mutually related time-dependent Schrödinger equation for a Hermitian and non-Hermitian spin 1/2, 1 and 3/2 model with time-independent and time-dependent metric, respectively. In all models the overdetermined coupled system of equations for the Dyson map can be decoupled algebraic manipulations and reduces to simple linear differential equations and an equation that can be converted into the non-linear Ermakov-Pinney equation.We elaborate further on the metric representation that is obtained by transferring the time-dependence from a Hermitian Hamiltonian to the metric operator in a related non-Hermitian system. We provide further insight into the procedure on how to employ the time-dependent Dyson relation and the quasi-Hermiticity relation to solve time-dependent Hermitian Hamiltonian systems. By solving both equations separately we argue here that it is in general easier to solve the former. We solve the mutually related time-dependent Schrödinger equation for a Hermitian and non-Hermitian spin 1/2, 1 and 3/2 model with time-independent and time-dependent metric, respectively. In all models the overdetermined coupled system of equations for the Dyson map can be decoupled algebraic manipulations and reduces to simple linear differential equations and an equation that can be converted into the non-linear Ermakov-Pinney equation.

arXiv:1612.06122The European Physical Journal PlusFeb 1, 201814 citations

Spectrum of the Metric Operator of a Simple -Symmetric Model

Jakub Železný

International Journal of Theoretical PhysicsDec 21, 20104 citations

An open

U_q(sl_2)

-invariant spin chain of spin

S

and length

N

with inhomogeneous coupling is investigated as an example of a non-Hermitian (quasi-Hermitian) model. For several particular cases of such a chain, the ranges of the deformation parameter

\gamma

are determined for which the spectrum of the model is real. For a certain range of

\gamma

, a universal metric operator is constructed, and thus, the quasi-Hermitian nature of the model is established. This universal metric operator is nondynamical, its structure is determined only by the symmetry of the model. The results apply, in particular, to all known homogeneous

U_q(sl_2)

-invariant integrable spin chains with nearest-neighbor interaction. In addition, the most general form of a metric operator for a quasi-Hermitian operator in finite-dimensional spaces is discussed.