Searching for non-Hermitian (parity-time) $\mathcal{PT}$ -symmetric Hamiltonians \cite{bender} with real spectra has been acquiring much interest for fourteen years. In this article, we have introduced a $\mathcal{PT}$ symmetric non-Hermitian Hamiltonian model which is given as $\hat{\mathcal{H}}=ω(\hat{b}^\dagger\hat{b}+\frac{1}{2})+ α(\hat{b}^{2}-(\hat{b}^\dagger)^{2})$ where $ω$ and $α$ are real constants, $\hat{b}$ and $\hat{b^\dagger}$ are first order differential operators. Moreover, Pseudo-Hermiticity that is a generalization of $\mathcal{PT}$ symmetry has been attracting a growing interest \cite{mos}. Because the Hamiltonian $\mathcal{H}$ is pseudo-Hermitian, we have obtained the Hermitian equivalent of $\mathcal{H}$ which is in Sturm- Liouville form leads to exactly solvable potential models which are effective screened potential and hyperbolic Rosen-Morse II potential. $\mathcal{H}$ is called pseudo-Hermitian, if there exists a Hermitian and invertible operator $η$ satisfying $\mathcal{H^\dagger}=η\mathcal{H} η^{-1}$ . For the Hermitian Hamiltonian $h$ , one can write $h=ρ\mathcal{H} ρ^{-1}$ where $ρ=\sqrtη$ is unitary. Using this $ρ$ we have obtained a physical Hamiltonian $h$ for each case. Then, the Schrödinger equation is solved exactly using Shape Invariance method of Supersymmetric Quantum Mechanics \cite{susy1}. Mapping function $ρ$ is obtained for each potential case.Searching for non-Hermitian (parity-time)$\mathcal{PT}$-symmetric Hamiltonians \cite{bender} with real spectra has been acquiring much interest for fourteen years. In this article, we have introduced a $\mathcal{PT}$ symmetric non-Hermitian Hamiltonian model which is given as $\hat{\mathcal{H}}=ω(\hat{b}^\dagger\hat{b}+\frac{1}{2})+ α(\hat{b}^{2}-(\hat{b}^\dagger)^{2})$ where $ω$ and $α$ are real constants, $\hat{b}$ and $\hat{b^\dagger}$ are first order differential operators. Moreover, Pseudo-Hermiticity that is a generalization of $\mathcal{PT}$ symmetry has been attracting a growing interest \cite{mos}. Because the Hamiltonian $\mathcal{H}$ is pseudo-Hermitian, we have obtained the Hermitian equivalent of $\mathcal{H}$ which is in Sturm- Liouville form leads to exactly solvable potential models which are effective screened potential and hyperbolic Rosen-Morse II potential. $\mathcal{H}$ is called pseudo-Hermitian, if there exists a Hermitian and invertible operator $η$ satisfying $\mathcal{H^\dagger}=η\mathcal{H} η^{-1}$. For the Hermitian Hamiltonian $h$, one can write $h=ρ\mathcal{H} ρ^{-1}$ where $ρ=\sqrtη$ is unitary. Using this $ρ$ we have obtained a physical Hamiltonian $h$ for each case. Then, the Schrödinger equation is solved exactly using Shape Invariance method of Supersymmetric Quantum Mechanics \cite{susy1}. Mapping function $ρ$ is obtained for each potential case.

arXiv:1406.3298Jun 12, 20140 citations

Algebraic method for pseudo-Hermitian Hamiltonians

Junqing Li, Yan-Gang Miao, Zhao Xue

Jul 25, 20115 citations

Non-Hermitian von Roos Hamiltonian's $\eta$ -weak-pseudo-Hermiticity and exact solvabilityNon-Hermitian von Roos Hamiltonian's $\eta$-weak-pseudo-Hermiticity and exact solvability

Omar Mustafa, S. Habib Mazharimousavi

A complexied von Roos Hamiltonian is considered and a Hermitian rst-order intertwining di¤erential operator is used to obtain the related position dependent mass -weak-pseudo-Hermitian Hamiltonians. Two user -friendly reference-target maps are introduced to serve for exactsolvability of some non-Hermitian -weak-pseudo-Hermitian position dependent mass Hamiltonians. A non-Hermitian PT -symmetric Scarf II and a non-Hermitian periodic-type PT -symmetric Samsonov-Roy potentials are used as reference models in a user-friendly reference-target map and the corresponding isospectral Hamiltonians are obtained. It is observed that for each exactly-solvable reference Hamiltonian there is a corresponding set of exactly-solvable target Hamiltonians. PACS numbers: 03.65.Ge, 03.65.Fd, 03.65.CaA complexied von Roos Hamiltonian is considered and a Hermitian rst-order intertwining di¤erential operator is used to obtain the related position dependent mass -weak-pseudo-Hermitian Hamiltonians. Two user -friendly reference-target maps are introduced to serve for exactsolvability of some non-Hermitian -weak-pseudo-Hermitian position dependent mass Hamiltonians. A non-Hermitian PT -symmetric Scarf II and a non-Hermitian periodic-type PT -symmetric Samsonov-Roy potentials are used as reference models in a user-friendly reference-target map and the corresponding isospectral Hamiltonians are obtained. It is observed that for each exactly-solvable reference Hamiltonian there is a corresponding set of exactly-solvable target Hamiltonians. PACS numbers: 03.65.Ge, 03.65.Fd, 03.65.Ca

Jul 25, 20070 citations

Mathematical Aspects of Quantum Systems with a Pseudo-Hermitian Hamiltonian

Natália Bebiano, João da Providência, João P. da Providência

Brazilian Journal of PhysicsJan 5, 201610 citations

Pseudo-Hermiticity versus PT Symmetry: The structure responsible for the reality of the spectrum of a non-Hermitian Hamiltonian

Alí Mostafazadeh

We introduce the notion of pseudo-Hermiticity and show that every Hamiltonian with a real spectrum is pseudo-Hermitian. We point out that all the PT-symmetric non-Hermitian Hamiltonians studied in the literature belong to the class of pseudo-Hermitian Hamiltonians, and argue that the basic structure responsible for the particular spectral properties of these Hamiltonians is their pseudo-Hermiticity. We explore the basic properties of general pseudo-Hermitian Hamiltonians, develop pseudo-supersymmetric quantum mechanics, and study some concrete examples, namely the Hamiltonian of the two-component Wheeler-DeWitt equation for the FRW-models coupled to a real massive scalar field and a class of pseudo-Hermitian Hamiltonians with a real spectrum.We introduce the notion of pseudo-Hermiticity and show that every Hamiltonian with a real spectrum is pseudo-Hermitian. We point out that all the PT-symmetric non-Hermitian Hamiltonians studied in the literature belong to the class of pseudo-Hermitian Hamiltonians, and argue that the basic structure responsible for the particular spectral properties of these Hamiltonians is their pseudo-Hermiticity. We explore the basic properties of general pseudo-Hermitian Hamiltonians, develop pseudo-supersymmetric quantum mechanics, and study some concrete examples, namely the Hamiltonian of the two-component Wheeler-DeWitt equation for the FRW-models coupled to a real massive scalar field and a class of pseudo-Hermitian Hamiltonians with a real spectrum.

Jul 1, 20014 citations

\mathcal{PT}

Symmetric Hamiltonian Model and Exactly Solvable Potentials

Searching for non-Hermitian (parity-time)

\mathcal{PT}

-symmetric Hamiltonians \cite{bender} with real spectra has been acquiring much interest for fourteen years. In this article, we have introduced a

\mathcal{PT}

symmetric non-Hermitian Hamiltonian model which is given as

\hat{\mathcal{H}}=ω(\hat{b}^\dagger\hat{b}+\frac{1}{2})+ α(\hat{b}^{2}-(\hat{b}^\dagger)^{2})

where

ω

and

α

are real constants,

\hat{b}

and

\hat{b^\dagger}

are first order differential operators. Moreover, Pseudo-Hermiticity that is a generalization of

\mathcal{PT}

symmetry has been attracting a growing interest \cite{mos}. Because the Hamiltonian

\mathcal{H}

is pseudo-Hermitian, we have obtained the Hermitian equivalent of

\mathcal{H}

which is in Sturm- Liouville form leads to exactly solvable potential models which are effective screened potential and hyperbolic Rosen-Morse II potential.

\mathcal{H}

is called pseudo-Hermitian, if there exists a Hermitian and invertible operator

η

satisfying

\mathcal{H^\dagger}=η\mathcal{H} η^{-1}

. For the Hermitian Hamiltonian

h

, one can write

h=ρ\mathcal{H} ρ^{-1}

where

ρ=\sqrtη

is unitary. Using this

ρ

we have obtained a physical Hamiltonian

h

for each case. Then, the Schrödinger equation is solved exactly using Shape Invariance method of Supersymmetric Quantum Mechanics \cite{susy1}. Mapping function

ρ

is obtained for each potential case.

Non-Hermitian von Roos Hamiltonian's

\eta

-weak-pseudo-Hermiticity and exact solvability