Generalized hermitian operators

Sun, Shiqi; Ma, Xuefeng

doi:10.1016/j.laa.2010.04.001

“…An operator satisfying ( 13) is called either quasi-Hermitian [23,63], S L -Hermitian [64][65][66], pseudo-Hermitian [67][68][69][70][71][72], crypto-Hermitian [73][74][75], or generalised Hermitian [76], depending on the properties of S L . However, in contrast to these works the concept of symmetrisation considered here does not require the symmetrisation operators to be necessarily invertible [24,77] or their kernels to be empty.…”

Section: Symmetrisationmentioning

confidence: 99%

Balancing gain and loss in symmetrised multi-well potentials

Dizdarevic

¹

,

Cartarius

²

,

Main

³

2020

J. Phys. A: Math. Theor.

0

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Balanced gain and loss leads to stationary dynamics in open systems. This occurs naturally in PT -symmetric systems, where the imaginary part of the potential describing gain and loss is perfectly antisymmetric. While this case seems intuitive, stationary dynamics are also possible in asymmetric open systems. Open multi-well quantum systems can possess completely or partly real spectra if their Hamiltonian is symmetrised or semi-symmetrised, respectively. In contrast to similar concepts, symmetrisation allows for the description of physical multi-well potentials with gain and loss. A simple matrix model for the description of two and three-mode systems is used as an example, for which analytical symmetrised solutions are derived. It is explicitly shown how symmetrisation can be used to systematically find two-mode systems with a stable, stationary ground state and why only PT -symmetric two-mode systems can have stationary excited states.

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“…Here is one comparison: a matrix is similar to its adjoint if and only if it is the product of two Hermitian matrices, while it is similar to a Hermitian matrix if and only if it is the product of a positive definite and a Hermitian [11,14]. See [12] for a modern treatment of generalized Hermitian operators.…”

Section: Noncommutative *-Varietiesmentioning

confidence: 99%

Matrices similar to partial isometries

Garcia

¹

,

Sherman

²

2017

Linear Algebra and its Applications

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“…Here is one comparison: a matrix is similar to its adjoint if and only if it is the product of two Hermitian matrices, while it is similar to a Hermitian matrix if and only if it is the product of a positive definite and a Hermitian [11,14]. See [12] for a modern treatment of generalized Hermitian operators.…”

Section: Noncommutative *-Varietiesmentioning

confidence: 99%